Finite difference methods for the solution of fractional...
Transcript of Finite difference methods for the solution of fractional...
Finite Difference Methods for the Solution of FractionalDiffusion Equations
Orlando Miguel Reis e Ribeiro Santos
Thesis to obtain the Master of Science Degree in
Aerospace Engineering
Supervisors: Prof. José Carlos Fernandes PereiraProf. José Manuel da Silva Chaves Ribeiro Pereira
Examination Committee
Chairperson: Prof. Filipe Szolnoky Ramos Pinto CunhaSupervisor: Prof. José Carlos Fernandes Pereira
Member of the Committee: Prof. Duarte Pedro Mata de Oliveira Valério
November 2016
ii
Dedicated to my family.
iii
iv
Acknowledgments
I would like to start by expressing my deepest gratitude to my supervisors, Prof. Jose Carlos Pereira
and Prof. Jose Chaves Pereira, for the all the knowledge, feedback and constant support that always
kept me motivated during the development of this thesis.
A big thank you to my colleagues at LASEF, for their help and for always making me feel welcome.
I am also grateful to all my friends, for all the shared laughs and moments that made this journey
much more joyful.
I would like to end by remembering all the support and encouragement I received from my family and
my girlfriend. Each in their own way, they have helped me become who I am and without them I would
not be writing this thesis.
v
vi
Resumo
O calculo fraccional e uma disciplina matematica que lida com integrais e derivadas de ordem ar-
bitraria que tem vindo a encontrar aplicacoes na fısica, processamento de sinais, engenharia, biociencias
e financas. A difusao anomala tem recebido bastante atencao por parte do calculo fraccional. Nas
equacoes fraccionais de difusao as derivadas normais sao substituıdas por derivadas de ordem frac-
cional, dando origem a equacoes fraccionais no tempo, espaco e tempo-espaco. Dado que a solucao
analıtica de equacoes de difusao fraccionarias e difıcil de obter, os metodos de diferencas finitas
tornaram-se bastante populares havendo um grande numero de esquemas recentemente publicado.
Foram seleccionados tres esquemas com ordens crescentes para cada um dos subtipos de equacao
de difusao fraccional. A construcao de cada um dos esquemas e sumarizada e cada um e implemen-
tado de modo a permitir a sua validacao e comparacao com os restantes.
Apesar do seu sucesso, as equacoes de difusao faccionais de ordem constante mostraram dificul-
dades na modelacao de fenomenos mais complexos. Para as ultrapassar, foram propostas derivadas
de ordem variavel, funcao do tempo e/ou espaco,sendo importante entender claramente como e que
a ordem variavel afecta o comportamento de um sistema difusivo. Uma equacao de difusao de ordem
variavel no tempo e resolvida atraves de um esquema de diferencas finitas e a sua forma matricial
e apresentada. A implementacao e validada e usada para estudar a ordem variavel como funcao do
espaco, tempo ou ate da solucao da equacao que sao comparadas com a ordem constante.
Palavras-chave: Calculo Fraccional, Derivada Fraccional, Equacao da Difusao Fraccional,
Difusao Anomala, Metodos de Differencas Finitas, Ordem Variavel
vii
viii
Abstract
Fractional calculus is a mathematical field dealing with integrals and derivatives of arbitrary order. In
recent times fractional calculus has found applications in physics, signal-processing, engineering, bio-
science, and finance. Anomalous diffusion has received particular interest in the framework of fractional
calculus applications. In fractional diffusion equations, standard derivatives are replaced by fractional
order counterparts, originating time, space and time-space fractional diffusion equations. Since the an-
alytical solution of fractional differential equations is hard to obtain, finite difference methods in particular
became very popular and a large number of schemes has been published very recently. Three different
schemes with increasing order of accuracy were selected for time, space and time-space fractional diffu-
sion equations. To fulfil the first objective of this work, the construction of these schemes is summarized
and then with numerical examples, solved through self-written code, a comparison is made in terms of
accuracy and computational cost.
Despite their success, constant fractional order differential equations showed difficulties in modelling
complex phenomena. To overcome these difficulties, variable order fractional derivatives, whose order is
function of time and/or space have been proposed, becoming important to understand how the variable
order behaviour affects a diffusive system. A variable order time fractional diffusion equation is solved
via a finite difference scheme and its matrix form is presented in detail. A self-written implementation is
validated and then used to study the influence of the variable order time derivative, function of time and
space, in the solution of variable order time fractional diffusion equations.
Keywords: Fractional Calculus, Anomalous Diffusion, Fractional Diffusion Equation, Finite Dif-
ference Methods, Fractional Derivative, Variable-Order.
ix
x
Contents
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Topic Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Finite Difference Solution of Fractional Diffusion Equations 9
2.1 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Finite difference approximations for fractional derivatives . . . . . . . . . . . . . . . . . . . 11
2.2.1 Approximations for time fractional derivatives . . . . . . . . . . . . . . . . . . . . . 11
2.2.1.1 The Grunwald-Letnikov Approximation . . . . . . . . . . . . . . . . . . . 11
2.2.1.2 L1 Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1.3 Third order weighted and shifted Grunwald difference approximation . . . 12
2.2.2 Approximations for space fractional derivatives . . . . . . . . . . . . . . . . . . . . 14
2.2.2.1 The shifted Grunwald approximation . . . . . . . . . . . . . . . . . . . . . 14
2.2.2.2 Second order approximation . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.2.3 Fourth order compact finite difference approximation . . . . . . . . . . . . 16
2.3 Finite difference approximations of integer order derivatives . . . . . . . . . . . . . . . . . 18
2.4 Time fractional diffusion equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.2 First order finite difference scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4.3 Second order implicit finite difference scheme . . . . . . . . . . . . . . . . . . . . . 21
2.4.4 Third order compact finite difference scheme . . . . . . . . . . . . . . . . . . . . . 23
xi
2.4.5 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5 Space fractional diffusion equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5.2 First order finite difference scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5.3 Second order finite difference scheme . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5.4 Fourth order quasi-compact finite difference scheme . . . . . . . . . . . . . . . . . 31
2.5.5 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.6 Time-space fractional diffusion equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.6.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.6.2 First order finite difference scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.6.3 Second order finite difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.6.4 Fourth order finite difference scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.6.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3 Influence of variable-order operators in the behaviour of sub-diffusive systems 47
3.1 Numerical solution of variable order time fractional diffusion equations . . . . . . . . . . . 47
3.1.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.1.2 Numerical Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.1.3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2 Influence of variable order differential operators in anomalous diffusion . . . . . . . . . . . 53
3.2.1 Constant fractional order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2.2 Time dependent fractional order . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2.3 Space dependent fractional order . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2.4 Solution dependent fractional order . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4 Conclusions 65
References 69
A Schemes for fractional diffusion equations in matrix form A.1
A.1 Time Fractional Diffusion Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1
A.1.1 First Order Weighted Average Scheme . . . . . . . . . . . . . . . . . . . . . . . . A.1
A.1.2 Second Order Finite Difference Scheme . . . . . . . . . . . . . . . . . . . . . . . . A.1
A.1.3 Third Order Finite Difference Scheme . . . . . . . . . . . . . . . . . . . . . . . . . A.2
A.2 Space Fractional Diffusion Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3
A.2.1 First Order Finite Difference Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . A.3
A.2.2 Second Order Finite Difference Scheme . . . . . . . . . . . . . . . . . . . . . . . . A.3
A.2.3 Fourth Order Finite Difference Scheme . . . . . . . . . . . . . . . . . . . . . . . . A.4
A.3 Time-space Fractional Diffusion Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5
A.3.1 First Order Finite Difference Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . A.5
A.3.2 Second Order Finite Difference Scheme . . . . . . . . . . . . . . . . . . . . . . . . A.5
xii
A.3.3 Fourth Order Finite Difference Scheme . . . . . . . . . . . . . . . . . . . . . . . . A.6
xiii
xiv
List of Tables
2.4.1L∞ error and its order of convergence with decrease of the temporal step size, for the
presented schemes for the time fractional diffusion equation. The results were taken with
h = 1/2000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4.2Time of computation for each scheme for each of the presented schemes. The results
correspond to a constant space step h = 1/2000. . . . . . . . . . . . . . . . . . . . . . . . 29
2.5.1L∞h,τ errors and their order of convergence with the refinement of the space step for equa-
tion (2.5.19)-(2.5.21) with µ = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.5.2L∞h,τ errors and their order of convergence with the refinement of the space step for equa-
tion (2.5.19)-(2.5.21) with µ = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.5.3L∞h,τ errors and their order of convergence with the refinement of the space step for equa-
tion (2.5.19)-(2.5.21) with µ = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.5.4Time of computation in the solution of problem (2.5.19)-(2.5.21) for the three schemes
presented with 1/τ = 20000 and 1/h = 128 for α = 1.5. . . . . . . . . . . . . . . . . . . . . 36
2.6.1L∞h,τ error and the respective order of convergence with space step refinement for the first
order in space scheme with a constant τ = 1/8000 . . . . . . . . . . . . . . . . . . . . . . 43
2.6.2L∞h,τ error and the respective order of convergence with time step refinement for the first
order in space scheme, h ≈ τ (2−γ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.6.3L∞h,τ error and the respective order of convergence with space step refinement for the
second order in space scheme, h = 1/1000. . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.6.4L∞h,τ error and the respective order of convergence with time step refinement for the sec-
ond order in space scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.6.5L∞h,τ error and the respective order of convergence with space step refinement for the
fourth order in space scheme, for t ∈ [0, 0.1] and τ = 1/100000. . . . . . . . . . . . . . . . 45
2.6.6L∞h,τ error and the respective order of convergence with time step refinement for the fourth
order in space scheme, h = 1/1000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.6.7Computing times with refinement of the time interval with the first, second and fourth order
in space schemes and h = 1/2000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.1.1Error behaviour with decreasing temporal gridsize, h = 1/500. . . . . . . . . . . . . . . . . 52
3.1.2Error behaviour with decreasing temporal gridsize, t = h2. . . . . . . . . . . . . . . . . . . 53
xv
xvi
List of Figures
2.1 Exact solution of problem (2.4.47)-(2.4.49) with γ = 0.5. . . . . . . . . . . . . . . . . . . . 26
2.2 Absolute errors for each numerical scheme for the solution of time fractional diffusion
equations. All the the plots shown correspond to τ = 1/2000, h = 1/512 and γ = 0.5. . . . 27
2.3 Analytical solutions of problem (2.5.19)-(2.5.21) for µ = 2, 3, 4. . . . . . . . . . . . . . . . . 33
2.4 Absolute error in the numerical solution of the problem (2.5.19)-(2.5.21) with µ = 4 for the
three schemes presented. The errors correspond to h = 1/128 and τ = 1/20000. . . . . . 34
2.5 Exact solution of problem (2.6.27)-(2.6.29) with γ = 0.5 . . . . . . . . . . . . . . . . . . . 41
3.1 Exact solution of problem (3.1.20)-(3.1.22). . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Absolute error in the solution of (3.1.20)-(3.1.22), h = 1/250 and τ = 1/500. . . . . . . . . 52
3.3 Solution of the standard diffusion equation. . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.4 Solution versus time plot at x = 5 with different constant fractional orders. . . . . . . . . . 56
3.5 Solution at t = 0.2 with different constant fractional orders. . . . . . . . . . . . . . . . . . . 56
3.6 Solution at t = 5 with different constant fractional orders. . . . . . . . . . . . . . . . . . . . 57
3.7 Time evolution in x = 5 with different time dependent fractional orders. . . . . . . . . . . . 58
3.8 The three space dependent fractional orders considered. . . . . . . . . . . . . . . . . . . 59
3.9 Time evolution at x = 5 modelled with different space dependent fractional orders. . . . . 60
3.10 Solution at t = 10 modelled with different space dependent fractional orders. . . . . . . . 61
3.11 Solution dependent fractional order. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.12 Time evolution in x = 5 with a solution dependent variable order model. . . . . . . . . . . 63
3.13 Solution t = 10 with a solution dependent variable order model. . . . . . . . . . . . . . . . 64
xvii
xviii
Nomenclature
α Space fractional derivative order.
δtun+1i Backward operator for the first order time derivative.
δtun+ 1
2i Central difference operator for the first order time derivative.
δ2xuni Central differences operator for the second order space derivative.
γ Time fractional derivative order.
CaD
αt Left Caputo fractional derivative.
CtD
αb Right Caputo fractional derivative.
GLδγt Grunwald-Letnikov difference operator for the Riemann-Liouville time fractional derivative.
L1δγt un L1 operator for the Caputo time fractional derivative.
L1δγt un L1 operator for the Riemann-Liouville time fractional derivative.
RLaD−αt Left Riemann-Liouville fractional derivative.
RLaD−αt Left Riemann-Liouville fractional integral.
RLtD−αb Right Riemann-Liouville fractional derivative.
RLtD−αb Right Riemann-Liouville fractional integral.
RZDαt Riesz fractional derivative.
WS2δαx,+ Second order weighted and shifted Grunwald difference operator for the left Riemann-Liouville
space derivative.
WS2δαx,− Second order weighted and shifted Grunwald difference operator for the right Riemann-Liouville
space derivative.
WS3δγt u(t) Weighted and shifted difference operator for the Riemann-Liouville time derivative.
WS4δαx,+ Weighted and shifted Grunwald difference operator for the fourth order approximation to the
left Riemann-Liouville space derivative .
xix
WS4δαx,− Weighted and shifted Grunwald difference operator for the fourth order approximation tos the
right Riemann-Liouville space derivative.
pδαx,+ Shifted Grunwald-Letnikov operator for the left Riemann-Liouville space derivative.
pδαx,− Shifted Grunwald-Letnikov operator for the right Riemann-Liouville space derivative.
τ Time interval size.
0Dα(x,t)t Coimbra variable-order time derivative.
h Space interval size.
K Diffusivity coefficient.
L∞h,τ Maximum error.
M Space mesh size.
N Time mesh size.
Rni Truncation error.
xx
Glossary
EOC Error order of convergence.
TOC Time of computation.
xxi
xxii
Chapter 1
Introduction
1.1 Motivation
Fractional Calculus is a mathematical field dealing with integrals and derivatives of arbitrary order.
Even though the concept dates back to 1695 [1], it was only on the last century that the most impressive
achievements were made. Particularly in the last three decades fractional calculus has found applica-
tions in physics, signal-processing, engineering, bio-science, and finance [2, 3, 4, 5, 6, 7].
The field of Aerospace Engineering has been an early adopter of fractional calculus and one may
find its application regarding viscoelasticity and modelling of unsteady aerodynamic forces in AIAA con-
ferences and papers since the 1980’s, see e.g. [8, 9, 10, 11, 12, 13, 14]. The topic has also captured
both the attention of NASA and ESA in the solution of viscoelastic[15] and astrophysical problems [16].
Aerospace engineering beeing such a multidisciplinary field, can find applications of fractional calculus
in a vast number of areas including acoustics [17], fracture mechanics [18], composite materials [19]
and control theory [20, 21]. Recent developments have been also made in the fields of heat conduction
[22, 23] and flow in porous media [24, 25]. Interest in fractional calculus is currently experiencing an un-
precedented growth and there is no doubt aerospace engineering will benefit from future technologies
potentiated by this mind opening mathematical theory.
Anomalous diffusion has received particular interest in the framework of fractional calculus applica-
tions, see e.g. [3, 26, 27, 28, 29, 30, 31, 32]. So far, constant order fractional differential equations have
been the most used in anomalous diffusion modelling. Even if successful, constant order models have
failed to describe more complex phenomena whose behaviour is dependent of time, space and system
properties. Variable order differential operators, on the other hand allow the order of the derivative to
be a function of time, space or even the function itself, providing the flexibility to solve many of these
phenomena. A significant portion of this research effort is however concentrated in the mathematical
theory or numerical solution of these schemes. On the other hand, since the analytical solutions of frac-
tional differential equations are difficult to obtain, numerical methods for the solution of these equations
become extremely important. Finite difference methods in particular became very popular and a large
number of schemes has been published very recently. Consequently it becomes important to under-
1
stand how they compare in terms of accuracy, stability and computing times. The majority of the works
related to the numeric solution of fractional differential equations provides numerical examples based on
manufactured solutions, through which is not possible to gain the much needed intuitive understanding
of how the order of variable order derivatives affects the system behaviour.
1.2 Topic Overview
In fractional diffusion equations, integer order derivatives are replaced by fractional order counter-
parts, originating what may be considered as three different types of equations: i) time fractional, ii)
space fractional and iii) space-time fractional equations. Enjoying non-local properties, fractional inte-
grals and derivatives may describe more accurately anomalous diffusion processes. For instance it has
been suggested that the probability density function u(x, t) that describes anomalous subdiffusion par-
ticles follows the time fractional subdiffusion equation [3, 33, 34, 35]. Naturally, each type of fractional
diffusion equation has attracted in its own right a considerable number of works regarding its solution.
Loking at time- fractional diffusion equations, implicit shemes are more favorable than their explicit
counterparts, see e.g. [36, 37, 38, 39, 40]. Compact schemes have also attracted many researchers
because of the advantadge of keeping the tridiagonal nature, see e.g. [41]. Gao and Sun [42] applied
the L1 approximation for the time-fractional derivative and developed a compact finite difference scheme
for the fractional sub-diffusion equation. Most of these methods focus on the improvement of the space
order accuracy. It is however remarked that other numerical methods have been sought to solve time
fractional diffusion equations namely finite element [43] and spectral methods [44, 45]. High order in
space and time was sought by Ji and Sun [46] have proposed a high order compact difference scheme
able to solve the time fractional diffusion equation with third order accuracy in time. Most recently, Hu and
Zhang [47] have also proposed a second order implicit finite difference method in time for the fractional
diffusion equation.
Concerning the space fractional diffusion equation, Meerschaert and Tadjeran [48] proposed a shifted
Grunwald formula to approximate the space fractional derivative, overcoming the instability of the stan-
dard formula and applied it in the construction of schemes for the space fractional diffusion equation
[49, 50]. Tadjeran et al. [51] have also presented the Taylor expansion of the error of the shifted Grunwald
formula that enabled the construction of many high order schemes. Tian et al. [52], constructed a class
of second-order finite difference approximations with weighted and shifted Grunwald difference approx-
imations for Riemann-Liouville derivatives. Combining these approximations with a compact technique,
Zhou et al. [53] then suggested a third order scheme. Following this work, Chen et al. [54] further pro-
posed a class of second, third and fourth order difference approximations to solve the space fractional
diffusion equations. Hao et al. [55] have also proposed a fourth-order difference approximation for the
Riemann-Liouville space derivatives combining the weighted average of the shifted Grunwald operators
with a compact technique. Noticing that the matrices for the solution of space fractional diffusion equa-
tions have a structure of Toeplitz type, a fast finite difference solver has been proposed, reducing storage
and computing cost [56].
2
Schemes for time-space fractional diffusion equations have recently received considerable attention.
Liu et al. [57] proposed an implicit finite difference approximation to the time–space fractional diffusion
equation with first order accuracy in time and space. Yang et al. [45] derived a novel numerical method
based on the matrix transfer technique in space and finite difference scheme (or Laplace transform) in
time to deal with the time–space fractional diffusion equations in two dimensions. Chen et al. [58] applied
the L1 approximation to the time fractional derivative and second-order finite difference discretizations
to the space fractional derivative for solving the two-dimensional time–space Caputo-Riesz fractional
diffusion equation with variable coefficients in a finite domain. Ding [59] recently presented a numerical
method for the space–time Caputo-Riesz fractional diffusion equation, discretizing the Riesz derivative
by a fourth- order fractional-compact difference scheme and changing the space–time fractional diffu-
sion equation into a fractional ordinary differential equation system. Sun et al. [60] proposed several
difference schemes for one-dimensional and two-dimensional space and time fractional Bloch-Torrey
equations with second and fourth order in space. Wang et al. [61] proposed an alternating direction im-
plicit scheme with second-order accuracy in both time and space that is also considered the time–space
fractional subdiffusion equation. Pang and Sun have also proposed a fourth order accurate compact
difference scheme in space, using the L1 approximation for the time fractional derivative.
From the brief survey above, there are a multitude of finite difference methods for the approximation of
fractional derivatives and their application on schemes for time, space and space-time fractional diffusion
equations. The non-local properties of fractional derivatives translate into approximations that have
much longer computing times than integer order derivatives. For this reason, the stability criteria of
explicit schemes for fractional diffusion equations may lead to prohibitively high computational costs and
difficulties in the analysis of the orders of accuracy. As such, this work focuses on implicit schemes that
are unconditionally stable. Three different schemes with increasing order of accuracy were selected for
time, space and time-space fractional diffusion equations. The objective is the comparison of these finite
difference schemes in terms of accuracy and computational cost.
For time fractional diffusion equations the compared schemes are: i) the weigted average scheme
developed by Yuste [40],with the classic Grunwald-Letnikov Approximation with first order accuracy in
time, ii) the recent scheme proposed by Hu and Zhang [47] with second order accuracy and iii) the
third-order in time compact finite difference scheme developed by Ji and Sun [46].
The comparison of space fractional diffusion schemes is made with the following works: i) the first
order in space scheme developed by Meerschaert and Tadjeran, using the shifted Grunwald difference
formula for space fractional derivatives ii) the second order in space scheme developed by Tian et
al. [62], that used a weighted combination of the shifted Grunwald difference operator of the previous
scheme and iii) the fourth order in space scheme that uses weighted and shifted Grunwald difference
operators together with a compact technique. These works are representative of the evolution that has
been seen on space fractional diffusion equations since going from the first suggestion of the shifted
Grunwald difference formula to the more recent trend of compact difference schemes for fractional diffu-
sion equations that includes the use of weighted and shifted Grunwald difference operators.
Regarding the schemes for time-space fractional diffusion equations the L1 method stands as the
3
most common approximation for time fractional derivatives while the search continues for high-order
finite difference schemes both in time and space. Serving as a needed common ground, a comparison
is made with schemes that use the L1 approximation for the Caputo time fractional derivative with time
accuracy (2 − γ). In the schemes that were chosen, the discretization of space fractional derivatives is
made with the three approximations that were used in space fractional diffusion equations. The selected
schemes are: i) the first order in space scheme proposed by Li et al. [57] ii)a second order in space
scheme combining the L1 method and the second order approximation for space fractional derivatives
iii) the fourth order in space scheme proposed by Pang et al. [63].
Even if successful, constant order models have failed to describe more complex phenomena whose
behaviour is dependent of time, space and system properties. Variable order models, on the other hand,
have received far less attention than constant fractional order ones. To tackle this problem, several
authors have proposed different definitions of variable order operators [64] and distributed order [65].
Random order fractional differential equations have also been considered by Sun et al. [66, 67, 68]
that concluded that each type of fractional differential operator has distinct advantages and potential
applications for the modelling of diffusion processes. Distributed order models are the best at describing
multi-scale diffusion processes while the variable order models suit the description of diffusion processes
whose diffusion pattern changes with time evolution or space variation. To describe diffusion processes
subjected to an oscillating field or unstable system parameters, the random order model may be more
adequate.
Important works involving variable order calculus have been reviewd by Samko [69] that has also
proposed the concept of variable order differential operator, investigating the properties of variable order
Riemann-Liouville integrals and derivatives [70]. Lorenzo and Hartley [71] studied some mathematical
properties of candidate variable-order operators . Coimbra et al. [64] has investigated the dynamics
and control of nonlinear viscoelasticity oscillator with variable order operators . With a time dependent
variable order operator, Ingman et al. [72, 73] modeled the viscoelastic deformation process. Pedro et
al. [74] modelled the motion of particles suspended in a viscous fluid where the drag force is calculated
recurring to variable order calculus. Kobolev et al. [75] studied the statistical physics of dynamic systems
with variable memory. Chechkin et al. [76] studied the evolution of a composite system consisting of two
separate regions with the time-fractional diffusion equation with a space variable fractional order time
derivative.
This work focuses on variable order fractional diffusion equations that present an important tool in
the study of complex anomalous diffusion phenomena. Going beyond the constant fractional order
exponent allows modelling of situations where the diffusive behaviour may change with time evolution,
space location, the concentration of the diffusing species as well as system parameters. Very often
these issues are mainly addressed in the mathematical framework without discussion of effect of the
variable order on the solution. Attending to the vast number of possibilities, attention on this work will
be focused for simplicity on the variable order time fractional diffusion equation, capable of depicting
sub-diffusive processes in which temporal fractional derivatives is solution of i)time, ii) space and iii)
the dependent solution itself. The results of the different variable order functions are compared and the
4
results discussed. The diffusion equation is considered on the form
0Dγ(x,t)t u(x, t) = K
∂2u(x, t)
∂x2+ f(x, t), x ∈ [0, L]t ∈ [0, T ] (1.2.1)
where K > 0 is a generalized diffusion coefficient, the variable u(x, t) is a physical quantity of interest
such as temperature, concentration or survival propability of a particle and 0Dγ(x,t)t denotes the Coimbra
variable order derivative [64].
Variable time dependent fractional order diffusion equations are useful in the modelling of processes
for which the diffusive behaviour changes with time. There are situations for instance, where processes
tend to Fickian diffusion with the evolution of time [77, 78, 79]. This behaviour can found in biology,
plasma physics and economy. The opposite behaviour can also be observed, with diffusion rates de-
creasing with time evolution. Conventional solutions to these problems are often found through inte-
ger order differential equations using time dependent diffusion exponents [80, 81]. Good data fitting is
sometimes provided by such methods, but they cannot achieve a general formulation for time dependent
diffusion processes because they do not capture the origin of these problems [68].
The space dependent variable order time derivative may be thought of as the memory rate depending
on the space location in the diffusive system. While the constant order diffusion models seem to be ade-
quate to model homogeneous media, that is not the case for inhomogeneous and isotropic situations. In
this case, the diffusive behaviour changes with spacial location making the case for a space dependent
variable order. In recent years, anomalous diffusion in complex media has captured the attention of
many scholars in fields such as geophysics, environmental science, hydrology and biology [82]. Frac-
tional diffusion equations have here helped to model heat conduction and fluid flow in porous media
seismic waves and protein dynamics. Modelling of these problems is often made through nonlinear dy-
namics, statistical mechanics and memory formalisms [83]. With the significance of these problems, the
necessity for the investigation of diffusive behaviour in porous systems becomes apparent.
Solution dependent variable order diffusion equation allows the memory rate to vary along with the
solution the diffusive system, capturing information that would otherwise be coded in a complex expres-
sion of the diffusion coefficient. There are situations in physics, chemistry and biology where concentra-
tion plays the key role in diffusive behaviour [84]. Examples of this behaviour are the diffusive transport of
macromolecules in biological tissue and diffusion processes associated with chemical reactions where
the concentration of reactant will determine the characteristics of the chemical diffusion process. The
most common approaches to deal with these situations involve nonlinear or variable coefficient partial
differential equations [85, 86]. In these cases the expression adopted for the diffusion coefficient often
presents parameters that are difficult to physically analyse or obtain experimentally [87].
Finite difference methods stand as the most popular solution methods for fractional calculus. Nonethe-
less other methods have also been sought for the solution of variable order fractional differential equa-
tions, namely spectral methods [88]. Coimbra et al. [64] proposed a first order accurate approximation
for variable order differential equations. Soon et al. [89] have employed a second-order Runge-Kutta
method consisting of an explicit Euler predictor step followed by an implicit Euler corrector step to nu-
5
merically integrate the variable order differential equation. Zhuang et al. [90] constructed explicit and
implicit Euler approximations for the variable-order fractional advection-diffusion equation with a nonlin-
ear source term and several other approximations have been proposed.
Lin et al. [91] constructed an explicit finite difference scheme for spatial VO fractional differential
equation with a generalized Riesz fractional derivative of variable order α(x, t), (1 < α(x, t) ≤ 2) with
linear convergence on both time and space. Chen et al. [92] proposed a numerical scheme with first
order temporal accuracy and fourth order spatial accuracy for the fractional sub-diffusion equation. In
[93], the variable-order nonlinear Stokes’ first problem for a heated generalized second grade fluid with
a fourth order accurate numerical scheme is studied. The variable-order nonlinear reaction-subdiffusion
equation was considered by [94]. In [95] an implicit scheme for the variable order space fractional
diffusion equation in two dimensions was proposed and in [96] an alternating direction implicit method
for new two-dimensional variable-order fractional percolation equation with variable coefficients. Sun et
al. [97] developed three finite difference schemes for the variable-order fractional sub-diffusion equation
and suggested [98] a finite difference scheme with first order accuracy in time and second order in
space for the fractional subdiffusion equation. Shen et al. [99] developed a numerical scheme for the
variable order advection-diffusion equation with a nonlinear source term with first order accuracy. Zhang
et al. [100] proposed an implicit difference method, first order accurate in time and space, for the time
fractional variable order mobile-immobile-advection-dispersion equation.
The finite difference schemes that have been mentioned for variable order fractional diffusion equa-
tions exhibit first order convergence in time. Zhao et al. [101] derives two second- order approximation
formulas for the variable-order fractional time derivatives involved in anomalous diffusion and wave prop-
agation. It should be noted that not all the schemes here mentioned adopt the same definition of variable
order fractional derivative. The Coimbra [102] definition of fractional derivative is adopted in this work.
Several other authors adopt this definition, see e.g. [98, 101, 103].
1.3 Objectives
The scientific community’s interest in fractional calculus is undergoing exponential growth, many
applications have been found so far but the majority is yet to be unravelled. For fractional calculus to
succeed in engineering applications, a proper understanding of the underlying mathematical theory and
the tools to deal with it have to be acquired. As such this work represents and effort in the building of a
bridge between the mathematical and engineering standpoints, laying the ground for future applications.
This purpose is fulfilled with two objectives. Firstly, through the implementation and comparison of
several finite difference schemes for fractional diffusion equations, a good grasp on particularities of this
numerical solution technique, applied to fractional calculus, is intended. Secondly, an intuitive insight
on how constant and variable order fractional differential operators affect the solution of system is to be
gained that will certainly prove valuable in the development of an engineering application.
6
1.4 Thesis Outline
This document is organized as follows. Chapter 2 will concern the comparison of finite difference
schemes for fractional diffusion equations. Section 2.1 begins with a brief overview of the most important
mathematical definitions of fractional integrals and derivatives.
In section 2.2 the finite difference approximations that will be used to in the construction of schemes
for fractional diffusion equations will be provided. For time fractional derivatives the Grunwald-Letnikov,
L1 and a third order weighted and shifted Grunwald approximation aproximation will be presented. For
space fractional derivatives the first order shifted Grunwald difference approximation, the second order
weighted and shifted Grunwald difference approximation and a fourth order compace difference approx-
imation are presented. In section 2.3 finite difference approximations for the first order time derivative
and second order space derivative are given.
Section 2.4 studies the behaviour of three schemes with increasing order of accuracy for the time
fractional diffusion equation. The initial-boundary value problem is stated in section 2.4.1. In section
2.4.2 a first order weighted average finite difference method is given, section 2.4.3 presents a second
order scheme and section 2.4.4 will deal with a third order in time compact difference scheme. In
section 2.4.5 a numerical example is solved with the three schemes, with different time fractional orders
and time interval refinement, serving both for code validation and to compare the schemes in terms of
convergence order and computing time.
With a similar structure to section 2.4, section 2.5 studies the behaviour of three schemes with in-
creasing order of accuracy for the space fractional diffusion equation. The initial-boundary value problem
is stated in section 2.5.1. In section 2.5.2 a first order in space finite difference method is given, section
2.5.3 refers to a second order scheme and section 2.5.4 will deal with a fourth order in space compact
difference scheme. In section 2.5.5 a numerical example is solved with the three schemes, with different
space fractional orders and space interval refinement, serving both for code validation and to compare
the schemes in terms of convergence order and computing time.
Section 2.6 will deal with schemes for the time-space fractional diffusion equation under a common
fractional time derivative approximation. As in the two previous sections, the initial-boundary value
problem is stated in section 2.6.1. In section 2.6.2 a first order in space finite difference method is given,
section 2.6.3 refers to a second order scheme and section 2.6.4 will deal with a fourth order in space
compact difference scheme. In section 2.6.5 a numerical example is solved with the three schemes
for time-space, with different space and time fractional orders and refinement of the space and time
intervals, serving both for code validation and to compare the schemes in terms of convergence order
and computing time.
In chapter 3 an investigation is made into the effects of variable order differentiation in the behaviour
of sub-diffusive systems. Section 3.1 introduces a scheme able to solve variable order time-fractional
sub-diffusion equations. The initial-boundary value problem is stated in section 3.1.1. In section 3.1.2 a
difference scheme able to solve time fractional diffusion equations with variable coefficients dependent
on time and space [98] is implemented and provided to the reader in matrix form. The convergence
7
of the scheme is then validated with a numerical example against the analytic solution in section 3.1.3.
In section 3.2 the effects of order dependence on time, space and the solution itself will be analysed
through a numerical example. Departure is made in section 3.2.1 from the comparison of the standard
diffusion equation with constant order fractional diffusion which is then taken as reference for the analysis
of the behaviour of anomalously diffusive systems with variable order. Sections 3.2.2, 3.2.3 and 3.2.4,
study of the behaviour of a sub-diffusive with variable orders dependent of time, space and the system
solution, respectively.
In chapter 4 important conclusions are made regarding both the schemes for the different types
of fractional diffusion equations and the behaviour of the solution in the case of a variable order time
fractional diffusion equations. The main achievements are stated and suggestions for future works are
made.
8
Chapter 2
Finite Difference Solution of
Fractional Diffusion Equations
This purpose of this chapter is to compare finite difference schemes for fractional diffusion equations.
Section 2.1 begins with a brief overview of the most important mathematical definitions of fractional
integrals and derivatives.
In section 2.2 the finite difference approximations that will be used to in the construction of schemes
for fractional diffusion equations will be provided. For time fractional derivatives the Grunwald-Letnikov,
L1 and a third order weighted and shifted Grunwald approximation aproximation will be presented. For
space fractional derivatives the first order shifted Grunwald difference approximation, the second order
weighted and shifted Grunwald difference approximation and a fourth order compace difference approx-
imation are presented. In section 2.3 finite difference approximations for the first order time derivative
and second order space derivative are given.
Section 2.4 studies the behaviour of three schemes with increasing order of accuracy for the time
fractional diffusion equation. The initial-boundary value problem is stated in section 2.4.1. In section
2.4.2 a first order weighted average finite difference method is given, section 2.4.3 presents a second
order scheme and section 2.4.4 will deal with a third order in time compact difference scheme. In
section 2.4.5 a numerical example is solved with the three schemes, with different time fractional orders
and time interval refinement, serving both for code validation and to compare the schemes in terms of
convergence order and computing time.
With a similar structure to section 2.4, section 2.5 studies the behaviour of three schemes with in-
creasing order of accuracy for the space fractional diffusion equation. The initial-boundary value problem
is stated in section 2.5.1. In section 2.5.2 a first order in space finite difference method is given, section
2.5.3 refers to a second order scheme and section 2.5.4 will deal with a fourth order in space compact
difference scheme. In section 2.5.5 a numerical example is solved with the three schemes, with different
space fractional orders and space interval refinement, serving both for code validation and to compare
the schemes in terms of convergence order and computing time.
Section 2.6 will deal with schemes for the time-space fractional diffusion equation under a common
9
fractional time derivative approximation. As in the two previous sections, the initial-boundary value
problem is stated in section 2.6.1. In section 2.6.2 a first order in space finite difference method is given,
section 2.6.3 refers to a second order scheme and section 2.6.4 will deal with a fourth order in space
compact difference scheme. In section 2.6.5 a numerical example is solved with the three schemes
for time-space, with different space and time fractional orders and refinement of the space and time
intervals, serving both for code validation and to compare the schemes in terms of convergence order
and computing time.
2.1 Mathematical Preliminaries
In this chapter, the mathematical definitions of fractional integrals and derivatives used throughout
this paper are introduced [32].
Definition 2.1. The left and right fractional Riemann-Liouville integrals of order α > 0 of a given function
f(t), t ∈ (a, b) are defined as
RLaD−αt f(t) =
1
Γ(α)
∫ t
a
(t− s)α−1f(s)ds (2.1.1)
RLtD−αb f(t) =
1
Γ(α)
∫ b
t
(s− t)α−1f(s)ds (2.1.2)
respectively, where Γ(·) denotes Euler’s gamma function.
Definition 2.2. The left and right Riemann-Liouville derivatives with order α > 0 of the function f(t),
t ∈ (a, b) are defined as
RLaD
αt f(t) =
dm
dtm[D−(m−α)a,t f(t)] =
1
Γ(m− α)
dm
dtm
∫ t
a
(t− s)m−α−1f(s)ds (2.1.3)
RLtD
αb f(t) = (−1)m
dm
dtm[D−(m−α)t,b f(t)] =
(−1)m
Γ(m− α)
dm
dtm
∫ b
t
(s− t)m−α−1f(s)ds (2.1.4)
respectively, where m is a positive integer satisfying m− 1 ≤ α < m and Γ(·) is Euler’s gamma function.
Definition 2.3. The left and right Caputo derivatives with order α > 0 of the function f(t), t ∈ (a, b) are
defined as
CaD
αt f(t) = RL
aD−(m−α)t [f (m)(t)] =
1
Γ(m− α)
∫ t
a
(t− s)m−α−1f (m)(s)ds (2.1.5)
CtD
αb f(t) =
(−1)m
Γ(m− α)
∫ b
t
(s− t)m−α−1f (m)(s)ds (2.1.6)
respectively, where m is a positive integer satisfying m− 1 ≤ α < m and Γ(·) is Euler’s gamma function.
Although the definitions of the Riemann-Liouville and of the Caputo derivatives cannot be assumed
equal, they do have the following relationship
10
RLaD
αt f(t) = C
aDα
t f(t) +
m−1∑k=0
f (k)(a)(t− a)k−α
Γ(k + 1− α)(2.1.7)
where m − 1 < α < m is a positive integer and fm is integrable on [a, t]. On the special case where
fk(0) = 0 with k = 0, 1, 2, · · · ,m− 1,m− 1 < α < m the Riemann-Liouville and Caputo derivatives are
equivalent.
Furthermore, if the definitions of fractional integral and derivative are compared, it can be seen that
the Caputo derivative of order α is equivalent to the fractional integral of order (m − α) of f (m)(t), with
m− 1 < α < m [104].
Definition 2.4. The Riesz derivative of order α > 0 for a given function f(t), t ∈ (a, b) is defined as
RZDαt f(t) = cα(RLaD
αt f(t) + RL
tDαb f(t)) (2.1.8)
where cα = − 12 cos(απ/2) and α 6= 2k + 1, k = 0, 1, · · ·.
Definition 2.5. The Coimbra variable order time derivative of order α(x, t) ∈ [0, 1] for a given function
f(x, t), t ∈ (a, b) is defined as
0Dα(x,t)t f(x, t) =
1
Γ(1− α(x, t))
∫ t
0
(t− σ)−α(x,t)∂f(x, σ)
∂σdσ +
(f(x, 0+)− f(x, 0−))t−α(x,t)
Γ(1− α(x, t)(2.1.9)
2.2 Finite difference approximations for fractional derivatives
2.2.1 Approximations for time fractional derivatives
2.2.1.1 The Grunwald-Letnikov Approximation
The Grunwald-Letnikov approximation [40] is one of the most used for time fractional derivatives. Let
t ∈ [0, T ], τ = T/N so that tn = nτ , Ωτ = tn|0 ≤ n ≤ N and u(tn) = un. If u(t) is suitably smooth, the
left Riemann-Liouville derivative can be approximated with first order accuracy by
[RL0 Dγ
t u(t)]t=tn
= GLδγt un +O(τ) (2.2.1)
where the left Grunwald-Letnikov difference operator is given by
GLδγt un =1
τγ
n∑k=0
ω(γ)j un−k (2.2.2)
The Grunwald-Letnikov wheights ω(γ)k = (−1)k
(γk
), with k ≥ 0, are the coefficients of the power
series of the generating function (1− z)γ =∑∞k=0 ω
(γ)k zk. These weights satisfy the recursive formula
ω(γ)k =
(1− γ + 1
k
)ω(γ)k−1, w
(γ)0 = 1. (2.2.3)
11
However, other formulas for the calculation of these weights exit, leading to higher order approximations
[105, 32].
2.2.1.2 L1 Approximation
The L1 method [36] is another popular choice for the approximation of time fractional derivatives.
This approximation is found in many unconditionally stable schemes and is suitable for (0 < γ < 1).
Nevertheless, similar methods exist for 1 < γ < 2.
Let t ∈ [0, T ], τ = T/N so that tn = nτ and Ωτ = tn|0 ≤ n ≤ N. Additionally, let u(tn) = un. The
left Riemann-Liouville derivative can be discretized with (2− γ)th order accuracy by
[RL
0Dγ
t u(t)]t=tn
= L1δγt un +O(τ2−γ) (2.2.4)
where the L1δγt un operator is defined as
L1δγt un =τ−γ
Γ(2− γ)
n−1∑k=0
bn−k−1 [uk+1 − uk] +u0t−γn
Γ(1− γ)(2.2.5)
with
bk =[(k + 1)1−γ − k1−γ
](2.2.6)
If, on the other hand the Caputo definition of time fractional derivative is considered then the scheme
can be approximated with (2− γ)th order accuracy by
[C0D
γ
t u(t)]t=tn
= L1Cδ
γt un +O(τ2−γ) (2.2.7)
where the L1Cδ
γt un operator is defined as
L1Cδ
γt un =
τ−γ
Γ(2− γ)
n−1∑k=0
bn−k−1 [uk+1 − uk] (2.2.8)
2.2.1.3 Third order weighted and shifted Grunwald difference approximation
In [46] Ji and Sun developed a third order accurate weighted and shifted Grunwald difference operator
for the Riemann-Liouville derivative, that they used to construct in a compact difference scheme for the
time fractional diffusion equation. The construction of this approximation is now summarized.
Let t ∈ [0, T ], τ = T/N so that tn = nτ and Ωτ = tn|0 ≤ n ≤ N. Additionally, let u(tn) = un.
Supposing that u ∈ L1(R) ∩ Cγ+1(R), the Riemann-Liouville derivative (2.1.3) evaluated from negative
infinity (a = −∞) can be approximated with first order accuracy by
[RL−∞D
γt u(t)
]t=tn
= pδ(γ)t un +O(τ) (2.2.9)
12
where pδ(γ)t is the shifted Grunwald difference operator [48], defined as
pδ(γ)t un =
1
τγ
∞∑k=0
ω(γ)k un−(k−p) (2.2.10)
uniformly for t ∈ R as τ → 0. The integer p corresponds to the number of shifts and as in the Grunwald-
Letnikov approximation, the weights ω(γ)k are the coefficients of the power series of the generating func-
tion (1− z)γ , given in equation (2.2.3).
Moreover, if u(t) ∈ L1(R), −∞Dγ+3t u(t) and its Fourier transform belong to L1(R), then the operator
in (2.2.10) can be used to construct a third order approximation for RL−∞D
γt u(t)
[RL−∞D
γ
tu(t)
]t=tn
= p,q,rδ(γ)t un +O(τ3) (2.2.11)
where p,q,rδ(γ)t is a weighted and shifted Grunwald difference operator defined by
p,q,rδ(γ)t un = ρ1 pδ
(γ)t un + ρ2 qδ
(γ)t un + ρ3 rδ
(γ)t un (2.2.12)
with shifts p,q and r are defined in [46] as (p, q, r) = (0,−1,−2) and
ρ1 =12qr − (6q + 6r + 1)γ + 3γ2
12(qr − pq − pr + p2), ρ2 =
12pr − (6p+ 6r + 1)γ + 3γ2
12(pr − pq − qr + q2),
ρ3 =12pq − (6p+ 6q + 1)γ + 3γ2
12(pq − pr − qr + r2)
Introducing now,
u(t) =
u(t), t ∈ [0, T ]
0, t ∈ [−∞, 0]
(2.2.13)
it naturally occurs that RL0Dγ
t u(t) = RL−∞D
γ
tu(t) and therefore RL
0Dγ
t u(t) can be approximated with third
order accuracy by [RL
0Dγ
t u(t)]t=tn
= WS3δγt un + +O(τ3) (2.2.14)
where the weighted and shifted difference operator WS3δγt u(t) is defined as
WS3δγt un =1
τγ
[ρ1
n∑k=0
ω(γ)k un−k + ρ2
n−1∑k=0
ω(γ)k un−(k+1) + ρ3
n−2∑k=0
ω(γ)k un−(k+2)
](2.2.15)
For simplicity, the WD3δγt operator can be written as
WD3δγt un =1
τγ
n∑k=0
g(γ)k un−k, n = 2, 3, ..., N (2.2.16)
13
where g(γ)0 = ρ1ω
(γ)0
g(γ)1 = ρ1ω
(γ)1 + ρ2ω
(γ)0
g(γ)k = ρ1ω
(γ)k + ρ2ω
(γ)k−1 + ρ3ω
(γ)k−2, k ≥ 2
(2.2.17)
2.2.2 Approximations for space fractional derivatives
2.2.2.1 The shifted Grunwald approximation
The first approximation considered for space fractional derivatives will be the shifted Grunwald ap-
proximation, used by Meerschaert and Tadjeran [49] to construct a first order in space scheme for the
solution of space fractional diffusion equations. For (1 < α < 2), the standard approximation leads to
unstable numerical schemes, this problem is solved if the shifted approximation is chosen. Moreover,
the shifted approximation can be employed in the construction of higher order approximations through
the weighted combination of different shifts, as will later be demonstrated.
Let x ∈ [a, b], h = (b − a)/M so that xi = ih and Ωh = xi|0 ≤ i ≤ M. Additionally, let u(xi) = ui.
Similarly to the shifted Grunwald approximation introduced in section 2.2.1.3, let u(x) ∈ L1(R)∩Cγ+1(R).
The left and right Riemann-Liouville derivatives (2.1.3) evaluated with (a = −∞) and (b = +∞) can be
approximated with first order accuracy by
[RL−∞D
α
xu(x)
]x=xi
= Gpδαx,+ui +O(h) (2.2.18)[
RLxD
α
∞u(x)]x=xi
= Gpδαx,−ui = +O(h) (2.2.19)
where the shifted left and right Grunwald operators are defined as
Gpδαx,+ui =
1
hα
∞∑k=0
ω(α)k ui−k+p (2.2.20)
Gpδαx,−ui =
1
hα
∞∑k=0
ω(α)k ui+k−p (2.2.21)
respectively, as h→ 0 and where p ∈ Z is the number of shifts. It was found that optimum performance
comes from p = 1 when 1 < γ ≤ 2 [49].
If a zero extension of the function is made,
u(x) =
u(x), t ∈ [a, b]
0, x ∈ [−∞, a] ∪ [b,+∞]
(2.2.22)
it occurs that on the interval x ∈ [a, b], RLaDαxu(x) = RL
−∞Dγ
xu(x) and RL
xDαb u(x) = RL
xDγ
∞u(x). Hence,
the use of (2.2.30) and (2.2.31) to approximate RLaD
αxu(x) and RL
xDαb u(x) results in
[RLaD
α
xu(x)]x=xi
= pδαx,+ui +O(h) (2.2.23)
14
[RLxD
α
b u(x)]x=xi
= pδαx,−ui = +O(h) (2.2.24)
where the shifted left and right Grunwald-Letnikov operators are defined as
pδαx,+ui =
1
hα
i+p∑k=0
ω(α)k ui−k+p (2.2.25)
pδαx,−ui =
1
hα
M−i+p∑k=0
ω(α)k ui+k−p (2.2.26)
respectively. These approximations are also referred to as the Grunwald-Letnikov approximations. One
drawback of these is that they lack first order accuracy when the values at the boundaries are not zero
[106]. For instance, considering the left sided derivative, if u(a) 6= 0, first order accuracy can be achieved
with
[RLaD
α
xu(x)]x=xi
=[RLaD
α
x [u(x)− u(a)]]x=xi
+u(a)x−αiΓ(1− α)
=1
hα
i+p∑k=0
ω(α)k (ui−k+p − u(a)) +
u(a)x−αiΓ(1− α)
+O(h)
(2.2.27)
2.2.2.2 Second order approximation
Zhou, Tian and Deng [62] develop second order approximations for left and right Riemann-Liouville
derivatives and use them on the same paper for the construction of schemes for the solution of the
space fractional diffusion equation. These approximations are made with weighted and shifted Grunwald
difference operators, inspired in the shifted Grunwald difference operator introduced in the previous
section. A summary of the construction of such operators will be made, departing from the shifted
operators introduced in the previous section.
As before, let x ∈ [a, b], h = (b − a)/M so that xi = ih and Ωh = xi|0 ≤ i ≤ M. Additionally, let
u(xi) = ui. Assuming that u ∈ L1(R), RL−∞Dα+2
xu and its Fourier transform belong to L1(R) the weighted
and shifted Grunwald difference operators can be defined as
Gp,qδ
α
x,+ui =
α− 2q
2(p− q)Gpδα
x,+ui +
2p− α2(p− q)
Gqδα
x,+ui (2.2.28)
Gp,qδ
α
x,−ui =α− 2q
2(p− q)Gpδα
x,−ui +2p− α
2(p− q)Gqδα
x,−ui (2.2.29)
allowing the left and right Riemann-Liouville derivatives (2.1.3) and (2.1.4) to be evaluated for (a = −∞)
and (b = +∞) with second order accuracy
[RL−∞D
α
xu(x)
]x=xi
= Gp,qδ
α
x,+ui +O(h2) (2.2.30)
[RLxD
α
∞u(x)]x=xi
= Gp,qδ
α
x,−ui +O(h2) (2.2.31)
15
uniformly for x ∈ R, where p and q are integers with p 6= q.
Considering that the function u is well defined in the interval [a, b], if u(a) = 0 = u(b) = 0, a zero
extension of u(x) can be made for (x < a ∪ x > b). In this manner the Riemann-Liouville derivatives of
order α of u(x) can be approximated by (2.2.28) and (2.2.29) with second order accuracy, resulting in
[RLaD
α
xu(x)]x=xi
=µ1
hα
[ x−ah ]+p∑k=0
ω(α)k ui−(k−p) +
µ2
hα
[ x−ah ]+q∑k=0
ω(α)k ui−(k−q) +O(h2)
=µ1
hα pδαx,+ui +
µ2
hα qδαx,+ui +O(h2)
(2.2.32)
[RLxD
α
b u(x)]x=xi
=µ1
hα
[ b−xh ]+p∑k=0
ω(α)k ui+(k−p) +
µ2
hα
[ b−xh ]+q∑k=0
ω(α)k ui+(k−q) +O(h2)
=µ1
hα pδαx,−ui +
µ2
hα qδαx,−ui + o(h2)
(2.2.33)
where µ1 = α−2q2(p−q) , µ2 = 2p−α
2(p−q) and pδαx,+ and pδ
αx,− are defined in equations (2.2.25) and (2.2.26).
The choice of p and q in (2.2.32) and (2.2.33) must satisfy |p| ≤ 1 and |q| ≤ 1, ensuring that
the nodes at which the values of u needed in (2.2.32) and (2.2.33) are within the bounded interval,
when employing the finite difference method with weighted and shifted Grunwald difference formulas
for numerically solving non-periodic fractional differential equations on bounded intervals. Otherwise,
an alternative discretization method is necessary when x is near a boundary. Having the authors of
the approximation already concluded that (p, q) = (0,−1) is unstable for time dependent problems,
only (p, q) = (1, 0) and (p, q) = (1,−1) remain for for the construction of the quasi-compact difference
approximations. Further conclusions on these coefficients will be provided upon the derivation of the
difference scheme for space fractional diffusion equations employing the weighted and shifted Grunwald
operators in (2.2.34) and (2.2.35). Equations (2.2.32) and (2.2.33) can be simplified to yield
[RLaD
α
xu(x)]x=xi
=1
hα
i+1∑k=0
g(α)k ui−k+1 +O(h2) = WS2δαx,+ui +O(h2) (2.2.34)
[RLxD
α
b u(x)]x=xi
=1
hα
N−i+1∑k=0
g(α)k ui+k−1 +O(h2) = WS2δαx,−ui +O(h2) (2.2.35)
where(p, q) = (1, 0), g
(α)0 =
α
2ω(α)0 , g
(α)k =
α
2ω(α)k +
2− α2
ω(α)k−1, k ≥ 1
(p, q) = (1,−1), g(α)0 =
2 + α
4ω(α)0 , g
(α)1 =
2 + α
4ω(α)1 , g
(α)k =
2 + α
4ω(α)k +
2− α4
ω(α)k−2, k ≥ 2
(2.2.36)
2.2.2.3 Fourth order compact finite difference approximation
Recently, Hao, Sun and Cao [55] developed a fourth-order approximation for Riemann-Liouville frac-
tional derivatives. Yet again, this approximation departs from the use of a weighted average of shifted
16
Grunwald operators with different shifts combining the compact technique. The main idea is to vanish
the low order leading terms in asymptotic expansions for the truncation errors by means of a weighted
average.
Tadjerran et al. [107] provided the Taylor expansion for the error in the shifted Grunwald finite differ-
ence formula, fundamental to many high order schemes. If 1 < α < 2 and u ∈ Cn+3(R) such that all
derivatives of u up to order n+ 3 belong to L1(R), it can be obtained for any integer r ≥ 0 that
Gpδα
x,+u(x) = RL
−∞Dα
xu(x) +
n−1∑l=1
cα,rlRL−∞D
α+l
xu(x)hl +O(hn) (2.2.37)
uniformly for x ∈ R, where Gpδα
x,+was defined in (2.2.25) and cα,rl are the coefficients of the power series
expansion of function Wr(z) = ( 1−e−zz )αerz. The condition that u ∈ Cn+3(R) can however be weakened
to u ∈ Cn+α(R) .
As usual, let x ∈ [a, b], h = (b − a)/M so that xi = ih and Ωh = xi|0 ≤ i ≤ M. Additionally, let
u(xi) = ui, u ∈ L1(R) and f ∈ C4+α(R). For current case, the weighted and shifted Grunwald difference
operators for 1 < α ≤ 2,are defined by
WSG4δα
x,+ui = λ1G1 δ
α
x,+ui + λ0G0 δ
α
x,+ui + λ−1G−1δ
α
x,+ui, (2.2.38)
WSG4δα
x,−ui = λ1G1δα
x,−ui + λ0G0δα
x,−ui + λ−1G−1δ
α
x,−ui (2.2.39)
respectively, where the shifted Grunwald difference operators for Riemann-Liouville fractional derivatives
are given by (2.2.20) and (2.2.21) and
λ1 =α2 + 3α+ 2
12, λ0 =
4− α2
6, λ−1 =
α2 − 3α+ 2
12(2.2.40)
In [55], it was showed that the operators in (2.2.38) and (2.2.39) have second order accuracy for
approximating Riemann-Liouville fractional derivatives. Considering the second order central difference
operator in (2.3.3), the following difference operator is defined
Aαui = (1 + cαh2δ2x)ui with cα =
−α2 + α+ 4
24, (2.2.41)
Applying Aα to RL−∞D
α
xu(x) and RL
x Dα
+∞u(x) Hao et al. reach fourth-order approximations , this
operator will naturally have to be applied to the remaining of th equation when deriving the scheme.
Letting u(x) ∈ L1(R) and u(x) ∈ C4+α(R), one obtains
Aα(RL−∞Dα
xu(x)) = δαx,+u(x) +O(h4) (2.2.42)
Aα(RLx Dα
+∞u(x)) = δαx,−u(x) +O(h4) (2.2.43)
Combining (2.2.38) and (2.2.39) with (2.2.20) and (2.2.21), the weighted and shifted difference oper-
ators can be written in an abbreviated form
17
WSG4δα
x,+ui =1
hα
+∞∑k=0
g(α)k ui−(k−1) (2.2.44)
WSG4δα
x,−ui =1
hα
+∞∑k=0
g(α)k ui+(k−1) (2.2.45)
where g(α)0 = λ1ω
(α)0
g(α)1 = λ1ω
(α)1 + λ0ω
(α)0
g(α)k = λ1ω
(α)k + λ0ω
(α)k−1 + λ−1ω
(α)k−2, k ≥ 2
(2.2.46)
and the weights ω(α)k are given by equation (2.2.3).
If u(x) ∈ C[a, b] with u(a) = u(b) = 0, a zero extension of u can be made. Supposing u(x) ∈ C4+α(R),
equations (2.2.42) and (2.2.43) lead to
Aα(RLaDα
xui) =1
hα
i∑k=0
g(α)k ui−(k−1) +O(h4) = WS4δ
α
x,+ui +O(h4) (2.2.47)
Aα(RLxDα
b ui) =1
hα
M−i∑k=0
g(α)k ui+(k−1) +O(h4) = WS4δ
α
x,−ui +O(h4) (2.2.48)
2.3 Finite difference approximations of integer order derivatives
In addition to fractional derivatives, integer order derivative will also need to be approximated through-
out the coming sections. First order time derivatives will appear in space fractional diffusion equations
and are approximated either by central or backward difference operators. On the other hand, second
order difference operators are used to approximate second order derivatives in time fractional diffusion
equations and in the construction of high-order schemes. Hence, this section introduces the operators
used to approximate integer order derivatives, common to several of the coming schemes.
Take two positive integers M, N and let h = (b − a)/M and τ = T/N . Define xi = ih(0 ≤ i ≤ M),
tn = nτ(0 ≤ n ≤ N), Ωh = xi|0 ≤ i ≤M and Ωτ = tn|0 ≤ n ≤ N. The computational domain
[a, b]× [0, T ] is then covered by Ωτh = Ωh × Ωτ . Moreover, let uni = u(xi, tn).
At time level n+ 1/2 it holds that
∂u
∂t
∣∣∣∣xi,tn+1
2
= δtun+ 1
2i +O(τ2), where δtu
n+ 12
i =un+1i − uni
τ(2.3.1)
which holds a similar result, apart from the truncation error, to the approximation of the first time deriva-
tive at time t = (n+ 1)τ with backward differences
∂u
∂t
∣∣∣∣xi,tn+1
= δtun+1i +O(τ), where δtu
n+1i =
un+1i − uni
τ(2.3.2)
18
The second order space derivative can be approximated at x = ih with
∂2u
∂x2
∣∣∣∣xi,tn
= δ2xuni +O(h2), where δ2xu
ni =
uni−1 − 2uni + uni+1
h2(2.3.3)
2.4 Time fractional diffusion equations
2.4.1 Problem statement
Attention will now be devoted to the development of schemes for time fractional diffusion equations.
To this purpose, the two most common forms of these equations are presented. Throughout this section
take two positive integers M, N and let h = (b − a)/M and τ = T/N . Define xi = ih(0 ≤ i ≤ M),
tn = nτ(0 ≤ n ≤ N), Ωh = xi|0 ≤ i ≤M and Ωτ = tn|0 ≤ n ≤ N. The computational domain
[a, b]× [0, T ] is then covered by Ωτh = Ωh × Ωτ . Moreover, let uni = u(xi, tn).
Equations (2.4.1)-(2.4.3) give the first form of time fractional diffusion equations, where the time
fractional derivative is of the Riemann-Liouville type. This form of the time fractional diffusion equation
is used in the schemes within sections 2.4.2 and 2.4.3.
∂u
∂t= RL
0D1−γt
[Kγ
∂2u
∂x2
]+ f(x, t), x ∈ [a, b], t ∈ [0, T ] (2.4.1)
u(0, t) = φ(t), u(L, t) = Φ(t), t ∈ [0, T ] (2.4.2)
u(x, 0) = 0, x ∈ [a, b] (2.4.3)
where Kγ is the diffusion coefficient and RL0D
1−γt is the Riemann-Liouville derivative of order (1 − γ) of
function u as defined in section 2.1.
Alternatively, the Caputo fractional derivative can be used, in which case the initial-boundary value
problem of the form (2.4.4)-(2.4.6). Time fractional derivatives of the Caputo type will be used in the
scheme of section 2.4.4.
C0D
γt u(x, t) = Kγ
∂2u(x, t)
∂x2+ F (x, t), x ∈ [0, L], t ∈ [0, T ] (2.4.4)
u(0, t) = φ(t), u(L, t) = Φ(t), t ∈ [0, T ] (2.4.5)
u(x, 0) = 0, x ∈ [a, b] (2.4.6)
once again, Kγ is the diffusion coefficient and C0 D
γt is the Caputo derivative with order γ of the function
as defined in section 2.1.
The conversion between these two forms can be made in a straightforward manner when u(x, t =
0) = 0. In this case, if a Riemann-Liouville integration of order (1− γ) is made on both sides of equation
(2.4.1), the Caputo derivative naturally comes up on the left side because the Caputo derivative of order
γ is equivalent to the fractional integral of order (1 − γ) of u(1)(t), with (0 < γ < 1) (see section 2.1).
In this situation, the Riemann-Liouville fractional derivative on the left hand side of (2.4.1) vanishes and
19
the source term in equation (2.4.4) is given by the Riemann-Liouville fractional integral of order (1 − γ)
of f(x, t), denoted F (x, t).
2.4.2 First order finite difference scheme
The first scheme here shown for the time fractional diffusion equation was developed by Yuste in
[40], to which a source term was added. This scheme can be thought of as an extension of the weighted
average schemes for integer order differential equations.
Let us consider, equation (2.4.1) at the off-lattice point (xi, tn+ 12)
∂
∂tun+1/2i = Kγ
RL0D
(1−γ)t
(∂2
∂x2un+1/2i
)+ f
n+1/2i = 0 (2.4.7)
The integer order time and space derivatives in this equation are now replaced by the three-point centred
operator (2.3.1), for the first order time derivative and a weighted average of the three-point centred
finite difference operator in (2.3.3), evaluated at times tn and tn+1. Furthermore, the Riemman-Liouville
derivative is substituted by the Grunwald-Letnikov difference operator defined in (2.2.2).
δtun+1/2i −
[θKγδ
1−γt δ2xu
ni + (1− θ)Kγδ
1−γt δ2xu
n+1i + θfni + (1− θ)fn+1
i
]= T
n+1/2j (2.4.8)
Neglecting the truncation error and expanding the difference operators using equations (2.3.1),
(2.3.3) and (2.2.2) a computable finite difference scheme is achieved
− Sun+1j−1 + (1 + 2S)un+1
j − Sun+1j = R, 1 ≤ i ≤M − 1, 0 ≤ n ≤ N − 1 (2.4.9)
U0i = 0, 1 ≤ i ≤M − 1 (2.4.10)
Un0 = φ(tn), unM = Φ(tn), 0 ≤ n ≤ N (2.4.11)
where
S = (1− θ)ω(1−γ)0 S, S = Kγ
(τ)γ
(h)2(2.4.12)
and
R = unj + S
n∑k=0
[(1− θ)ω(1−γ)
k+1 + θω(1−γ)k
] [un−kj−1 − 2un−kj + un−kj+1
]+ τγ
[θfni + (1− θfn+1
i
]. (2.4.13)
Though the scheme is in general implicit, some particular cases are to be pointed out. If θ = 1 the
scheme is fully explicit while for θ = 0 the scheme is fully explicit. For θ = 1/2, a Crank-Nicholson type
scheme is achieved.
In [40], Yuste concluded that the truncation error Tn+1/2j in equation 2.4.8 is O(h2 + τ q), with q = 1 if
θ 6= 12 and q = 2 if θ = 1
2 and a second order discretization scheme is used for the fractional derivative.
This means that if the scheme is used, as given in [40] there is no significant improvement between the
20
semi-implicit (θ = 1/2) and fully implicit (θ = 1).
On the same paper, it was proven through a von Neumann stability analysis that the stability of
the method is strongly dependent on the chosen θ, being unconditionally stable for 0 ≤ θ ≤ 12 and
conditionally stable for 12 ≤ θ ≤ 1. This criterion is summarized in equation (2.4.14).
1
S≥ 1
S×≡ 2(2θ − 1)W (−1, 1− γ) (2.4.14)
where W (z, γ) is the generating function of the coefficients, in this caseW (z, γ) = (1− z)γ .
2.4.3 Second order implicit finite difference scheme
In [47] Hu and Zhang develop, through an integration method, a second order difference scheme for
the time fractional diffusion equation. In the derivation of the scheme they make use of the following
lemma.
Lemma 2.4.1. [92] If u(x, t) is sufficiently smooth, then we have
u(xi, t)−(tn+1 − t)u(xi, tn)− (t− tn)u(xi, tn+1)
τ=
1
2
∂2u(xi, tn)
∂t2(tn − t)(tn+1 − t) ≤ C1τ
2,
C1 = max
∣∣∣∣12 ∂2u(x, t)
∂t2
∣∣∣∣ (2.4.15)
Integrating equation 2.4.1, one gets
u(xi, tn+1)− u(xi, tn) =Kγ
Γ(γ)
∫ tn+1
0
uxx(xi, ξ)
(tn+1 − ξ)1−γdξ − Kγ
Γ(γ)
∫ tn
0
uxx(xi, ξ)
(tn − ξ)1−γdξ +
∫ tn+1
tn
f(xi, ξ)dξ
= I1 − I2 + I3
(2.4.16)
Applying Lemma 2.4.1 and the central difference formula for the second order space derivative,
I1,I2,I3 are discretized
I1 =Kγ
Γ(γ)
n∑k=0
∫ tk+1
tk
[(tk+1 − ξ)uxx(xi, tk) + (ξ − tk)uxx(xi, tk+1)
τ
]1
(tn+1 − ξ)1−γdξ +Rn+1
1i
= r
n∑k=0
[ω(γ)k δ2xu(xi, tn−k) + υ
(γ)k δ2xu(xi, tn−k+1)
]+Rn+1
1i +Rn+12i
(2.4.17)
I2 = r
n−1∑k=0
[ω(γ)k δ2xu(xi, tn−k−1) + υ
(γ)k δ2xu(xi, tn−k)
]+Rn1i +Rn2i (2.4.18)
I3 =τ
2[f(xi, tn) + f(xi, tn+1] +Rn+1
3i (2.4.19)
where
r =Kγτ
γ
Γ(1 + γ)(2.4.20)
21
ω(γ)i =
1
1 + γ
[(i+ 1)1+γ − i1+γ
](2.4.21)
υ(γ)i =
1
1 + γ
[(i+ 1)1+γ − i1+γ
]− iγ (2.4.22)
and
Rn1i =Kγ
Γ(γ)
n−1∑k=0
∫ tk+1
tk
[uxx(xi, ξ)−
(tk+1 − ξ)uxx(xi, tk) + (ξ − tk)Lu(xi, tk+1)
τ
]1
(tn − ξ)1−γdξ
(2.4.23)
Rn2i = r
n−1∑k=0
[ω(γ)k (uxx(xi, tn−k−1)− δ2xu(xi, tn−k−1)) + υ
(γ)k (uxx(xi, tn−k)− δ2xu(xi, tn−k))
](2.4.24)
After some handling of the equation, defining ω(γ)−1 = 0 and replacing uni by its numerical approxima-
tion Uni , it is possible to obtain
Un+1i − rυ(γ)0 δ2xU
n+1i = Uni + r
n−1∑k=0
[(ω(γ)k − ω(γ)
k−1 + υ(γ)k+1 − υ
(γ)k
)δ2xU
n−ki
]+ r
(ωγ)n − ω
γ)n−1
)δ2xu
0i + τf
n+ 12
i , 1 ≤ i ≤M − 1, 1 ≤ n ≤ N
(2.4.25)
Un0 = φ(tn), unM = ψ(tn), 1 ≤ n ≤ N (2.4.26)
U0i = 0, 0 ≤ i ≤M (2.4.27)
where the omitted truncation error Rn+1i is equal to
Rn+1i = Rn+1
1i −Rn1i +Rn+12i −Rn2i +Rn+1
3i (2.4.28)
Hu and Zhang have also estimated that |Rn+1i | ≤ Cr(τ2 + h2)(ω
(γ)n + υ(γ)). In the same paper, the
stability of the scheme was proven and summarized in the following theorem.
Theorem 2.4.1. When 0 < γ ≤ log2 3−1 the scheme is stable to the initial data and the inhomogeneous
term in the L∞ norm, defined as
||u||∞ = max1≤i≤M−1
|ui| (2.4.29)
The convergence of the scheme was also proved and the following theorem holds
Theorem 2.4.2. Let u(x, t) ∈ C4,3x,t ([0, L] × [0, T ]) be the solution of the problem (2.4.1)-(2.4.3) and
Uni |0 ≤ i ≤M, 0 ≤ n ≤ N be the solution of the scheme (2.4.25)-(2.4.27). Denote eni = u(xi, tn)−Uni ,
0 ≤ i ≤M , 0 ≤ n ≤ N . Then for nτ ≤ T and 0 < γ ≤ log2 3− 1, there exists a positive constant C, such
that ||en||∞ ≤ C(τ2 + h2).
Though the stability and convergence of this scheme is only proven for γ ∈ [0, log2 3 − 1], Hu and
Zhang point out that numerical experiments show evidence of unconditional stability and convergence,
leaving this proof an open problem.
22
2.4.4 Third order compact finite difference scheme
The final scheme here presented for time fractional diffusion equations was developed by Ji and Sun
in [46]. This high-order compact difference scheme uses the third order accurate time weighted and
shifted Grunwald difference operator for time discretization defined in (2.2.15). For the spacial direction,
a compact technique is employed. Even though Ji and Sun derive an approximation for the Riemann-
Liouville derivatives, they then focus on the particular cases where there is equivalence between the
Riemann-Liouville and Caputo forms of the time fractional diffusion problem, developing a scheme for
the Caputo form of the time fractional initial-boundary value problem in equations (2.4.4)-(2.4.6). Before
proceeding further with the discretization there are two lemmas in [46] which are fundamental to the
development of the scheme that will now be stated.
Lemma 2.4.2. If u(0)=0, then it holds that 0D−γt (C0 D
γt u(t)) = u(t) for 0 < γ < 1.
Lemma 2.4.3. Define θ(s) = (1− s)3 = [5− 3(1− s)2]. if g(x) ∈ C6[a, b],h = (b− a)/M , xi = a+ ih(0 ≤
i ≤M) it holds that
1
12[g′′(xi−1) + 10g′′(xi) + g′′(xi+1)]
=g(xi−1 − 2g(xi) + g(xi+1)
h2+
h4
360
∫ 1
0
[g(6)(xi − sh) + g(6)(xi + sh)]θ(s)ds, 1 ≤ i ≤M − 1
(2.4.30)
In addition, let an average operator be defined as
Auni =
112 (uni−1 + 10uni + uni+1) = (I + h2
12 δ2x)uni , 1 ≤ i ≤M − 1
uni , i = 0 or M
(2.4.31)
Looking at the structure of the time weighted and shifted Grunwald difference operator in (2.2.15),
it can be seen that the discretization of the first time level for equation (2.4.4) needs to be considered
separately from the second to the Nth time levels. It will be further assumed that u(x, t) ∈ C6,5x,t ([a, b] ×
[0, T ]) and ∂ku(x,0)∂tk
= 0 for k = 0, 1, ..., 5, which allows for C0Dγt u(xi, tn) = RL
0Dγt u(xi, tn)).
The discretization for time levels with 2 ≤ n ≤ N will be first considered. At grid point (xi, tn) equation
2.4.4 givesC0D
γt u
ni = Kγ
∂2uni∂x2
+ fni , 0 ≤ i ≤M, 2 ≤ n ≤ N (2.4.32)
If the weighted and shifted Grunwald difference operator is chosen to approximate C0D
γt u(xi, tn) ,
followed by the application of the average operator A to both sides of the equation and using Lemma
2.4.3 will lead to
1
τγ
n∑k=0
g(γ)k Au
n−ki = Kγδ
2xu
ni +Afni +Rni , 1 ≤ i ≤M − 1, 2 ≤ n ≤ N (2.4.33)
where
|Rni | ≤ C1(τ3 + h4), 1 ≤ i ≤M − 1, 2 ≤ n ≤ N (2.4.34)
To obtain the discretization at the first time step, the Riemann-Liouville integral operator RL0D−γt is
23
applied on both sides of equation (2.4.4). Making use of lemma 2.4.2 one obtains
u(x, t1) =Kγ
Γ(γ)
∫ t1
0
uxx(x, ξ)
(t1 − ξ)1−γdξ + F (x, t1) (2.4.35)
using uxx(x, 0), uxxt(x, 0) and uxx(x, t1) to make an Hermite interpolation of uxx(x, ξ) on the interval
[0, t1], it follows that
P (x, ξ) = uxx(x, 0) + uxxt(x, 0)(ξ − 0) +uxx(x, t1)− uxx(x, 0)− τuxxt(x, 0)
τ2(ξ − 0)2 (2.4.36)
If u(x, 0) = 0 and ut(x, 0) = 0 one obtains
u(x, t1) ≈ u(x, t1) =Kγ
Γ(γ)
∫ t1
0
P (x, ξ)
(t1 − ξ)1−γdξ + F (x, t1) =
2Kγ
Γ(γ + 3)τγuxx(x, t1) + F (x, t1) (2.4.37)
where F (x, t) =RL0 D−γf(x, t)
Once again, applying the space average operator A and using Lemma 2.4.3 gives
1
τγAu1i =
2Kγ
Γ(γ + 3)δ2xu
1i +
1
τγAF (xi, t1) +R1
i , 1 ≤ i ≤M − 1 (2.4.38)
where
|R1i | ≤ C3(τ3 + h4), 1 ≤ i ≤M − 1 (2.4.39)
Finally, omitting the error terms Rni and replacing uni with the numerical approximation Uni the final
scheme is1
τγ
n∑k=0
g(γ)k AU
n−ki = Kγδ
2xU
ni +Afni , 1 ≤ i ≤M − 1, 2 ≤ n ≤ N (2.4.40)
1
τγAU1
i =2Kγ
Γ(γ + 3)δ2xU
1i +
1
τγAF (xi, t1), 1 ≤ i ≤M − 1 (2.4.41)
U0i = 0, 1 ≤ i ≤M − 1 (2.4.42)
Un0 = φ(tn), unM = Φ(tn), 0 ≤ n ≤ N (2.4.43)
Equations (2.4.40) and (2.4.41) are systems of linear diagonally dominant equations, having a unique
solution and easily solved. Having discretized the scheme and provided an error estimation Ji and Sun
proceed with the stability and convergence analysis of the scheme resulting in
Theorem 2.4.3. The difference scheme (2.4.40)-(2.4.43) is unconditionally stable to the right hand therm
and initial value for all γ ∈ [0, γ∗], with γ∗ = 0.9569347.
Theorem 2.4.4. Assume that u(x, t) ∈ C6,5x,t ([a, b]×[0, T ]) is the solution of problem (2.4.4) to (2.4.6), and
uni |0 ≤ i ≤M, 0 ≤ n ≤ N is the solution of the finite difference scheme (2.4.40) to (2.4.43). Suppose
∂ku(x, 0)
δtk= 0, k = 0, 1, ..., 5 (2.4.44)
24
Denote
eni = u(xi, tn)− uni , 0 ≤ i ≤M, 0 ≤ n ≤ N (2.4.45)
Then when Nτ ≤ T it holds that
τ
N∑m=1
‖em‖∞ ≤=b− a
22
√C4T (c21T + C2
3 )(τ3 + h4). (2.4.46)
2.4.5 Numerical examples
In this section numerical, the behaviour of the selected schemes for the time fractional diffusion equa-
tions is studied. A comparison of the schemes is made, confronting the solutions of the same problem
given by different schemes, in terms of convergence order, error and computational cost. Consider the
following time fractional diffusion equation of the form (2.4.1)-(2.4.3)
∂u
∂t= RL
0D1−γt
[K∂2u
∂x2
]+ f(x, t), x ∈ [0, L], t ∈ [0, T ] (2.4.47)
u(x, t = 0) = 0, x ∈ [0, L] (2.4.48)
u(x = 0, t) = 0, U(x = L, t) = t4−γ sin(1), t ∈ [0, t] (2.4.49)
where K = 1, L = 1, T = 1 and source term f(x, t) given by
f(x, t) = sin(x)
[(4− γ)t3−γ
Γ(5− γ)t3
6
](2.4.50)
The exact solution of the problem is u(x, t) = t4−γ sin(x), depicted on Figure 2.1 for γ = 0.5. Since
u(x, t = 0) = 0, equation (2.4.47) can easily be converted to the form of equation (2.4.4) form by means
of the procedure described in section 2.4.1.
Figure 2.2, shows the absolute errors for γ = 0.5, for each of the schemes. As expected the highest
errors are present on the first order scheme followed by the second order (thousands of times smaller)
and the compact third order scheme which shows a maximum error in the order of 10−10.
Table 2.4.1 lists the results of the time convergence analysis of the three schemes that were anal-
ysed. For each scheme three different fractional orders (γ = 0.2, γ = 0.5 and γ = 0.8) were considered
and the L∞h,τ error was registered with the refinement of the time interval, allowing the computation of
the convergence order. The L∞h,τ error is defined as follows
L∞h,τ = max |Uni − uni | , 0 ≤ i ≤M, 0 ≤ n ≤ N (2.4.51)
Where uni is the exact and Uni the numerical solution of problem (2.4.47)-(2.4.49), with the mesh
step-sizes h and τ at the grid point (xi, tn). If h τa/b the order of convergence EOC in time of an error
E(h, τ) may be calculated by
25
Figure 2.1: Exact solution of problem (2.4.47)-(2.4.49) with γ = 0.5.
EOC = log τ1τ2
(E(h, τ1)
E(h, τ2)
)(2.4.52)
The time step was kept at value h = 1/2000, guaranteeing that in every test case the contribution
of the space truncation error to the solution is minimal when compared with the time contribution. The
first, second and third order schemes were shown, in the previous sections, to have errors O(τ + h2),
O(τ2 + h2) and O(τ3 + h4), respectively and any h ≤ τ will result for the three cases in a smaller
contribution of the space error.
The results for the first order scheme, listed in the table correspond to θ = 0, the fully implicit situation.
Other values of θ were tested namely θ = 1/2, but use of first order weights in the computation of the
time fractional derivative prevents that higher than first order convergence is achieved. Consequently the
results with other θ values display similar results. The first order scheme clearly shows the expected first
order of convergence,with the halving of the maximum error with the halving of the time step. Moreover,
a slight error decrease can be seen with the increase in γ, all within the same order of magnitude.
The second order scheme follows also the theoretical second order predictions for error convergence.
This scheme naturally allows a significant reduction of the maximum error when compared with the first
order scheme. A slight error decrease with the increase of γ is also observed in this case.
The third order scheme, even if it can be said that it behaves according to the theoretical results,
shows a need for smaller time steps to reach asymptotic convergence in the case of γ = 0.8. On the
other hand, the results for γ = 0.2 and γ = 0.5 show third order convergence even with the coarse
grids. Contrary to the two previous schemes, a slight increase in the error is observed with the increase
γ. Naturally, the maximum errors observed are orders of magnitude smaller than with the two previous
26
(a) Absolute error in the numerical solution of the problem (2.4.47)-(2.4.49) with the first order implicit scheme.
(b) Absolute error in the numerical solution of the problem (2.4.47)-(2.4.49) with the second order implicit scheme.
(c) Absolute error in the numerical solution of the problem (2.4.47)-(2.4.49) with the third order compact scheme.
Figure 2.2: Absolute errors for each numerical scheme for the solution of time fractional diffusion equa-tions. All the the plots shown correspond to τ = 1/2000, h = 1/512 and γ = 0.5.
27
schemes.
Table 2.4.2 lists the computing times of each of the solutions used to produce the results of Table
2.4.1 with (γ = 0.5) and no significant differences were observed for other γ values. The results on Table
2.4.1 show that the time of computation is decreasing with the order of the scheme partly because the
computation of time fractional derivatives involves all previous time steps, regardless of the order of the
scheme the number of time steps involved will be the same. It was seen that this decrease is mainly due
to the computing times of the right side matrix, when building the scheme. The matrix associated with
the implicit time step showed similar spectral radius for every scheme and the times for the solution of
the system of equations at each time step also revealed to be identical. The first order scheme requires
the computation of the weighted average of the time fractional derivative of the space derivative in two
time steps, while the second order scheme evaluates the fractional derivative at only one time step, thus
explaining the observed drop in computing time. The third order scheme also reveals a considerable
decrease of computing times with respect to the second order schemes that may be related with the
solution of the fractional diffusion equation in the Caputo form.
These results indicate that high order schemes for the solution of the time fractional diffusion equation
are the best choice and the ability of reaching the same order of magnitude with less time steps is an
enormous advantage, with significant reduction of computing times.
Table 2.4.1: L∞ error and its order of convergence with decrease of the temporal step size, for thepresented schemes for the time fractional diffusion equation. The results were taken with h = 1/2000.
γ = 0.2 γ = 0.5 γ = 0.8
1/τ L∞h,τ EOC L∞h,τ EOC L∞h,τ EOC
1stOrder 8 2.696E-02 - 2.438E-02 - 2.162E-02 -16 1.327E-02 1.02 1.223E-02 1.00 1.100E-02 0.9732 6.581E-03 1.01 6.123E-03 1.00 5.550E-03 0.9964 3.277E-03 1.01 3.064E-03 1.00 2.787E-03 0.99128 1.635E-03 1.00 1.533E-03 1.00 1.397E-03 1.00256 8.168E-04 1.00 7.664E-04 1.00 6.991E-04 1.00512 4.082E-04 1.00 3.832E-04 1.00 3.497E-04 1.00
2ndOrder 8 1.223E-03 - 8.840E-04 - 8.613E-04 -16 2.939E-04 2.06 2.156E-04 2.04 2.101E-04 2.0432 7.097E-05 2.05 5.301E-05 2.02 5.164E-05 2.0264 1.721E-05 2.04 1.310E-05 2.02 1.276E-05 2.02128 4.187E-06 2.04 3.248E-06 2.01 3.164E-06 2.01256 1.022E-06 2.03 8.079E-07 2.01 7.871E-07 2.01512 2.510E-07 2.03 2.019E-07 2.00 1.967E-07 2.00
3rdOrder 8 5.792E-05 - 1.164E-04 - 2.053E-04 -16 7.518E-06 2.95 1.461E-05 2.99 3.357E-05 2.6132 9.568E-07 2.97 1.829E-06 3.00 4.982E-06 2.7564 1.207E-07 2.99 2.288E-07 3.00 7.362E-07 2.76128 1.517E-08 2.99 2.863E-08 3.00 1.007E-07 2.87256 1.926E-09 2.98 3.608E-09 2.99 1.337E-08 2.91512 2.691E-10 2.84 4.915E-10 2.88 1.772E-09 2.92
28
Table 2.4.2: Time of computation for each scheme for each of the presented schemes. The resultscorrespond to a constant space step h = 1/2000.
Time Of Computation (s)
1/τ 1stOrder 2ndOrder 3rdOrder
8 0.095 0.090 0.33216 0.105 0.093 0.36332 0.149 0.116 0.36764 0.288 0.225 0.400
128 0.949 0.672 0.636256 3.390 2.331 1.287512 14.125 9.124 4.209
2.5 Space fractional diffusion equations
2.5.1 Problem statement
On this section focus will be given to space fractional diffusion equations. The space fractional initial-
boundary value problem is considered as follows
∂u(x, t)
∂t= K1
RLaD
αxu(x, t) +K2
RLxD
αb u(x, t) + f(x, t), (x, t) ∈ [a, b]× [0, T ] (2.5.1)
u(x, 0) = ξ(x), x ∈ [a, b] (2.5.2)
u(a, t) = 0, u(b, t) = 0, t ∈ [0, T ] (2.5.3)
where RLaD
αxu and RL
xDαb u are the left and right Riemann-Liouville derivatives defined in equations
(2.1.3) and (2.1.4) with 1 < α < 2. The diffusion coefficients K1 and K2 are nonnegative constants with
K21 +K2
2 6= 0. If K1 6= 0 then ψ(t) = 0 and if K2 6= 0 then φ(t) = 0.
Throughout the section let xi = ih(0 ≤ i ≤ M), tn = nτ(0 ≤ n ≤ N), Ωh = xi|0 ≤ i ≤M and
Ωτ = tn|0 ≤ n ≤ N. The computational domain [a, b] × [0, T ] is then covered by Ωτh = Ωh × Ωτ .
Moreover, consider uni = u(xi, tn).
2.5.2 First order finite difference scheme
The first scheme considered for space fractional diffusion equations, first order in time and space,
was introduced by Meerschaert and Tadjeran [49]. Here the left and right Riemann-Liouville fractional
derivatives are approximated with the shifted Grunwald difference operators defined in (2.2.25) and
(2.2.26), respectively.
Equation (2.5.1) at gridpoint (xi, tn+1) gives
∂tun+1i = K1
RLa Dα
xun+1i +K2
RLx Dα
b un+1i + fn+1
i (2.5.4)
If first order backward differences are used for time and the the fractional derivatives are approxi-
29
mated with operators (2.2.25) and (2.2.26) with p=1, the optimal value, it is obtained that
un+1i − uni
τ=
1
hα
[K1
i+1∑k=0
ω(α)k un+1
i−k+1 +K2
N−i+1∑k=0
ω(α)k un+1
i+k−1
]+ fn+1
i +O(h+ τ) (2.5.5)
Solving for u at time level tn + 1, omitting the truncation error and denoting by Uni the numerical
approximation of uni yields,
Un+1i − τ
hα
[K1
i+1∑k=0
ω(α)k Un+1
i−k+1 +K2
N−i+1∑k=0
ω(α)k Un+1
i+k−1
]= Uni + τfn+1
i (2.5.6)
Meerschaert and Tadjeran proved in [49], that this implicit method is unconditionally stable for 1 ≤
α ≤ 2. As an addition, it is noted that if this method was used to solve the equation explicitly, the method
would be stable if τ/hα ≤ 1/[α(K1 +K2)]
2.5.3 Second order finite difference scheme
This section describes the construction of the scheme developed by Tian et al. in [62]. The scheme
is second order accurate in time and space, using the shifted Grunwald difference operators developed
in section 2.2.2.2. For this scheme it will be assumed that u ∈ L1(R) and u ∈ C2+α(R).
If the Crank-Nicholson technique is used for time discretization, it is obtained
δun+1/2i − 1
2
(K1(RLaD
αxu)ni +K1(RLaD
αxu)n+1
i +K2(RLxDαb u)ni +K2(RLxD
αb u)n+1
i
)= f
n+1/2i +O(τ2)
(2.5.7)
For space, the approximation of the left and right Riemann-Liouville derivatives with the weighted
and shifted operators WS2δα
x,+ and WS2δα
x,− with (p, q) = (1, 0)or(1,−1) defined in section 2.2.2.2 leads
to
δun+1/2i − 1
2
(K1
WS2δα
x,+uni +K1
WS2δα
x,+un+1i +K2
WS2δα
x,−uni +K2
WS2δα
x,−un+1i
)= f
n+1/2i +O(τ2 + h2)
(2.5.8)
Substitution of the fractional difference operator with equations (2.2.34) and (2.2.35), separation of
time layers, omission of the truncation error and denoting by Uni the numerical approximation of uni gives
Un+1i − K1τ
2hα
i+1∑k=0
g(α)k Un+1
i−k+1 −K2τ
2hα
N−i+1∑k=0
g(α)k Un+1
i+k−1
= Uni +K1τ
2hα
i+1∑k=0
g(α)k Uni−k+1 +
K2τ
2hα
N−i+1∑k=0
g(α)k Uni+k−1 +
τ
2(fn+1i + fni )
(2.5.9)
Following the construction of the finite difference scheme Tian et al. proved that the scheme in
30
unconditionally stable and convergent with order O(h2 + τ2).
2.5.4 Fourth order quasi-compact finite difference scheme
The construction of the high order scheme made by Hao et al. in [55] is summarized in this section.
This scheme employs the fourth-order approximations for left and right Riemann-Liouville derivatives
introduced in section 2.2.2.3. For this scheme it will be assumed that u ∈ L1(R) and u ∈ C4+α(R).
For convenience,let
Dαx = (K1
RLaD
αx +K2
RLxD
αb ), δαx = (K1
WS4δα
x,+ +K2WS4δ
α
x,−) (2.5.10)
where the weighted difference operators WS4δα
x,+ and WS4δα
x,− are defined by (2.2.47) and (2.2.48),
respectively. Considering equation (2.5.1) at point (xi, t) gives
ut(xi, t) = Dαxu(xi, t) + f(xi, t), 0 ≤ i ≤M (2.5.11)
Applying now the average difference operator in equation (2.2.41) to equation (2.5.11)
Aut(xi, t) = ADαxu(xi, t) +Af(xi, t), 1 ≤ i ≤M − 1 (2.5.12)
Equations (2.2.47) and (2.2.48) allow to write this equation as
Aut(xi, t) = δαxu(xi, t) +Af(xi, t) +O(h4), 1 ≤ i ≤M − 1 (2.5.13)
The above equation is now averaged at time levels t = tn and t = tn+1, yielding
Aδtun+1/2i − δαxu
n+1/2i = Afn+1/2
i +Rni , 1 ≤ i ≤M − 1, 0 ≤ n ≤ N − 1 (2.5.14)
where Rni is the truncation error with |Rni | ≤ c1(τ2 + h4)
Omitting the truncation error and replacing uni by its numerical approximation Uni the following finite
difference scheme is obtained
AδtUn+1/2i − δαxU
n+1/2i = Afn+1/2
i , 1 ≤ i ≤M − 1, 0 ≤ n ≤ N − 1 (2.5.15)
Un0 = φa(x0), unM = φb(xM ), 1 ≤ n ≤ N (2.5.16)
U0i = u0(xi), 0 ≤ i ≤M (2.5.17)
Hao, Sun and Cao proved in [55] that the difference scheme (2.5.15)-(2.5.17) is unconditionally stable
to the initial value u0 and right hand term f for all 1 < α ≤ 2. They further proved that the scheme is
convergent and the estimate
||un − Un|| ≤ 3c1√b− aT (τ2 + h4), 1 ≤ n ≤ N (2.5.18)
31
holds for all 1 < α ≤ 2 where uni is the exact solution of the problem (2.5.1)-(2.5.3) and Uni is the
corresponding solution of the scheme (2.5.15)-(2.5.17).
2.5.5 Numerical examples
The following space fractional initial-boundary value problem, of type (2.5.1)-(2.5.3) is considered.
∂u(x, t)
∂t= RL
0Dα
xu(x, t) + RLxD
α
1u(x, t) + f(x, t), (x, t) ∈ [0, 1]× [0, T ] (2.5.19)
u(x, t = 0) = xµ(1− x)µ, x ∈ [a, b] (2.5.20)
u(x = 0, t) = u(x = 1, t) = 0, t ∈ [0, T ] (2.5.21)
with source term
f(x, t) = e−t(u(x, 0)− RL0D
αxu(x, t)− RL
xDα1u(x, t)) (2.5.22)
and exact solution u(x, t) = e−txµ(1− x)µ.
Considering µ = 2, 3, 4 will allow some conclusion regarding the smoothness requirements for the
schemes to behave according to the theoretical predictions. In figure 2.3 the plots of the analytical
solutions for µ = 2, 3, 4 are shown. The comparison of these three plots clearly shows the increase of
the smoothness in the boundaries with increasing µ.
Figure 2.4 depicts the absolute error in the solution of the three schemes for µ = 4, with the same
grid (τ = 1/20000 and h = 1/128). Oscillations in the absolute errors can be seen in the space direction,
whose frequency increases with the order of the schemes. Moreover, it is seen that with the increase in
the order of the scheme there is a local maximum of the error approaching the boundaries.
If a numerical scheme has a truncation error O(τa + hb) then optimal step sizes will yield τa ≈ hb. If
τ hb/a, the contribution of time discretization of the overall error will be minimal when compared with
the space discretization contribution. Bearing this conclusion in mind, a fixed time step τ = 1/20000 will
be considered to analyse the space convergence.
In each of the three following Tables 2.5.1 , 2.5.2 and 2.5.3, a convergence analysis in the space
direction is made for the three schemes under different fractional orders, with each table corresponding
to a different value of µ. Tables 2.5.1, 2.5.2 and 2.5.3 correspond to µ = 2, µ = 3 and µ = 4.
Regarding the first order scheme, it can be seen that asymptotic convergence requires smaller space
intervals than the remaining schemes and that the theoretical first order prediction is achieved. As α
increases from α = 1.2 to α = 1.8 there is also a clear reduction in the maximum error which may reach
two orders of magnitude and the increased smoothness leads to smaller errors.
Focusing now on the second order scheme, the tables show that second order convergence is
achieved for every µ. The second order scheme has two possibilities, (p, q) = (1, 0) and (p, q) = (1,−1),
that lead to unconditional stability being the former case the one listed in the tables. Only a slightly better
convergence rate was observed than for the latter. In this case, asymptotic convergence reached with
courser space grids. As in the first order scheme, an increase of µ leads to smaller maximum errors and
32
(a) µ = 2
(b) µ = 3
(c) µ = 4
Figure 2.3: Analytical solutions of problem (2.5.19)-(2.5.21) for µ = 2, 3, 4.
33
(a) Absolute error in the numerical computation with the first order scheme.
(b) Absolute error in the numerical computation with the second order scheme.
(c) Absolute error in the numerical computation with the fourth order scheme.
Figure 2.4: Absolute error in the numerical solution of the problem (2.5.19)-(2.5.21) with µ = 4 for thethree schemes presented. The errors correspond to h = 1/128 and τ = 1/20000.
34
no significant changes in the maximum error with the variation of the fractional order α.
Looking now at the fourth order scheme, the first thing that meets the eye is the change in behaviour
with µ. It is seen that with µ = 2 second order of convergence is achieved, third order with µ = 3 and
fourth order with µ = 4. This undoubtedly proves that the importance of the smoothness criteria in the
numerical solution, particularly for high-order schemes. Looking for instance at Table 2.5.3, with µ = 4, it
can be proved that the integer order space derivatives of u(x, t) are continuous up to the third derivative
at the boundaries (x ∈ [0, 1]), therefore the function u ∈ C3 in space. Numerical results then suggest
that the continuity requirements may be eased past the C4+α(R) that was assumed in the theoretical
analysis. As the value of µ decreases, so does the convergence rate of the fourth order scheme. It may
then be said that the fourth order scheme is convergent with order µ for µ ≤ 4. This means that for the
fourth order scheme the maximum error greatly increases with the decrease of µ ≤ 4 without significant
variations across different values of α.
Table 2.5.1: L∞h,τ errors and their order of convergence with the refinement of the space step for equation(2.5.19)-(2.5.21) with µ = 2
α = 1.2 α = 1.5 α = 1.8
1/h L∞h,τ EOC L∞h,τ EOC L∞h,τ EOC
First Order Scheme 8 1.3921E-02 - 4.0300E-03 - 1.5458E-03 -16 8.4348E-03 0.723 2.6013E-03 0.632 1.8557E-04 3.05832 4.6923E-03 0.846 1.4432E-03 0.850 1.0165E-04 0.86864 2.4808E-03 0.919 7.5459E-04 0.936 8.8970E-05 0.192
128 1.2747E-03 0.961 3.8462E-04 0.972 5.3671E-05 0.729256 6.4533E-04 0.982 1.9388E-04 0.988 2.9041E-05 0.886512 3.2439E-04 0.992 9.7260E-05 0.995 1.5049E-05 0.948
1024 1.6254E-04 0.997 4.8693E-05 0.998 7.6515E-06 0.976
Second Order Scheme 8 1.8813E-03 - 2.9161E-03 - 3.3964E-03 -16 4.6022E-04 2.031 7.0247E-04 2.054 8.3020E-04 2.03232 1.5325E-04 1.586 1.6817E-04 2.062 2.0174E-04 2.04164 4.3826E-05 1.806 4.0196E-05 2.065 4.8889E-05 2.045
128 1.1775E-05 1.896 9.6206E-06 2.063 1.1836E-05 2.046
Fourth Order Scheme 8 2.0239E-04 - 4.5958E-04 - 8.3249E-04 -16 5.9058E-05 1.777 1.1855E-04 1.955 2.0970E-04 1.98932 1.6822E-05 1.812 3.1574E-05 1.909 5.4195E-05 1.95264 4.6011E-06 1.870 8.2943E-06 1.929 1.4035E-05 1.949
128 1.2216E-06 1.913 2.1446E-06 1.951 3.5974E-06 1.964
Table 2.5.4 lists the times of computation for the numerical solutions with three different values of µ
for α = 1.5 , 1/τ = 20000 and 1/h = 128. The fourth order scheme to gives slightly better results for the
less smooth µ function than the second order scheme. The homogeneity in computing times is however
to be expected, since the matrices used to compute the left and right space fractional derivatives are
full. In other words, since the value of the space fractional derivatives depends on all the space values
at a given instant, the higher order scheme has no significant increase in computational cost. If a large
space domain is to be analysed with small errors, the high order scheme becomes even more important
in fractional diffusion equations since it will allow the same error magnitude with a courser space grid.
The fourth order finite difference scheme, for the same absolute error magnitude presents a smaller
computing cost than the first and second order schemes because less points may be considered.
35
Table 2.5.2: L∞h,τ errors and their order of convergence with the refinement of the space step for equation(2.5.19)-(2.5.21) with µ = 3
α = 1.2 α = 1.5 α = 1.8
1/h L∞h,τ EOC L∞h,τ EOC L∞h,τ EOC
First Order Scheme 8 3.6888E-03 - 1.1387E-03 - 2.6918E-04 -16 2.2923E-03 0.686 7.1007E-04 0.681 5.3609E-05 2.32832 1.3136E-03 0.803 3.9075E-04 0.862 3.9861E-05 0.42864 7.1174E-04 0.884 2.0443E-04 0.935 2.6149E-05 0.608
128 3.7183E-04 0.937 1.0447E-04 0.969 1.4615E-05 0.839256 1.9020E-04 0.967 5.2786E-05 0.985 7.6915E-06 0.926512 9.6201E-05 0.983 2.6528E-05 0.993 3.9415E-06 0.965
1024 4.8377E-05 0.992 1.3297E-05 0.996 1.9946E-06 0.983
Second Order Scheme 8 3.1605E-04 - 3.3891E-04 - 2.8112E-04 -16 8.4564E-05 1.902 8.7296E-05 1.957 7.0401E-05 1.99832 2.2056E-05 1.939 2.2411E-05 1.962 1.7812E-05 1.98364 5.6565E-06 1.963 5.7032E-06 1.974 4.4948E-06 1.987
128 1.4344E-06 1.979 1.4405E-06 1.985 1.1300E-06 1.992
Fourth Order Scheme 8 7.7491E-04 - 4.9583E-04 - 6.1105E-04 -16 1.0862E-04 2.835 6.0868E-05 3.026 1.0479E-04 2.54432 1.3956E-05 2.960 5.4423E-06 3.483 1.3285E-05 2.98064 1.8088E-06 2.948 8.2337E-07 2.725 1.7370E-06 2.935
128 2.3531E-07 2.942 1.1825E-07 2.800 2.0022E-07 3.117
Table 2.5.3: L∞h,τ errors and their order of convergence with the refinement of the space step for equation(2.5.19)-(2.5.21) with µ = 4
α = 1.2 α = 1.5 α = 1.8
1/h L∞h,τ EOC L∞h,τ EOC L∞h,τ EOC
First Order Scheme 8 9.6878E-04 - 2.9516E-04 - 8.2201E-05 -16 6.2081E-04 0.642 1.8834E-04 0.648 1.6002E-05 2.36132 3.6606E-04 0.762 1.0585E-04 0.831 9.3134E-06 0.78164 2.0230E-04 0.856 5.6071E-05 0.917 6.6894E-06 0.477
128 1.0696E-04 0.919 2.8850E-05 0.959 3.8588E-06 0.794256 5.5078E-05 0.957 1.4629E-05 0.980 2.0570E-06 0.908512 2.7953E-05 0.978 7.3626E-06 0.991 1.0589E-06 0.958
1024 1.4076E-05 0.990 3.6905E-06 0.996 5.3542E-07 0.984
Second Order Scheme 8 2.1098E-04 - 2.4372E-04 - 2.0179E-04 -16 5.4244E-05 1.960 6.1119E-05 1.996 5.0377E-05 2.00232 1.3887E-05 1.966 1.5467E-05 1.982 1.2660E-05 1.99364 3.5249E-06 1.978 3.9019E-06 1.987 3.1748E-06 1.995
128 8.8876E-07 1.988 9.8070E-07 1.992 7.9501E-07 1.998
Fourth Order Scheme 8 9.4148E-06 - 1.8968E-05 - 2.6585E-05 -16 8.3720E-07 3.491 1.5792E-06 3.586 2.0566E-06 3.69232 5.9107E-08 3.824 1.0861E-07 3.862 1.3693E-07 3.90964 3.8954E-09 3.924 6.9461E-09 3.967 8.6772E-09 3.980
128 2.8474E-10 3.774 4.3124E-10 4.010 5.3745E-10 4.013
Table 2.5.4: Time of computation in the solution of problem (2.5.19)-(2.5.21) for the three schemespresented with 1/τ = 20000 and 1/h = 128 for α = 1.5.
1st Order Scheme 2nd Order Scheme 4th Order Scheme
TOC (s) L∞h,τ TOC (s) L∞h,τ TOC (s) L∞h,τ
µ = 2 3.49 3.8462E-04 3.58 9.6206E-06 4.42 2.1446E-06µ = 3 3.47 1.0447E-04 3.27 1.4405E-06 4.23 1.1825E-07µ = 4 3.61 2.8850E-05 3.25 9.8070E-07 4.32 4.3124E-10
36
2.6 Time-space fractional diffusion equations
2.6.1 Problem statement
On this section focus will be given to space fractional diffusion equations. The space fractional initial-
boundary value problem is considered as follows
C0D
γ
t u(x, t) = K1RLaD
α
xu(x, t) +K2RLxD
α
b u(x, t) + f(x, t), (x, t) ∈ [a, b]× [0, T ] (2.6.1)
u(x, 0) = ξ(x), x ∈ [a, b] (2.6.2)
u(a, t) = 0, u(b, t) = 0, t ∈ [0, T ] (2.6.3)
where RLa Dα
xu and RLx Dα
b u are the left and right Riemann-Liouville derivatives defined in equations
(2.1.3) and (2.1.4) with 1 < α < 2. The time fractional derivative C0D
γ
t u(x, t), is considered to be of the
Caputo type as defined in (2.1.5) with 0 ≤ γ ≤ 1. The diffusion coefficients K1 and K2 are nonnegative
constants with K21 +K2
2 6= 0. If K1 6= 0 then ψ(t) = 0 and if K2 6= 0 then φ(t) = 0.
Throughout the section let xi = ih(0 ≤ i ≤ M), tn = nτ(0 ≤ n ≤ N), Ωh = xi|0 ≤ i ≤M and
Ωτ = tn|0 ≤ n ≤ N. The computational domain [a, b] × [0, T ] is then covered by Ωτh = Ωh × Ωτ .
Moreover, consider uni = u(xi, tn).
An extensive research effort revealed that the L1 method stands as the most common approximation
for time fractional derivatives in time-space fractional diffusion equations. The search for high-order finite
difference schemes both in time and space, with stability and convergence analysis failed to produce any
results. Bearing these facts in mind and serving as a needed common ground for the comparison of the
schemes all the finite difference schemes now introduced will use the L1 approximation for the Caputo
time fractional derivative found in section 2.2.1.2. For the discretization of space fractional derivatives
the three approximations that were previously given for Riemann-Liouville space fractional derivatives
will be used. Consequently, three schemes will be available with first, second and fourth order in space
and (2 − γ)th order in time and since they will all use the same approximation for the time derivative,
they will be named after their space convergence order.
2.6.2 First order finite difference scheme
The first finite difference scheme for time-space fractional diffusion equations that will be studied was
developed by Liu et al. [57], to which the right time fractional derivative and the source term were added.
This scheme will make use of the L1 method for the discretization of the Caputo time derivative and
the Riemann-Liouville derivatives will be approximated through the shifted Grunwald operators, already
used in the first order scheme for space fractional diffusion equations. Consider equation (2.6.1), at
(xi, tn) it becomes
C0D
γ
t uni = K1
RLaD
α
xuni +K2
RLxD
α
b uni + fni , (x, t) ∈ [a, b]× [0, T ] (2.6.4)
37
Approximating the Caputo time derivative with the L1 operator as in equation (2.2.7) and the Riemann-
Liouville space derivatives with the shifted Grunwald operators defined in (2.2.23) and (2.2.24), with
p = 1, leads to
L1Cδ
γ
t uni = K11δ
αx,+u
ni +K21δ
αx,−u
ni + fni (2.6.5)
Substitution of the fractional difference operators with equations (2.2.8), (2.2.25) and (2.2.26) yields
τ−γ
Γ(2− γ)
n−1∑k=0
bn−k−1(uk+1i − uki ) =
K1
hα
i+1∑k=0
ωαk uni−k+1 +
K2
hα
M−i+1∑k=0
ωαk uni+k−1 + fni (2.6.6)
Development of the left side of the equation will result in
τ−γ
Γ(2− γ)
[uni −
n−1∑k=1
(bn−k−1 − bn−k)uki − bn−1u0i
]
=K1
hα
i+1∑k=0
ωαk uni−k+1 +
K2
hα
M−i+1∑k=0
ωαk uni+k−1 + fni
(2.6.7)
Rearrangement and substitution of uni by its numerical approximation Uni gives
Uni − µ1
i+1∑k=0
ωαkUni−k+1 − µ2
M−i+1∑k=0
ωαkUni+k−1
=
n−1∑k=1
(bn−k−1 − bn−k)Uki + bn−1U0i + µff
ni , 1 ≤ i ≤M − 1, 1 ≤ n ≤ N − 1
(2.6.8)
Un0 = 0, unM = 0, 1 ≤ n ≤ N (2.6.9)
U0i = u0(xi), 0 ≤ i ≤M (2.6.10)
where µ1 =K1τ
γΓ(2− γ)
hα, µ2 =
K2τγΓ(2− γ)
hαand µf = τγΓ(2− γ).
After constructing the scheme, Liu et al. have made a stability analysis, concluding that it is uncon-
ditionally stable . They have also determined that the scheme is convergent, existing a positive constant
C such that |uni − Uni | ≤ C(τ + h).
2.6.3 Second order finite difference
A second order in space scheme for time-space fractional diffusion equations will now be developed.
Similarly to the first order scheme, this scheme will use the L1 method for the discretization of the Caputo
time derivative and the Riemann-Liouville derivatives will be approximated through the weighted shifted
Grunwald operators defined in section 2.2.2.2, used in the second order scheme for space fractional
diffusion equations. Considering equation (2.6.1) at (xi, tn) gives
C0D
γ
t uni = K1
RLaD
α
xuni +K2
RLxD
α
b uni + fni (2.6.11)
Approximating the Caputo time derivative with the L1 operator as in equation (2.2.7) and the Riemann-
38
Liouville space derivatives with the wheighted and shifted Grunwald operators defined in (2.2.34) and
(2.2.35), with (p, q) = (1, 0), leads to
L1Cδ
γ
t uni = K1
WS2δα
x,+uni +K2
WS2δα
x,−uni + fni , (x, t) ∈ [a, b]× [0, T ] (2.6.12)
Substitution of the fractional difference operators with equations (2.2.8), (2.2.25) and (2.2.26) yields
τ−γ
Γ(2− γ)
n−1∑k=0
bn−k−1(uk+1i − uki ) =
K1
hα
i+1∑k=0
gαk uni−k+1 +
K2
hα
M−i+1∑k=0
gαk uni+k−1 + fni (2.6.13)
Development of the left side of the equation will result in
τ−γ
Γ(2− γ)
[uni −
n−1∑k=1
(bn−k−1 − bn−k)uki − bn−1u0i
]
=K1
hα
i+1∑k=0
gαk uni−k+1 +
K2
hα
M−i+1∑k=0
gαk uni+k−1 + fni
(2.6.14)
Rearrangement and substitution of uni by its numerical approximation Uni gives
Uni − µ1
i+1∑k=0
gαkUni−k+1 − µ2
M−i+1∑k=0
gαkUni+k−1
=
n−1∑k=1
(bn−k−1 − bn−k)Uki + bn−1U0i + µff
ni , 1 ≤ i ≤M − 1, 1 ≤ n ≤ N − 1
(2.6.15)
Un0 = 0, unM = 0, 1 ≤ n ≤ N (2.6.16)
U0i = u0(xi), 0 ≤ i ≤M (2.6.17)
where µ1 =K1τ
γΓ(2− γ)
hα, µ2 =
K2τγΓ(2− γ)
hαand µf = τγΓ(2− γ). As it can be seen, equation
(2.6.15) is very similar to equation (2.6.8), with the only difference being the replacement of the first
order weights in space fractional derivatives by second order ones.
2.6.4 Fourth order finite difference scheme
In this section the difference scheme recently developed by Pang et al. [63] is visited. This scheme
combines the L1 method for the approximation of the temporal derivative with the compact difference
scheme developed by Hao et al. [55] for space fractional derivatives, seen on section 2.5.4. The tech-
nique to reach fourth-order approximations for left and right Riemann-Liouville derivatives was intro-
duced in section 2.2.2.3.
As in section 2.5.4, conveniently let
Dαx = (K1
RLaD
αx +K2
RLxD
αb ), δαx = (K1
WS4δα
x,+ +K2WS4δ
α
x,−) (2.6.18)
where the weighted difference operators WS4δα
x,+ and WS4δα
x,− are defined by (2.2.47) and (2.2.48),
respectively.
39
Considering equation (2.6.1) at point (xi, tn) gives
C0D
γ
t uni = Dα
xuni + fni , 0 ≤ i ≤M, 1 ≤ n ≤ N (2.6.19)
The average difference operator in (2.2.41) is now applied to equation (2.6.19) to give
A(C0Dγ
t uni ) = A(Dα
xuni ) +A(fni ), 1 ≤ i ≤M − 1, 1 ≤ n ≤ N (2.6.20)
Equations (2.2.47) and (2.2.48) allow to write this equation as
A(C0Dγ
t uni ) = δαxu
ni +A(fni ) +O(h4) (2.6.21)
Approximating the Caputo time derivative with the L1 operator as in equation (2.2.7) yields
A(L1Cδγ
t uni ) = δαxu
ni +A(fni ) +O(h4 + τ2−γ) (2.6.22)
Substitution of the L1 operator with (2.2.8) leads to
τ−γ
Γ(2− γ)A
[uni −
n−1∑k=1
(bn−k−1 − bn−k)uki − bn−1u0i
]= δαxu
ni +A(fni ) +O(h4 + τ2−γ) (2.6.23)
Omitting the truncation error, replacing uni by its numerical approximation Uni , and denoting µ =
τγΓ(2− γ) the following finite difference scheme is obtained
A
[Uni −
n−1∑k=1
(bn−k−1 − bn−k)Uki − bn−1U0i
]= µδαxU
ni + µA(fni ), 1 ≤ i ≤M − 1, 1 ≤ n ≤ N − 1
(2.6.24)
Un0 = 0, unM = 0, 1 ≤ n ≤ N (2.6.25)
U0i = u0(xi), 0 ≤ i ≤M (2.6.26)
Pang et al., after studying the stability and convergence of the scheme have concluded that for all
0 < γ < 1 and 1 < α < 2 the scheme is unconditionally stable that ||eni || = ||uni − uni || ≤ C(τ2−γ + h4).
2.6.5 Numerical Examples
The following time-space fractional initial-boundary value problem of type (2.6.1)-(2.6.3) is consid-
ered.
C0D
γt u(x, t) = RL
0Dαxu(x, t) + RL
xDα1u(x, t) + f(x, t), (x, t) ∈ [0, 1]× [0, T ] (2.6.27)
u(x, t = 0) = (x4(1− x)4), x ∈ [0, 1] (2.6.28)
40
u(x = 0, t) = u(x = 1, t) = 0, t ∈ [0, T ] (2.6.29)
with source term
f(x, t) =
(Γ(3 + γ)
Γ(3)t2 +
Γ(2)
Γ(2− γ)
)x4(1− x)4 −
(t2−γ + t+ 2
) [ Γ(9)
Γ(9− α)x8−α
− 4Γ(8)
Γ(8− α)x7−α +
6Γ(7)
Γ(7− α)x6−α − 4Γ(6)
Γ(6− α)x5−α +
Γ(5)
Γ(5− α)x4−α
]−(t2−γ + t+ 2
) [ Γ(9)
Γ(9− α)(1− x)8−α − 4Γ(8)
Γ(8− α)(1− x)7−α +
6Γ(7)
Γ(7− α)(1− x)6−α
− 4Γ(6)
Γ(6− α)(1− x)5−α +
Γ(5)
Γ(5− α)(1− x)4−α
](2.6.30)
The exact solution of the problem is given by u(x, t) = (t2+γ + t+ 2)x4(1− x)4 and it is depicted on
Figure 2.5 for the case γ = 0.5
Figure 2.5: Exact solution of problem (2.6.27)-(2.6.29) with γ = 0.5
To validate and compare the implemented schemes, the L∞h,τ error and its order of convergence,
already defined in (2.4.51) and (2.4.52), were calculated for each of the schemes with the refinement of
the space and time intervals.
Tables 2.6.1 and 2.6.2 list the L∞h,τ error along with the respective order of convergence with the
refinement of the space and time intervals, respectively, for the first order in space scheme. The results
listed on Table 2.6.1 were taken with a constant value τ = 1/8000, guaranteeing a smaller contribution
of the space truncation error when compared with the time contribution. It can be seen that in this case,
the error and its convergence order are insensitive to variations in γ. Although first order convergence
can be observed, the higher value of α seems to require smaller space steps to reach asymptotic
41
convergence than the lower values of α. In Table 2.6.2 the results of the time convergence analysis for
the first order in space scheme are listed. This analysis was made optimizing the space interval with
h ≈ τ2−γ because the attempt to verify first order convergence with a fixed h τ2−γ failed to provide a
satisfactory outcome. The results show that in this manner the (2−γ) order of accuracy is achieved, with
higher values of γ naturally showing larger errors. At the same time the error magnitude is decreasing
with the increase of α. This increase is due to the higher orders of convergence in coarser grids seen
with high α, before asymptotic convergence is achieved with time step refinement.
Tables 2.6.3 and 2.6.4 list the L∞h,τ error along with the respective order of convergence with the re-
finement of the space and time intervals, respectively, for the second order in space scheme. The results
of Table 2.6.3 were taken with a constant τ = 1/80000 and reveal a perfect second order convergence
with space step refinement, while remaining relatively in sensitive to variations in both α and γ.
Table 2.6.3 lists the maximum error and its order of convergence with time step refinement for h =
1/1000. It is seen that for γ = 0.5 and γ = 0.8, the orders of convergence are very close to (2 − γ). On
the other hand, it is seen that for γ = 0.2, the results seem to show a significantly lower convergence
order than what was expected. The error however is still decreasing with the increase of γ.
Tables 2.6.5 and 2.6.6 list the L∞h,τ error along with the respective order of convergence with the
refinement of the space and time intervals, respectively, for the fourth order in space scheme. In Table
2.6.5 the space convergence of the L∞h,τ scheme had to be carried out with t ∈ [0, 0.1] for reasons of
computational cost. Since the scheme is of order O(τ2−γ , h4) the computational cost of keeping the time
contribution to the error lower than the space contribution became too high to carry such an extensive
analysis. It can bee seen that fourth order convergence space is achieved, although a slight increase in
error is seen with increasing α. Furthermore, it is seen that the effect of γ in the error order of magnitude
is very small. Table 2.6.6, was built using a constant value of h = 1/1000 and the results exhibit (2− γ)
convergence order. A slight increase on convergence order is also seen with increasing α, that explains
the decreasing errors observed.
Table 2.6.7 lists the errors and computing times that were verified with the refinement of the time
interval for a fixed h = 1/2000. Here the error for the first order scheme shows to be insensitive to the
refinement of the space interval and order γ. Leading to smaller errors than the first order scheme, the
second and third order schemes exhibit errors of the same order of magnitude. Due to the similarity in
the construction and solution of the schemes the first and second order in time schemes are exhibiting
similar computing times. On the other hand, the fourth order scheme is exhibiting computing times more
than ten times larger than the second order scheme. This huge increase in the computing time for the
high order scheme is here due to the application of the averaging operator of the compact scheme to the
time fractional derivative, greatly increasing the computational cost. These conclusions indicate that the
most efficient choice for time-space fractional diffusion equation is the second order scheme. To unlock
the potential of a high order approximation in space, a high order approximation should also be used for
the time fractional derivative, enabling the use a coarser time grid to get the same error magnitude.
42
Table 2.6.1: L∞h,τ error and the respective order of convergence with space step refinement for the firstorder in space scheme with a constant τ = 1/8000
γ = 0.2 γ = 0.5 γ = 0.8
α 1/h L∞h,τ EOC L∞h,τ EOC L∞h,τ EOC
1.2 4 6.9409E-03 - 6.9470E-03 - 6.9741E-03 -8 5.1888E-03 0.420 5.1900E-03 0.421 5.2069E-03 0.42216 3.3637E-03 0.625 3.3625E-03 0.626 3.3699E-03 0.62832 1.9769E-03 0.767 1.9755E-03 0.767 1.9782E-03 0.76864 1.0860E-03 0.864 1.0851E-03 0.864 1.0861E-03 0.865128 5.7173E-04 0.926 5.7121E-04 0.926 5.7157E-04 0.926
1.5 4 1.4609E-03 - 1.4628E-03 - 1.4677E-03 -8 1.4461E-03 0.015 1.4476E-03 0.015 1.4510E-03 0.01712 9.2856E-04 0.639 9.2946E-04 0.639 9.3140E-04 0.64016 5.2280E-04 0.829 5.2329E-04 0.829 5.2433E-04 0.82964 2.7707E-04 0.916 2.7732E-04 0.916 2.7785E-04 0.916128 1.4258E-04 0.958 1.4271E-04 0.958 1.4297E-04 0.959
1.8 4 1.9696E-03 - 1.9704E-03 - 1.9713E-03 -8 3.5892E-04 2.456 3.5904E-04 2.456 3.5914E-04 2.45712 7.3973E-05 2.279 7.4015E-05 2.278 7.4061E-05 2.27816 4.3100E-05 0.779 4.3138E-05 0.779 4.3174E-05 0.77964 3.1069E-05 0.472 3.1091E-05 0.472 3.1104E-05 0.473128 1.7931E-05 0.793 1.7943E-05 0.793 1.7942E-05 0.794
Table 2.6.2: L∞h,τ error and the respective order of convergence with time step refinement for the firstorder in space scheme, h ≈ τ (2−γ).
α = 1.2 α = 1.5 α = 1.8
γ 1/τ L∞h,τ EOC L∞h,τ EOC L∞h,τ EOC
0.2 4 4.05E-03 - 1.13E-03 - 1.44E-04 -8 1.56E-03 1.374 4.06E-04 1.477 3.87E-05 1.89016 4.99E-04 1.646 1.24E-04 1.713 1.56E-05 1.31632 1.48E-04 1.751 3.62E-05 1.775 4.84E-06 1.68664 4.30E-05 1.786 1.05E-05 1.793 1.42E-06 1.770128 1.24E-05 1.796 3.01E-06 1.798 4.09E-07 1.793
0.5 4 5.12E-03 - 1.42E-03 - 3.53E-04 -8 2.58E-03 0.985 6.98E-04 1.020 3.31E-05 3.41716 1.07E-03 1.269 2.72E-04 1.358 2.86E-05 0.21132 4.05E-04 1.403 9.99E-05 1.447 1.23E-05 1.22064 1.47E-04 1.463 3.58E-05 1.481 4.60E-06 1.413128 5.24E-05 1.487 1.27E-05 1.494 1.66E-06 1.471
0.8 4 5.72E-03 - 1.40E-03 - 1.13E-03 -8 3.97E-03 0.529 1.09E-03 0.362 1.32E-04 3.09516 2.18E-03 0.861 5.77E-04 0.917 3.17E-05 2.05832 1.06E-03 1.044 2.67E-04 1.114 2.52E-05 0.33264 4.89E-04 1.116 1.20E-04 1.153 1.33E-05 0.922128 2.19E-04 1.160 5.30E-05 1.178 6.25E-06 1.091
43
Table 2.6.3: L∞h,τ error and the respective order of convergence with space step refinement for thesecond order in space scheme, h = 1/1000.
γ = 0.2 γ = 0.5 γ = 0.8
α 1/h L∞h,τ EOC L∞h,τ EOC L∞h,τ EOC
1.2 4 4.1593E-03 - 4.1602E-03 - 4.1767E-03 -8 1.0548E-03 1.979 1.0554E-03 1.979 1.0584E-03 1.98116 2.7021E-04 1.965 2.7036E-04 1.965 2.7105E-04 1.96532 6.9060E-05 1.968 6.9101E-05 1.968 6.9332E-05 1.96764 1.7509E-05 1.980 1.7522E-05 1.980 1.7645E-05 1.974128 4.4120E-06 1.989 4.4180E-06 1.988 4.5129E-06 1.967
1.5 4 4.7018E-03 - 4.7060E-03 - 4.7122E-03 -8 1.1158E-03 2.075 1.1166E-03 2.075 1.1178E-03 2.07612 2.8216E-04 1.984 2.8235E-04 1.984 2.8267E-04 1.98316 7.1518E-05 1.980 7.1565E-05 1.980 7.1673E-05 1.98064 1.8036E-05 1.987 1.8049E-05 1.987 1.8102E-05 1.985128 4.5310E-06 1.993 4.5352E-06 1.993 4.5742E-06 1.985
1.8 4 3.7413E-03 - 3.7426E-03 - 3.7437E-03 -8 8.8624E-04 2.078 8.8654E-04 2.078 8.8686E-04 2.07812 2.2559E-04 1.974 2.2568E-04 1.974 2.2579E-04 1.97416 5.6921E-05 1.987 5.6943E-05 1.987 5.6984E-05 1.98664 1.4285E-05 1.994 1.4292E-05 1.994 1.4315E-05 1.993128 3.5775E-06 1.998 3.5795E-06 1.997 3.5987E-06 1.992
Table 2.6.4: L∞h,τ error and the respective order of convergence with time step refinement for the secondorder in space scheme.
α = 1.2 α = 1.5 α = 1.8
γ 1/τ L∞h,τ EOC L∞h,τ EOC L∞h,τ EOC
0.2 4 1.6593E-05 - 6.2447E-06 - 3.1952E-06 -8 5.3092E-06 1.644 2.0004E-06 1.642 1.0261E-06 1.63916 1.6680E-06 1.670 6.3548E-07 1.654 3.2931E-07 1.64032 5.2425E-07 1.670 2.0763E-07 1.614 1.1103E-07 1.56864 1.7156E-07 1.612 7.5866E-08 1.452 4.3841E-08 1.341
0.5 4 9.2840E-05 - 3.5656E-05 - 1.8281E-05 -8 3.5368E-05 1.392 1.3511E-05 1.400 6.9145E-06 1.40312 1.3116E-05 1.431 4.9999E-06 1.434 2.5588E-06 1.43416 4.7930E-06 1.452 1.8305E-06 1.450 9.3936E-07 1.44664 1.7413E-06 1.461 6.7158E-07 1.447 3.4775E-07 1.434
0.8 4 3.1483E-04 - 1.2645E-04 - 6.5451E-05 -8 1.4469E-04 1.122 5.8155E-05 1.121 3.0021E-05 1.12412 6.4925E-05 1.156 2.6065E-05 1.158 1.3436E-05 1.16016 2.8749E-05 1.175 1.1533E-05 1.176 5.9425E-06 1.17764 1.2641E-05 1.185 5.0730E-06 1.185 2.6155E-06 1.184
44
Table 2.6.5: L∞h,τ error and the respective order of convergence with space step refinement for the fourthorder in space scheme, for t ∈ [0, 0.1] and τ = 1/100000.
γ = 0.2 γ = 0.5 γ = 0.8
α 1/h L∞h,τ EOC L∞h,τ EOC L∞h,τ EOC
1.2 4 1.6450E-04 - 1.1525E-04 - 4.4287E-05 -8 2.0260E-05 3.021 1.2400E-05 3.216 4.4206E-06 3.32516 1.8460E-06 3.456 1.2015E-06 3.367 4.8230E-07 3.19632 1.3077E-07 3.819 8.7423E-08 3.781 5.2136E-08 3.21064 8.6077E-09 3.925 6.9323E-09 3.657 6.0357E-09 3.111
1.5 4 3.6896E-04 - 2.9775E-04 - 2.0223E-04 -8 4.2480E-05 3.119 3.5738E-05 3.059 2.2893E-05 3.14312 3.5837E-06 3.567 3.0039E-06 3.573 1.9867E-06 3.52616 2.4483E-07 3.872 2.0874E-07 3.847 1.4237E-07 3.80364 1.5664E-08 3.966 1.3370E-08 3.965 9.2127E-09 3.950
1.8 4 5.6067E-04 - 4.9140E-04 - 3.6127E-04 -8 5.9297E-05 3.241 5.4091E-05 3.183 4.4163E-05 3.03212 4.6038E-06 3.687 4.1167E-06 3.716 3.2129E-06 3.78116 3.0638E-07 3.909 2.7752E-07 3.891 2.2345E-07 3.84664 1.9423E-08 3.979 1.7562E-08 3.982 1.4168E-08 3.979
Table 2.6.6: L∞h,τ error and the respective order of convergence with time step refinement for the fourthorder in space scheme, h = 1/1000.
α = 1.2 α = 1.5 α = 1.8
γ 1/τ L∞h,τ EOC L∞h,τ EOC L∞h,τ EOC
0.2 4 1.6575E-05 - 6.2261E-06 - 3.1805E-06 -8 5.2911E-06 1.647 1.9817E-06 1.652 1.0114E-06 1.65316 1.6498E-06 1.681 6.1683E-07 1.684 3.1463E-07 1.68532 5.0604E-07 1.705 1.8898E-07 1.707 9.6357E-08 1.70764 1.5335E-07 1.722 5.7218E-08 1.724 2.9166E-08 1.724
0.5 4 9.2823E-05 - 3.5637E-05 - 1.8266E-05 -8 3.5350E-05 1.393 1.3493E-05 1.401 6.8998E-06 1.40512 1.3098E-05 1.432 4.9812E-06 1.438 2.5441E-06 1.43916 4.7748E-06 1.456 1.8118E-06 1.459 9.2469E-07 1.46064 1.7231E-06 1.470 6.5292E-07 1.472 3.3307E-07 1.473
0.8 4 3.1481E-04 - 1.2643E-04 - 6.5437E-05 -8 1.4467E-04 1.122 5.8137E-05 1.121 3.0006E-05 1.12512 6.4907E-05 1.156 2.6047E-05 1.158 1.3421E-05 1.16116 2.8731E-05 1.176 1.1514E-05 1.178 5.9278E-06 1.17964 1.2623E-05 1.187 5.0544E-06 1.188 2.6008E-06 1.189
45
Table 2.6.7: Computing times with refinement of the time interval with the first, second and fourth orderin space schemes and h = 1/2000.
1st Order 2nd Order 4th Orderγ 1/τ L∞h,τ TOC(s) L∞h,τ TOC(s) L∞h,τ TOC(s)
0.2 16 8.753E-06 1.345E+00 6.355E-07 1.356E+00 6.168E-07 9.160E+0132 9.181E-06 2.414E+00 2.076E-07 2.453E+00 1.890E-07 9.227E+0164 9.313E-06 4.746E+00 7.587E-08 4.726E+00 5.722E-08 9.539E+01
128 9.353E-06 9.162E+00 3.582E-08 9.518E+00 1.717E-08 1.001E+02
0.5 16 4.781E-06 1.275E+00 5.000E-06 1.346E+00 4.981E-06 9.090E+0132 7.568E-06 2.376E+00 1.830E-06 2.446E+00 1.812E-06 9.207E+0164 8.726E-06 4.594E+00 6.716E-07 4.799E+00 6.529E-07 9.448E+01
128 9.145E-06 8.949E+00 2.525E-07 9.451E+00 2.339E-07 9.943E+01
0.8 16 1.768E-05 1.296E+00 2.607E-05 1.302E+00 2.605E-05 9.223E+0132 8.106E-06 2.289E+00 1.153E-05 2.532E+00 1.151E-05 9.059E+0164 4.801E-06 4.544E+00 5.073E-06 4.760E+00 5.054E-06 9.079E+01
128 7.190E-06 8.993E+00 2.229E-06 9.397E+00 2.210E-06 9.557E+01
46
Chapter 3
Influence of variable-order operators
in the behaviour of sub-diffusive
systems
In this chapter an investigation is made into the effects of variable order differentiation in the be-
haviour of sub-diffusive systems. Section 3.1 introduces a scheme able to solve variable order time-
fractional sub-diffusion equations. The initial-boundary value problem is stated in section 3.1.1. In
section 3.1.2 a difference scheme able to solve time fractional diffusion equations with variable coeffi-
cients dependent on time and space [98] is implemented and provided to the reader in matrix form. The
convergence of the scheme is then validated with a numerical example against the analytic solution in
section 3.1.3. In section 3.2 the effects of order dependence on time, space and the solution itself will
be analysed through numerical examples. Departure is made in section 3.2.1 from the comparison of
the standard diffusion equation with constant order fractional diffusion which is then taken as reference
for the analysis of the behaviour of anomalously diffusive systems with variable order. Sections 3.2.2,
3.2.3 and 3.2.4, study of the behaviour of a sub-diffusive with variable orders dependent of time, space
and the system solution, respectively.
3.1 Numerical solution of variable order time fractional diffusion
equations
3.1.1 Problem Statement
The following variable order time fractional diffusion equation will be considered [98]
0Dγ(x,t)t u(x, t) = K
∂2u(x, t)
∂x2+ f(x, t), x ∈ [0, L]t ∈ [0, T ] (3.1.1)
47
with the following initial and boundary conditions
u(x, 0) = ξ(x), x ∈ [0, L] (3.1.2)
u(0, t) = ψ(t); u(L, t) = φ(t), t ∈ [0, T ] (3.1.3)
In equation (3.1.1), K > 0 is a generalized diffusion coefficient, the variable u(x, t) is a physical
quantity of interest such as the concentration and 0Dγ(x,t)t denotes the Coimbra [64] variable order
fractional derivative
0Dγ(x,t)t u(x, t) =
1
Γ(1− γ(x, t))
∫ t
0
(t− σ)−γ(x,t)∂u(x, σ)
∂σdσ +
(u(x, 0+)− u(x, 0−))t−γ(x,t)
Γ(1− γ(x, t))(3.1.4)
where the order γ(x, t) ∈ [0, 1] may be a function of time, space or both.
If u(x, 0+) = u(x, 0−) then the Coimbra definition is equivalent to the Caputo type definition [97]
C0 D
γ(x,t)t =
1
Γ(1− γ(x, t))
∫ t
0
(t− σ)−γ(x,t)∂u(x, σ)
∂σdσ (3.1.5)
However, it is pointed out that there are more definitions for the variable order fractional differential
operator, namely definitions that account for the memory of the history of the derivative [108, 109].
3.1.2 Numerical Scheme
In this section, the main steps regarding the scheme developed by Shen et al. in [98] will be stated.
The matrix form of the scheme is then provided to expedite future implementations.
Let the solution u(x, t) of problem (3.1.1)-(3.1.3) be an adequately smooth function. In addition let,
the domain be represented by an equally spaced mesh with M + 1 points in the spacial domain and
N + 1 points in the temporal domain. That is, xi = ih, i = 0, 1, ...,M and tn = nτ, n = 0, 1, ..., N , where
the spacial and temporal grid sizes are h = L/M and τ = T/N , respectively.
The second order space derivative in equation (3.1.1) will be approximated by following central finite
difference formula, denoting by δ2x the second order difference operator
∂2u(xi, tn)
∂x2= δ2xu(xi, tn) +O(h)2,
δ2xu(xi, tn) =u(xi−1, tn)− 2u(xi, tn) + u(xi+1, tn)
h2
(3.1.6)
Following the scheme given in [98], the variable order time fractional derivative can be approximated
by
0Dγ(xi,tn)t u(xi, tn) =
τ−γ(xi,tn)
Γ(2− γ(xi, tn))
n−1∑j=0
di,j,n[u(xi, tj+1)− u(xi, tj)] + rni (3.1.7)
where
di,j,n = (n− j)1−γ(xi,tn) − (n− j − 1)1−γ(xi,tn), j = 0, 1, ..., n− 1 (3.1.8)
48
and rni is the truncation error
rni =C · τ2−γ(xi,tn)
Γ(2− γ(xi, tn))n1−γ(xi,tn) ≤ C · T 1−γ(xi,tn)
Γ(2− γ(xi, tn))· τ ≤ C · τ (3.1.9)
Substituting the derivatives in equation (3.1.1) by their respective numerical approximations and de-
noting by Uni the numerical approximation of u(xi, tn), f(xi, tn) by fni and γ(xi, tn) by γni results in
τ−γni
Γ(2− γni )
n−1∑j=0
di,j,n
(U j+1i − U ji
)= K
Uni−1 − 2Uni + Uni+1
h2+ fni (3.1.10)
Leading to
Uni −KΓ(2− γni )τγ
ni
h2(Uni−1 − 2Uni + Uni+1) =
Un−1i −n−2∑j=0
di,j,n(U j+1i − U ji ) + Γ(2− γni )τγ
ni fni
i = 1, 2, ...,M − 1, n = 1, 2, .., N
(3.1.11)
Discretization of the initial and boundary conditions completes the scheme
U0i = ξ(xi), i = 0, 1, ...,M (3.1.12)
Un0 = ψ(tn), UnM = φ(tn), n = 0, 1...N (3.1.13)
Equation (3.1.11) can be written in matrix for as
(E − K
h2GnS
)Un = Un−1 −
n−2∑j=0
Dnj
(U j+1 − U j
)+Gnfn +
K
h2XGnbU
nb (3.1.14)
where Un = [Un1 , Un2 , ..., U
nM−1]T , Unb = [Un0 , U
nM ]T , fn = [fn1 , f
n2 , ..., f
nM−1]T . S, Gn and Dn
j are diagonal
matrices of size (M − 1)× (M − 1)
S =
−2 1 0 0 · · · 0 0
1 −2 1 0 · · · 0 0
0 1 −2 1 · · · 0 0...
......
.... . .
......
0 0 0 0 · · · −2 1
0 0 0 0 · · · 1 −2
(M−1)×(M1)
(3.1.15)
49
Gn =
Γ(2− γn1 )τγ
n1 0 · · · 0
0 Γ(2− γn2 )τγn2 · · · 0
......
. . ....
0 0 · · · Γ(2− γnM−1)τγnM−1
(M−1)×(M1)
(3.1.16)
Dnj =
d1,j,n 0 · · · 0
0 d2,j,n · · · 0...
.... . .
...
0 0 · · · dM−1,j,n
(M−1)×(M1)
(3.1.17)
Furthermore, the boundary terms can be included through the following matrices
X =
1 0...
...
0 1
(M−1)×2
(3.1.18)
Gnb =
Γ(2− γn1 )τγn1 0
0 Γ(2− γnM−1)τγnM−1
2×2
(3.1.19)
Unconditional stability of the scheme (3.1.11)-(3.1.13) was proved in [98] through Fourier analysis.
The scheme also proved to be both convergent with order O(τ + h2) and uniquely solvable.
3.1.3 Numerical Example
In order to carry out a numerical test of the convergence order of scheme (3.1.11)-(3.1.13), consider
the following initial and boundary value problem of the type (3.1.1)-(3.1.3) [98]
0Dγ(x,t)t u(x, t) = K
∂2u(x, t)
∂x2+ f(x, t), x ∈ [0, L], t ∈ [0, T ] (3.1.20)
with the following initial and boundary conditions
u(x, 0) = 10x2(1− x), x ∈ [0, 1] (3.1.21)
u(0, t) = u(1, t) = 0, t ∈ [0, 1] (3.1.22)
with order γ(x, t) =2 + sin(xt)
4and source term f(x, t) given by
f(x, t) = 20x2(1− x)
[t2−γ(x,t)
Γ(3− γ(x, t))+
t1−γ(x,t)
Γ(2− γ(x, t))
]− 20(t+ 1)2(1− 3x) (3.1.23)
50
The exact solution, depicted in Figure 3.1, is
u(x, t) = 10x2(1− x)(t+ 1)2 (3.1.24)
Figure 3.1: Exact solution of problem (3.1.20)-(3.1.22).
Let, u(xi, tn) and Uni be the exact and numeric solutions, respectively. Then the L2h,τ and L∞h,τ errors
are defined as
L2h,τ =
√√√√ 1
P
M∑i=0
N∑n=0
[u(xi, tn)− Uni ]2 (3.1.25)
L∞h,τ = max |u(xi, tn)− Uni | (3.1.26)
where P is the total number of mesh points domain.
Furthermore, let the order of convergence of an error e, EOC, be given by the following expression
EOC = log2
eτeτ/2
(3.1.27)
In figure 3.2 the absolute error in each point of the domain can be observed. Although the plot
shown corresponds to the situation where h = 1/250 and τ = 1/500, the morphology remained constant
throughout all numerical test with an increasing in error with time and towards the centre of the space
interval.
51
Figure 3.2: Absolute error in the solution of (3.1.20)-(3.1.22), h = 1/250 and τ = 1/500.
To numerically verify the order of convergence of the scheme, Tables 3.1.1 and 3.1.2 were produced,
where the L2 and L∞ errors can be found along with the respective orders of convergence. The L2 error
presents an average of the error in the entirety of the domain, while the L∞ error depicts the error in the
most critical situation.
Table 3.1.1: Error behaviour with decreasing temporal gridsize, h = 1/500.
1/τ L2 EOC(L2) L∞ EOC(L∞)
8 2.0911× 10−3 - 4.1270× 10−3 -
16 8.1127× 10−4 1.37 1.6213× 10−3 1.35
32 3.1097× 10−4 1.38 6.3430× 10−4 1.35
36 1.1838× 10−4 1.39 2.4759× 10−4 1.36
128 4.4901× 10−5 1.40 9.6527× 10−5 1.36
256 1.7002× 10−5 1.40 3.7614× 10−5 1.36
The error behaviour with the reduction of the temporal grid size is shown on Table 3.1.1. The nu-
merical tests were carried out with h = 1/500, a value that guarantees a negligible contribution of the
spacial discretization to the error because the scheme has second order in space. The L2 error order
of convergence is just slightly superior to the L∞ with both errors having the same order of magnitude.
This similarity suggests a homogeneous error behaviour, a conclusion supported by additional numerical
tests at different time steps and positions where the variable fractional order did not appear to change
the rate of convergence. The EOC exhibits an order in between 1.35 and 1.40 , nonetheless, consider-
ably above first order the theoretical prediction. In fact, the observed EOC is much closer to 2 − γavg,
52
with γavg being the average value of the order γ across the computational domain, thus resembling the
order observed in the L1 scheme for contant order fractional diffusion equations [63].
Table 3.1.2: Error behaviour with decreasing temporal gridsize, t = h2.
1/h L2 EOC(L2) L∞ EOC(L∞)
4 7.5920× 10−4 - 1.6252× 10−3 -
8 1.1306× 10−4 2.75 2.4936× 10−4 2.70
16 1.6561× 10−5 2.77 3.7585× 10−5 2.73
32 2.4031× 10−6 2.78 5.7123× 10−6 2.72
In Table 3.1.2 lists the L2 and L∞ errors and their rate of convergence with spacial grid size reduction.
As in the previous case, the L2 and L∞ errors exhibit the same magnitude. The order of convergence
ranges between 2.70 and 2.78, also above the second order theoretical calculations.
Attending to the results of this numerical simulation, a successful implementation of a numerical
scheme able to solve variable order fractional diffusion equations was made. Furthermore, the scheme
is able to deal with a fractional order function dependent of time and space. This scheme can be used
to analyse the impact that different dependences of the fractional order of differentiation have in the
behaviour of the solution of the diffusion equation.
3.2 Influence of variable order differential operators in anomalous
diffusion
Let’s consider the one-dimensional diffusion equation in variable fractional order with the initial and
boundary value problem of the type (3.1.1)-(3.1.3). The initial distribution is given and so are the bound-
ary conditions, maintained at a zero constant value.
0Dγ(x,t)t u(x, t) =
∂2u(x, t)
∂x2, x ∈ [0, 10], t ∈ [0, T ] (3.2.1)
with the following initial and boundary conditions
u(x, 0) = sin2(xπ
10
), x ∈ [0, 10] (3.2.2)
u(0, t) = u(10, t) = 0, t ∈ [0, T ] (3.2.3)
where u may represent for instance the temperature or concentration and the diffusion coefficient was
set to K = 1. An analysis of the effect of the variable order will only require the adjustment of the order
γ for each of the intended cases. The initial and boundary conditions will remain the same for the time,
space and solution dependent cases. The following results were taken with a time and space steps
equal to τ = 1/25 and h = 1/25, respectively.
53
3.2.1 Constant fractional order
In order to investigate the effects of different order dependences of time derivatives in anomalous
diffusion modelling it is important to understand the behaviour of the solution under different constant
fractional orders and how this fractional behaviour differs from standard diffusion (γ = 1). The behaviour
of the constant order solution may then be taken as reference for the analysis of variable order modelling.
For the numerical solution of constant order fractional diffusion equations, the previously introduced
scheme can be used with a constant order γ and equation (3.2.1) will become
0Dγt u(x, t) = K
∂2u(x, t)
∂x2, x ∈ [0, 10], t ∈ [0, T ] (3.2.4)
Figure 3.3: Solution of the standard diffusion equation.
The scheme proved able to also solve the standard diffusion with γ = 1, the corresponding plot is
shown on Figure 3.3 with first order accuracy in time. Figure 3.4 shows the time evolution of the solution
of (3.2.4) for x = 5 with different fractional orders, along with the solution of the standard diffusion
equation (γ = 1) . As the order of the time fractional derivative increases so does the diffusion rate
and the solution approaches the standard diffusion equation. Figure 3.6 portrays the solution in space
for t = 5 for the same orders, where the results with the different fractional orders behave as expected,
again it is seen that with each increase in order the solution get closer to the standard diffusion equation.
However, one intriguing feature does stand out, in the initial time steps a decrease in the order of the
derivative corresponds to an increase in the diffusion rate, this behaviour can also be observed in Figure
54
3.5 which depicts the solution at t = 0.2. This behaviour is counter-intuitive when one is looking for
sub-diffusion, and some attempts to modify this initial behaviour were made. The inclusion of a constant
value history before the initial time step was considered and so was the reduction of to a zero initial value
problem, with persistence of this initial super-diffusive behaviour. After unfruitful checks to the initial
conditions and to the well-posedness of the problem, a dimensional analysis was made. From the last,
some conclusions are worth remarking. If a physical problem is considered, for instance the evolution
of the concentration given an initial distribution and boundary conditions, it is known that [u] = ML−3,
[∂u/∂t] = ML−3T−1 , [∂2u/∂x2] = ML−5 and thus [K] = L2T−1. Now, if the intent is to model diffusion
in a porous media, through the substitution of the integer order time derivative by the fractional one
and with the same diffusivity coefficient as would be used in the standard diffusion equation then the
dimensions of the right and left sides of equation (3.2.4) do not match. Indeed, if [Dγ0,tu] = ML−3T−γ
then the dimensions of the diffusivity coefficient should be [K] = L2T−γ . To account for this change, a
Riemann-Liouville integration [32] of order 1 − γ was made to the diffusion coefficient so that it would
have the desired dimensions, as in the following equation
0D−(1−γ)t K = Kγ(t) =
1
Γ(1− γ)
∫ t
0
(t− σ)−γKdσ =Kt1−γ
(1− γ)Γ(1− γ)(3.2.5)
Hence, the fractional time fractional derivative may require a temporal adjustment of the diffusivity
coefficient. As the order of the fractional derivative tends to the unity, the dependence on time disappears
and the diffusion coefficient tends to one, retrieving standard diffusion. On the other hand, as the order
of the fractional derivative tends to zero, Kγ approaches Kγ = Kt. In figure 3.4, the dashed lines
correspond to the computation of equation (3.2.4) after substitution of the standard diffusion coefficient
by the time varying coefficient. It can be seen that the super-diffusive behaviour disappears and that
as the order of the fractional derivative increases from 0 to 1, the diffusion rate increases towards the
standard diffusion solution as expected. It is noted that the overall solution decay with time is higher
than with a constant diffusion coefficient when comparing derivatives of the same order because the
time dependent diffusion coefficient increases with time. Despite this increase , it was verified that the
solution remains higher than the standard diffusion solution.
3.2.2 Time dependent fractional order
To evaluate the impact of a time varying order, equation (3.2.1) will be solved with a time only depen-
dent order becoming
0Dγ(t)t u(x, t) = K
∂2u(x, t)
∂x2, x ∈ [0, 10]t ∈ [0, T ] (3.2.6)
To analyse the behaviour of the solution in this case, the following three order functions will be
considered
γ1(t) = 0.5 + 0.25
(t
T
)(3.2.7)
γ2(t) = 0.75− 0.25
(t
T
)(3.2.8)
55
0 1 2 3 4 5 6 7 8 9 10
0.4
0.5
0.6
0.7
0.8
0.9
1
t
u(5,
t)
γ=1γ=0.25γ=0.5γ=0.75γ=0.25 with K
γ(t)
γ=0.5 with Kγ(t)
γ=0.75 with Kγ(t)
Figure 3.4: Solution versus time plot at x = 5 with different constant fractional orders.
0 1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
u(x,
0.2)
γ=1γ=0.25γ=0.5γ=0.75u(x,t=0)
Figure 3.5: Solution at t = 0.2 with different constant fractional orders.
56
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
x
u(x,
5)
γ=1γ=0.25γ=0.5γ=0.75γ=0.25 with K
γ(t)
γ=0.5 with Kγ(t)
γ=0.75 with Kγ(t)
Figure 3.6: Solution at t = 5 with different constant fractional orders.
57
γ3(t) = 0.5 + 0.25 sin
(πt
T
)(3.2.9)
where the maximum time considered is T = 10.
0 2 4 6 8 100.4
0.5
0.6
0.7
0.8
0.9
1
t
u(5,
t)
γ=0.5(t)γ=0.75γ1(t)
γ2(t)
γ3(t)
Figure 3.7: Time evolution in x = 5 with different time dependent fractional orders.
Figure 3.7 shows the evolution of the solution at x = 5. Focusing on γ1(t), the plot clearly demon-
strates a diffusion rate that follows the constant order solution with γ = 0.5 at initial time steps, again
demonstrating super-diffusion. Having chosen a variable order that increases linearly from 0.5 at t = 0
to 0.75, the diffusion rate gradually changes to the rate of the constant order γ = 0.75, at t = 10. With
different order functions it is possible to model decelerating diffusion processes or an even more com-
plex situation containing both behaviours. In the case of a decreasing linear order γ2(t), the behaviour
is analogous. The solution starts by behaving similarly to initial order, gradually slowing down with time.
It appears that, for the case of a linear order variation with time, the solution is bounded by the constant
order solutions equal to the maximum and minimum of the variable order solution. Attending now to
γ3(t), which corresponds to a variable order that increases until t = 5 decreasing afterwards up to t = 10
it is possible to conclude that the rate at which the derivative changes need careful consideration in the
modelling process. In Figure 3.7 it is possible to see that a sudden decrease of the derivative order may
produce non-physical results, with the solution increasing with time at its maximum value in space near
t = 10. This example, however simple, demonstrates that variable order fractional calculus is a viable
58
tool to model situations where the diffusive behaviour changes with time.
3.2.3 Space dependent fractional order
To evaluate the impact of a space varying order, equation (3.2.1) will become
0Dγ(x)t u(x, t) = K
∂2u(x, t)
∂x2, x ∈ [0, 10]t ∈ [0, T ] (3.2.10)
To better compare the effects of space varying order, the following three space dependent orders will
be modelled, as represented in Figure 3.8
γ1(x) = 0.25 +∣∣∣0.75
( xL− 0.5
)∣∣∣ (3.2.11)
γ2(t) =
0.25 + 0.75
x
L, 0 < x < L/2
0.25 + 0.75(
1− x
L
), L/2 ≤ x < L
(3.2.12)
γ3(x) = 0.25 + 0.25 cos
(4πx
10
)+∣∣∣0.75
( xL− 0.5
)∣∣∣ (3.2.13)
0 2 4 6 8 100.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x
γ(x)
γ1(x)
γ2(x)
γ3(x)
Figure 3.8: The three space dependent fractional orders considered.
Figure 3.9 shows the evolution of the solution at x = 5, the point where the highest value of the
solution occurs. It can be seen that super-diffusion also occurs on the initial time steps for this case.
Although the evolution with time tends to follow a constant order closer to the local value of the space
59
0 2 4 6 8 100.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
t
u(5,
t)
γ=0.25γ=0.65γ1(x)
γ2(x)
γ3(x)
Figure 3.9: Time evolution at x = 5 modelled with different space dependent fractional orders.
60
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
x
u(x,
10)
γ=0.25γ=0.65γ1(x)
γ2(x)
γ3(x)
Figure 3.10: Solution at t = 10 modelled with different space dependent fractional orders.
61
varying order, the effects of the order on the remaining domain are clearly noticeable. With the local
fractional order assuming a value γ1(5) = 0.25, the increase in order towards the boundaries of the
interval causes a higher effective diffusion rate at x = 5. Conversely, the order γ2(x) has a maximum
at x = 5, decreasing linearly towards the boundaries. Consequently, at x = 5, the effective diffusion is
rate lower than what would be observed if the order was constant, with the value corresponding to the
local value at x = 5. Also in Figure 3.9, one can see that at x = 5, the behaviour of an oscillating space
variable order can be seen with the solution for γ3(x). In this case, because of the oscillating effect, the
effective order at x = 5 is very close to the local fractional order γ3(5) = 0.5.
Figure 3.10 depicts the solution at t = 10. With this figure, the way the space varying order affects
the solution becomes even clearer. One can notice for instance that for x . 2 the solution with order
γ1(x) is higher than with a constant fractional order γ = 0.25, even if the solution tends to follow more
closely this order, corresponding to the local order γ1(x) at x = 5. The solution with γ3(x) may also be
found in Figure 3.10, allowing further conclusions into the behaviour with an oscillating variable order in
space. Near the boundaries the variable order γ3(x) assumes higher values than the order γ1(x), giving
higher solutions at the boundaries in respect to γ1(x). In the center of the interval γ3(x) is higher than
γ1(x) leading to a lower solution with a more rounded shape, in accordance with the behaviour of the
derivative of order γ3(x) in the center of the space interval.
3.2.4 Solution dependent fractional order
To evaluate the impact of a space varying order, equation 3.2.1 will become
0Dγ[u(x,t)]t u(x, t) = K
∂2u(x, t)
∂x2, x ∈ [0, 10]t ∈ [0, T ] (3.2.14)
The order γ[u(x, t)] was assumed to follow the following function, represented in Figure 3.11.
γ[u(x, t)] = 0.5 + 0.3u
Umax(3.2.15)
The value Umax is taken to be Umax = 1, the highest value given by the initial condition.
An iterative procedure was carried out to calculate the solution with a solution dependent time deriva-
tive. As an initial guess, the variable order was calculated with the results from the initial time step. The
final solution can then be used to calculate the derivative order, solving equation (3.2.14) a second time.
Following the same procedure more iterations can be made until the solution converges. After three
iterations, the maximum difference to the previous iteration had order 10−6 the iterative procedure was
terminated.
A solution dependent fractional order poses an even more complex problem, mixing the effects of
a space and time varying fractional orders. Figure 3.12, shows the solution at x = 5 which appears to
begin by following the constant fractional order γ = 0.8 but then, as the solution decreases, the diffusion
process slows down with the solution reaching t = 10 at approximately the same diffusion rate as the
constant order γ = 0.6. The combined effect of a space varying order can also seen on Figure 3.13. At
62
0
5
10
0
5
100.5
0.6
0.7
0.8
0.9
tx
γ[u(
x,t)
]
Figure 3.11: Solution dependent fractional order.
0 2 4 6 8 100.4
0.5
0.6
0.7
0.8
0.9
1
t
u(5,
t)
γ=0.6γ=0.8γ[u(x,t)]
Figure 3.12: Time evolution in x = 5 with a solution dependent variable order model.
63
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
x
u(x,
10)
γ=0.6γ=0.8γ[u(x,t)]
Figure 3.13: Solution t = 10 with a solution dependent variable order model.
x = 0.5 the solution assumes a value lower than the the local fractional order because of decay with time
of the solution, but higher that it would be if the solution did not vary with space, because the fractional
order decreases towards the boundaries, the same effect observed in space varying fractional order.
64
Chapter 4
Conclusions
This work presents intends to contribute in the building of a road between fractional calculus and its
engineering applications. A total of nine finite difference schemes are implemented and compared within
three different types of fractional diffusion equations, providing the tools and knowledge to properly apply
finite differences in the solution of fractional partial differential equations. An intuitive grasp on how the
order of the time fractional derivative impacts the solution of a sub-diffusive system was also achieved,
both for the constant and variable order cases, greatly increasing the odds of a successful application.
The result of this effort resulted in two papers [110, 111], currently under submission. The mastery of
a numerical solution method and the understanding of the effects of constant and variable order have
gathered the necessary and sufficient conditions to proceed with a real world application.
In chapter 2 the solutions time, space and time-space fractional diffusion equations were calculated
each with three different finite difference schemes. Numerical examples have validated the implemen-
tations and allowed the comparison of the different schemes in terms of accuracy and computing time.
Looking at time fractional diffusion equations, all schemes have shown to follow the theoretical pre-
dictions of their order of convergence. It was seen that weighted and shifted methods increase greatly
increase computing times by considering two time levels in the time fractional derivative and that high or-
der schemes for the solution of the time fractional diffusion equation are the best choice and the ability of
reaching the same order of magnitude with less time steps is an enormous advantage, with significant
reduction of computing times. The three schemes developed for space fractional diffusion equations
have also shown to follow the theoretical predictions regarding their accuracy. Since the matrices used
to compute the left and right space fractional derivatives are full, the first, second and fourth schemes
have shown little difference in computing times. However, higher order approximations for time fractional
derivatives come with increased smoothness requirements and, as it was seen in numerical examples,
these requirements change considerably the accuracy of these schemes. Despite these requirements,
the fourth order scheme has shown the smallest errors and since the computing time is almost the same
as the lower order schemes and therefore is considered the best for the solution of these equations.
Regarding time-space fractional diffusion equations, three different different schemes in increas-
ing spatial order of accuracy were implemented, using the same approximation for the time fractional
65
derivative. The numerical examples, analysing time and space refinement with different fractional orders
in time and space, have proved that the theoretical predictions for the order of accuracy were correct.
However the error for the first order scheme shows to be insensitive to the refinement of the space in-
terval and order γ, with a fixed time step. This situation was overcome by using h = τ2−γ . Due to the
similarities in the construction and solution of the schemes the first and second order in time schemes
show similar computing times for the same grid. On the other hand, the fourth order scheme shows
computing times more than ten times larger than the previous. This huge increase in the computing time
for the high order scheme is caused by the application of the averaging operator of the compact scheme
to the time fractional derivative, greatly increasing the computational cost. The results indicate that the
most efficient choice for time-space fractional diffusion equation is the second order scheme. To unlock
all the potential of a high order scheme, a high order approximation should also be used for the time
fractional derivative, enabling the use a coarser time grid for the same error magnitude.
In chapter 3, a scheme able to solve time and space dependent variable order fractional diffusion
equations was implemented and tested. The matrix forms of the scheme were also constructed, allowing
for immediate implementation. With the help of this scheme, the effect of the order dependence on time,
space and the solution itself on the modelling of time fractional diffusion equations was studied.
The examples of variable order time fractional diffusion equations helped in achieving better under-
standing of how the order of the derivative affects the behaviour of a physical system that are relevant
for modelling real world problems. A comparison of the constant fractional order case with standard
diffusion provided an intuitive grasp on the effect a fractional order effect. A clear evolution towards
the standard diffusion equation was seen with the increase of the time fractional derivative for orders
0 ≤ γ ≤ 1. A super-diffusive behaviour was noticed at initial times and dimensional analysis led to a
time fractional diffusion coefficient based for the modelling based on the standard diffusion coefficient,
eliminating this behaviour. A linear time evolution of the fractional order led to solutions bounded by the
constant order solutions with the initial and final values of time variable order. It also seen that care has
rapid changed in the time variable order may lead to non-physical solutions. For a variable order that is
function of space it was seen that the solution at each point in space follows an effective order that is
lower than its local value that depends of the order in the remaining of space. For variable orders that
are function of the solution, the combined effects of space and time dependent orders were observed.
All the developed algorithms were coded and the programs developed from scratch by the author. In
addition the numerical methods are presented in matrix form that may help others to programme these
schemes more quickly.
There is plenty future work to be done in the realm of fractional calculus. Schemes regarding other
types of equation can be solved and the problem can be extended to the two-dimensional case. A higher
order scheme in time and space should also be developed for the time-space fractional diffusion equa-
tion. Regarding variable order diffusion equations, a higher order scheme could also be implemented
and the analysis of the order effects could be extended to the space derivative or to the fractional
diffusion-wave equation. As the purpose of this work is also paving the way to future engineering ap-
plications one could now start by a simple study of diffusion, for instance in porous media, using the
66
results obtained with a commercial software to do a parametric study of the order of the derivative in the
diffusion equation.
67
68
References
[1] K. B. Oldham and J. Spanier. The Fractional Calculus. Academic Press, Inc., New York.
[2] E. Barkai, R. Metzler, and J. Klafter. From continuous time random walks to the fractional Fokker-
Planck equation. Physical Review E, 61(1):132–138, 2000. ISSN 1063-651X. doi: 10.1103/
PhysRevE.61.132. URL http://pre.aps.org/abstract/PRE/v61/i1/p132_1%5Cnhttp://
link.aps.org/doi/10.1103/PhysRevE.61.132.
[3] R. Metzler and J. Klafter. The random walk’s guide to anomalous diffusion: a fractional dynamics
approach. Physics Reports, 339(1):1–77, 2000. ISSN 03701573. doi: 10.1016/S0370-1573(00)
00070-3.
[4] G. M. Zaslavsky. Chaos, fractional kinetics, and anomalous transport, 2002. ISSN 03701573.
[5] A. Kilbas, H. Srivastava, and J. Trujillo. Theory and applications of fractional differential equations.
Elsevier, Boston, 2006. ISBN 978-0-444-51832-3.
[6] R. L. Magin. Fractional calculus models of complex dynamics in biological tissues. Computers and
Mathematics with Applications, 59(5):1586–1593, 2010. ISSN 08981221. doi: 10.1016/j.camwa.
2009.08.039. URL http://dx.doi.org/10.1016/j.camwa.2009.08.039.
[7] C. Li and F. Zeng. Finite Difference Methods for Fractional Differential Equations. Inter-
national Journal of Bifurcation and Chaos, 22(04):1230014, 2012. ISSN 0218-1274. doi:
10.1142/S0218127412300145.
[8] R. L. Bagley and J. Torvik. Fractional calculus - A different approach to the analysis of vis-
coelastically damped structures. In AIAA Journal, volume 21, pages 741–748. may 1983. doi:
10.2514/3.8142. URL http://arc.aiaa.org/doi/abs/10.2514/3.8142.
[9] R. L. Bagley and P. J. Torvik. Fractional calculus in the transient analysis of viscoelastically damped
structures. AIAA Journal, 23(6):918–925, jun 1985. ISSN 0001-1452. doi: 10.2514/3.9007. URL
http://arc.aiaa.org/doi/abs/10.2514/3.9007.
[10] M. Enelund and B. L. Josefson. Time-Domain Finite Element Analysis of Viscoelastic Structures
with Fractional Derivatives Constitutive Relations. AIAA Journal, 35(10):1630–1637, oct 1997.
ISSN 0001-1452. doi: 10.2514/2.2. URL http://arc.aiaa.org/doi/abs/10.2514/2.2.
69
[11] A. Wilson, M. Enelund, and B. Josefson. Dynamic substructuring of viscoelastic structures
with fractional derivatives constitutive relations. In 39th AIAA/ASME/ASCE/AHS/ASC Struc-
tures, Structural Dynamics, and Materials Conference and Exhibit, Reston, Virigina, apr 1998.
American Institute of Aeronautics and Astronautics. doi: 10.2514/6.1998-1862. URL http:
//arc.aiaa.org/doi/10.2514/6.1998-1862.
[12] K. Adolfsson, M. Enelund, and P. Olsson. A large deformation viscoelastic model with fractional
order rate laws. In 19th AIAA Applied Aerodynamics Conference, Reston, Virigina, jun 2001.
American Institute of Aeronautics and Astronautics. doi: 10.2514/6.2001-1517. URL http://
arc.aiaa.org/doi/10.2514/6.2001-1517.
[13] T. Beda and Y. Chevalier. Identification of Viscoelastic Fractional Complex Modulus. AIAA Journal,
42(7):1450–1456, jul 2004. ISSN 0001-1452. doi: 10.2514/1.11883. URL http://arc.aiaa.org/
doi/10.2514/1.11883.
[14] R. Bagley, D. Swinney, and K. Griffin. Fractional calculus - A new approach to modeling unsteady
aerodynamic forces. In 29th Aerospace Sciences Meeting, Reston, Virigina, jan 1991. American
Institute of Aeronautics and Astronautics. doi: 10.2514/6.1991-748. URL http://arc.aiaa.org/
doi/10.2514/6.1991-748.
[15] K. Diethehn and Y. Luchko. Fractional-Order Viscoelasticity (FOV): Constitutive Development Us-
ing the Fractional Calculus: First Annual Report. Technical Report December, NASA, 2002.
[16] A. A. Stanislavsky. Astrophysical Applications of Fractional Calculus. Technical report, 2010. URL
http://link.springer.com/10.1007/978-3-642-03325-4_8.
[17] Z. E. A. Fellah, C. Depollier, and M. Fellah. Application of Fractional Calculus to the Sound Waves
Propagation in Rigid Porous Materials: Validation Via Ultrasonic Measurements. Acta Acustica,
88:34 – 39, 2002. ISSN 14367947. URL http://www.european-acoustics.net/products/
acta-acustica/most-cited/acta_88_2002_Fellah.pdf.
[18] A. Carpinteri, B. Chiaia, and P. Cornetti. On the mechanics of quasi-brittle materials with a fractal
microstructure. Engineering Fracture Mechanics, 70(16):2321–2349, 2003. ISSN 00137944. doi:
10.1016/S0013-7944(02)00220-5.
[19] M. Mateos, L. Gornet, H. Zabala, L. Aretxabaleta, P. Rozycki, and P. Cartraud. Hysteretic Be-
haviour Of Fibre-Reinforced Composites. 2012.
[20] G. Sun and Z. H. Zhu. Fractional-Order Tension Control Law for Deployment of Space Tether
System. Journal of Guidance, Control, and Dynamics, 37(6):2057–2062, nov 2014. ISSN 0731-
5090. doi: 10.2514/1.G000496. URL http://arc.aiaa.org/doi/abs/10.2514/1.G000496.
[21] G. P. K. Mand, R. Ghosh, and D. Ghose. A fractional order proportional integral controller for
path following and trajectory tracking of miniature air vehicles. Proceedings of the Institution of
Mechanical Engineers, Part G: Journal of Aerospace Engineering, 228(8):1389–1402, jun 2014.
70
ISSN 0954-4100. doi: 10.1177/0954410013493055. URL http://pig.sagepub.com/lookup/
doi/10.1177/0954410013493055.
[22] T. Mishra and K. Rai. Fractional single-phase-lagging heat conduction model for describing
anomalous diffusion. Propulsion and Power Research, 5(1):45–54, 2016. doi: 10.1016/j.jppr.
2016.01.003.
[23] A. Suzuki, S. A. Fomin, V. A. Chugunov, Y. Niibori, and T. Hashida. Fractional diffusion modeling
of heat transfer in porous and fractured media. International Journal of Heat and Mass Transfer,
103:611–618, 2016. ISSN 00179310. doi: 10.1016/j.ijheatmasstransfer.2016.08.002.
[24] A. Sapora, P. Cornetti, B. Chiaia, E. K. Lenzi, and L. R. Evangelista. Nonlocal Diffusion
in Porous Media: A Spatial Fractional Approach. Journal of Engineering Mechanics, page
D4016007, mar 2016. ISSN 0733-9399. doi: 10.1061/(ASCE)EM.1943-7889.0001105. URL
http://ascelibrary.org/doi/10.1061/%28ASCE%29EM.1943-7889.0001105.
[25] G. Alaimo and M. Zingales. Laminar flow through fractal porous materials: the fractional-order
transport equation. Communications in Nonlinear Science and Numerical Simulation, 22(1):889–
902, 2015. ISSN 10075704. doi: 10.1016/j.cnsns.2014.10.005.
[26] W. Chen and H. Sun. Multiscale Statistical Model Of Fully-Developed Turbulence Parti-
cle Accelerations. Modern Physics Letters B, 23(03):449–452, jan 2009. ISSN 0217-
9849. doi: 10.1142/S021798490901862X. URL http://www.worldscientific.com/doi/abs/
10.1142/S021798490901862X.
[27] Y. Zhang, M. M. Meerschaert, and B. Baeumer. Particle tracking for time-fractional diffusion. Phys-
ical Review E, 78(3):036705, sep 2008. ISSN 1539-3755. doi: 10.1103/PhysRevE.78.036705.
URL http://link.aps.org/doi/10.1103/PhysRevE.78.036705.
[28] Y. He, S. Burov, R. Metzler, and E. Barkai. Random Time-Scale Invariant Diffusion and Transport
Coefficients. Physical Review Letters, 101(5):058101, jul 2008. ISSN 0031-9007. doi: 10.1103/
PhysRevLett.101.058101. URL http://link.aps.org/doi/10.1103/PhysRevLett.101.058101.
[29] E. Scalas. The application of continuous-time random walks in finance and economics. Physica
A: Statistical Mechanics and its Applications, 362(2):225–239, 2006. ISSN 03784371. doi: 10.
1016/j.physa.2005.11.024.
[30] M. Dalir and M. Bashour. Applications of Fractional Calculus. Applied Mathematicals Sciencies, 4
(21):1021–1032, 2010. ISSN 0369-8211. doi: 10.1016/S0898-1221(99)00333-8.
[31] S. Samko, A. Kilbas, and O. Marichev. Fractional Integrals and Derivatives: Theory and Ap-
plications. Gordon and Breach, New York, 1993. ISBN 2881248640. URL http://tocs.ulb.
tu-darmstadt.de/32759916.pdf.
[32] I. Podlubny. Fractional Differential Equations. Academic Press, New York, 1999. ISBN 0 -1 2
S5H810 -2. doi: 10.1016/S0076-5392(99)80017-4.
71
[33] R. Gorenflo, F. Mainardi, D. Moretti, and P. Paradisi. Time Fractional Diffusion: A Discrete Random
Walk Approach. Nonlinear Dynamics, 29(1-4):129–143, 2002.
[34] S. B. Yuste and L. Acedo. Some exact results for the trapping of subdiffusive particles in one
dimension. Physica A: Statistical Mechanics and its Applications, 336(3-4):334–346, 2004. ISSN
03784371. doi: 10.1016/j.physa.2003.12.048.
[35] P. Zhuang, F. Liu, V. Anh, and I. Turner. New Solution And Analytical Techniques Of The Implicit
Numerical Method For The Anomalous Subdiffusion Equation. SIAM J. Numer. Anal., 46(2):1079–
1095, 2008.
[36] T. A. M. Langlands and B. I. Henry. The accuracy and stability of an implicit solution method for
the fractional diffusion equation. Journal of Computational Physics, 205(2):719–736, 2005. ISSN
00219991. doi: 10.1016/j.jcp.2004.11.025.
[37] Z. Z. Sun and X. Wu. A fully discrete difference scheme for a diffusion-wave system. Applied
Numerical Mathematics, 56(2):193–209, 2006. ISSN 01689274. doi: 10.1016/j.apnum.2005.03.
003.
[38] Y.-n. Zhang, Z.-z. Sun, and H.-w. Wu. Error Estimates of Crank–Nicolson-Type Difference
Schemes for the Subdiffusion Equation. SIAM Journal on Numerical Analysis, 49(6):2302–2322,
2011. ISSN 0036-1429. doi: 10.1137/100812707. URL http://epubs.siam.org/doi/abs/10.
1137/100812707.
[39] S. B. Yuste and L. Acedo. An Explicit Finite Difference Method and a New von Neumann-Type
Stability Analysis for Fractional Diffusion Equations. SIAM Journal on Numerical Analysis, 42(5):
1862–1874, 2005. ISSN 0036-1429. doi: 10.1137/030602666. URL http://epubs.siam.org/
doi/abs/10.1137/030602666.
[40] S. B. Yuste. Weighted average finite difference methods for fractional diffusion equations. Journal
of Computational Physics, 216(1):264–274, 2006. ISSN 00219991. doi: 10.1016/j.jcp.2005.12.
006.
[41] M. Cui. Compact finite difference method for the fractional diffusion equation. Journal of Com-
putational Physics, 228(20):7792–7804, 2009. ISSN 00219991. doi: 10.1016/j.jcp.2009.07.021.
URL http://dx.doi.org/10.1016/j.jcp.2009.07.021.
[42] G. H. Gao and Z. Z. Sun. A compact finite difference scheme for the fractional sub-diffusion
equations. Journal of Computational Physics, 230(3):586–595, 2011. ISSN 00219991. doi:
10.1016/j.jcp.2010.10.007.
[43] V. J. Ervin and J. P. Roop. Variational formulation for the stationary fractional advection disper-
sion equation. Numerical Methods for Partial Differential Equations, 22(3):558–576, 2006. ISSN
0749159X. doi: 10.1002/num.20112.
72
[44] Y. Lin and C. Xu. Finite difference/spectral approximations for the time-fractional diffusion
equation. Journal of Computational Physics, 225(2):1533–1552, 2007. ISSN 00219991. doi:
10.1016/j.jcp.2007.02.001.
[45] M. Zayernouri, M. Ainsworth, and G. E. Karniadakis. A unified Petrov-Galerkin spectral method
for fractional PDEs. Computer Methods in Applied Mechanics and Engineering, 283:1545–1569,
2015. ISSN 00457825. doi: 10.1016/j.cma.2014.10.051.
[46] C.-c. J. Z.-z. Sun. A High-Order Compact Finite Difference Scheme for the Fractional Sub-diffusion
Equation. Journal of Scientific Computing, 64(11271068):959–985, 2015. ISSN 0885-7474. doi:
10.1007/s10915-014-9956-4. URL http://dx.doi.org/10.1007/s10915-014-9956-4.
[47] X. Hu and L. Zhang. An analysis of a second order difference scheme for the fractional subdif-
fusion system. Applied Mathematical Modelling, 40(2):1634–1649, 2016. ISSN 0307904X. doi:
10.1016/j.apm.2015.08.010. URL http://dx.doi.org/10.1016/j.apm.2015.08.010.
[48] M. M. Meerschaert and C. Tadjeran. Finite difference approximations for fractional advection-
dispersion flow equations. Journal of Computational and Applied Mathematics, 172(1):65–77,
2004. ISSN 03770427. doi: 10.1016/j.cam.2004.01.033.
[49] M. M. Meerschaert and C. Tadjeran. Finite difference approximations for two-sided space-
fractional partial differential equations. Applied Numerical Mathematics, 56(1):80–90, 2006. ISSN
01689274. doi: 10.1016/j.apnum.2005.02.008.
[50] M. M. Meerschaert, H. P. Scheffler, and C. Tadjeran. Finite difference methods for two-dimensional
fractional dispersion equation. Journal of Computational Physics, 211(1):249–261, 2006. ISSN
00219991. doi: 10.1016/j.jcp.2005.05.017.
[51] C. Tadjeran, M. M. Meerschaert, and H.-P. Scheffler. A second-order accurate numerical approx-
imation for the fractional diffusion equation. Journal of Computational Physics, 213(1):205–213,
2006. ISSN 00219991. doi: 10.1016/j.jcp.2005.08.008.
[52] W. Tian, H. Zhou, and W. Deng. A class of second order difference approximations for solving
space fractional diffusion equations. Mathematics of Computation, 84(294):1703–1727, 2015. doi:
10.1090/S0025-5718-2015-02917-2.
[53] H. Zhou, W. Tian, and W. Deng. Quasi-Compact Finite Difference Schemes for Space Fractional
Diffusion Equations. Journal of Scientific Computing, 56(1):45–66, 2012. ISSN 0885-7474. doi:
10.1007/s10915-012-9661-0. URL http://link.springer.com/10.1007/s10915-012-9661-0.
[54] M. Chen and W. Deng. Fourth Order Accurate Scheme for the Space Fractional Diffusion Equa-
tions. SIAM Journal on Numerical Analysis, 52(3):1418–1438, 2014. ISSN 0036-1429. doi:
10.1137/130933447.
[55] Z. peng Hao, Z. zhong Sun, and W. rong Cao. A fourth-order approximation of fractional derivatives
with its applications. Journal of Computational Physics, 281(11271068):787–805, 2015. ISSN
73
10902716. doi: 10.1016/j.jcp.2014.10.053. URL http://dx.doi.org/10.1016/j.jcp.2014.10.
053.
[56] H. Wang and N. Du. A superfast-preconditioned iterative method for steady-state space-fractional
diffusion equations. Journal of Computational Physics, 240:49–57, 2013. ISSN 00219991. doi:
10.1016/j.jcp.2012.07.045.
[57] F. Liu, P. Zhuang, V. Anh, and I. Turner. A fractional-order implicit difference approximation for the
space-time fractional diffusion equation. ANZIAM Journal, 47:C48—-C68, 2006. ISSN 1446-8735.
URL http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/1030.
[58] M. Chen, W. Deng, and Y. Wu. Superlinearly convergent algorithms for the two-dimensional
space-time Caputo-Riesz fractional diffusion equation. Applied Numerical Mathematics, 70:22–
41, 2013. ISSN 01689274. doi: 10.1016/j.apnum.2013.03.006. URL http://dx.doi.org/10.
1016/j.apnum.2013.03.006.
[59] H. Ding. General Pade approximation method for time–space fractional diffusion equation.
Journal of Computational and Applied Mathematics, 299(11561060):221–228, 2016. ISSN
03770427. doi: 10.1016/j.cam.2015.11.043. URL http://linkinghub.elsevier.com/retrieve/
pii/S0377042715006032.
[60] H. Sun, Z. Z. Sun, and G. H. Gao. Some high order difference schemes for the space and time
fractional Bloch-Torrey equations. Applied Mathematics and Computation, 281:356–380, 2016.
ISSN 00963003. doi: 10.1016/j.amc.2016.01.044. URL http://dx.doi.org/10.1016/j.amc.
2016.01.044.
[61] Z. Wang, S. Vong, and S.-L. Lei. Finite difference schemes for two-dimensional time-space frac-
tional differential equations. International Journal of Computer Mathematics, 93(3):578–595, mar
2016. ISSN 0020-7160. doi: 10.1080/00207160.2015.1009902. URL http://www.tandfonline.
com/doi/full/10.1080/00207160.2015.1009902.
[62] W. Tian, H. Zhou, and W. Deng. Approximations for Solving Space Fractional Diffusion Equations.
5718:1–25, 2015.
[63] H. K. Pang and H. W. Sun. Fourth order finite difference schemes for time-space fractional sub-
diffusion equations. Computers and Mathematics with Applications, 71(6):1287–1302, 2016. ISSN
08981221. doi: 10.1016/j.camwa.2016.02.011. URL http://dx.doi.org/10.1016/j.camwa.
2016.02.011.
[64] C. F. M. Coimbra. Mechanics with variable-order differential operators. Annalen der Physik, 12
(1112):692–703, 2003. ISSN 0003-3804. doi: 10.1002/andp.200310032. URL http://doi.
wiley.com/10.1002/andp.200310032.
[65] M. Caputo. Mean fractional-order-derivatives differential equations and filters. Annali
dell’Universita di Ferrara, 41(1):73–84. ISSN 0430-3202. doi: 10.1007/BF02826009.
74
[66] H. G. Sun, W. Chen, H. Wei, and Y. Q. Chen. A comparative study of constant-order and variable-
order fractional models in characterizing memory property of systems. European Physical Journal:
Special Topics, 193(1):185–192, 2011. ISSN 19516355. doi: 10.1140/epjst/e2011-01390-6.
[67] H. Sun, W. Chen, C. Li, and Y. Chen. Fractional differential models for anomalous diffusion.
Physica A: Statistical Mechanics and its Applications, 389(14):2719–2724, 2010. ISSN 03784371.
doi: 10.1016/j.physa.2010.02.030. URL http://dx.doi.org/10.1016/j.physa.2010.02.030.
[68] H. Sun, W. Chen, and Y. Chen. Variable-order fractional differential operators in anomalous
diffusion modeling. Physica A: Statistical Mechanics and its Applications, 388(21):4586–4592,
2009. ISSN 03784371. doi: 10.1016/j.physa.2009.07.024. URL http://dx.doi.org/10.1016/
j.physa.2009.07.024.
[69] S. Samko. Fractional integration and differentiation of variable order: an overview. Nonlinear
Dynamics, 71(4):653–662, mar 2013. ISSN 0924-090X. doi: 10.1007/s11071-012-0485-0. URL
http://link.springer.com/10.1007/s11071-012-0485-0.
[70] S. G. Samko and B. Ross. Integration and differentiation to a variable fractional order.
Integral Transforms and Special Functions, 1(4):277–300, dec 1993. ISSN 1065-2469.
doi: 10.1080/10652469308819027. URL http://www.tandfonline.com/doi/abs/10.1080/
10652469308819027.
[71] C. F. Lorenzo and T. T. Hartley. Variable order and distributed order fractional operators. Nonlinear
Dynamics, 29(1-4):57–98, 2002. ISSN 0924090X. doi: 10.1023/A:1016586905654.
[72] D. Ingman, J. Suzdalnitsky, and M. Zeifman. Constitutive Dynamic-Order Model for Nonlin-
ear Contact Phenomena. Journal of Applied Mechanics, 67(2):383, 2000. ISSN 00218936.
doi: 10.1115/1.1304916. URL http://appliedmechanics.asmedigitalcollection.asme.org/
article.aspx?articleid=1393293.
[73] D. Ingman and J. Suzdalnitsky. Application of Differential Operator with Servo-
Order Function in Model of Viscoelastic Deformation Process. Journal of Engi-
neering Mechanics, 131(7):763–767, jul 2005. ISSN 0733-9399. doi: 10.1061/
(ASCE)0733-9399(2005)131:7(763). URL http://ascelibrary.org/doi/10.1061/
%2528ASCE%25290733-9399%25282005%2529131%253A7%2528763%2529.
[74] H. Pedro, M. Kobayashi, J. Pereira, and C. Coimbra. Variable Order Modeling of Diffusive-
convective Effects on the Oscillatory Flow Past a Sphere. Journal of Vibration and Control,
14(9-10):1659–1672, sep 2008. ISSN 1077-5463. doi: 10.1177/1077546307087397. URL
http://jvc.sagepub.com/cgi/doi/10.1177/1077546307087397.
[75] Y. L. Kobelev, L. Y. Kobelev, and Y. L. Klimontovich. Statistical physics of dynamic systems with
variable memory. Doklady Physics, 48(6):285–289, jun 2003. ISSN 1028-3358. doi: 10.1134/1.
1591315. URL http://link.springer.com/10.1134/1.1591315.
75
[76] A. V. Chechkin, R. Gorenflo, and I. M. Sokolov. Fractional diffusion in inhomogeneous media.
Journal of Physics A: Mathematical and General, 38(42):L679–L684, 2005. ISSN 0305-4470. doi:
10.1088/0305-4470/38/42/L03.
[77] P. S. Addison, B. Qu, A. S. Ndumu, and I. C. Pyrah2. A Particle Tracking Model for Non-Fickian
Subsurface Diffusion. Mathematical Geology, 30(6), 1998.
[78] I. Sokolov, A. Chechkin, and J. Klafter. Fractional diffusion equation for a power-law-truncated
Levy process. Physica A: Statistical Mechanics and its Applications, 336(3):245–251, 2004. ISSN
03784371. doi: 10.1016/j.physa.2003.12.044.
[79] A. V. Chechkin, V. Yu Gonchar, R. Gorenflo, N. Korabel, and I. M. Sokolov. Generalized fractional
diffusion equations for accelerating subdiffusion and truncated Levy flights. Physical Review E,
78:021111, 2008. doi: 10.1103/PhysRevE.78.021111.
[80] O. O. Vaneeva, A. G. Johnpillai, R. O. Popovych, C. Sophocleous, and S. G. Krantz. Enhanced
group analysis and conservation laws of variable coefficient reaction–diffusion equations with
power nonlinearities. J. Math. Anal. Appl. O.O. Vaneeva et al. J. Math. Anal. Appl, 330(330):
1363–1386, 2007. doi: 10.1016/j.jmaa.2006.08.056. URL www.elsevier.com/locate/jmaa.
[81] E. K. Lenzi, R. S. Mendes, K. Sau Fa, L. R. da Silva, and L. S. Lucena. Solutions for a fractional
nonlinear diffusion equation: Spatial time dependent diffusion coefficient and external forces.
Journal of Mathematical Physics J. Math. Phys. J. Math. Phys. J. Math. Phys, 45(46):103302–
83506, 2004. doi: 10.1063/1.1768619. URL http://dx.doi.org/10.1063/1.1768619http:
//scitation.aip.org/content/aip/journal/jmp/45/9?ver=pdfcov.
[82] V. Anh, J. Angulo, and M. Ruiz-Medina. Diffusion on multifractals. Nonlinear Analysis: Theory,
Methods & Applications, 63(5):e2043–e2056, 2005. ISSN 0362546X. doi: 10.1016/j.na.2005.02.
107.
[83] M. Caputo. Diffusion with space memory modelled with distributed order space fractional differen-
tial equations. Annals of Geophysics, 46(2), 2003. ISSN 2037-416X. doi: 10.4401/ag-3395. URL
http://www.annalsofgeophysics.eu/index.php/annals/article/view/3395.
[84] T. S. Choong, T. Wong, T. Chuah, and A. Idris. Film-pore-concentration-dependent surface diffu-
sion model for the adsorption of dye onto palm kernel shell activated carbon. Journal of Colloid
and Interface Science, 301(2):436–440, 2006. ISSN 00219797. doi: 10.1016/j.jcis.2006.05.033.
[85] S. Blackband and P. Mansfield. Diffusion in liquid-solid systems by NMR imaging. Journal of
Physics C: Solid State Physics, 19(2):L49–L52, jan 1986. ISSN 0022-3719. doi: 10.1088/
0022-3719/19/2/004. URL http://stacks.iop.org/0022-3719/19/i=2/a=004?key=crossref.
95f07962c8d2539f9633e9d439d53b69.
[86] E. N. De Azevedo, P. L. De Sousa, R. E. De Souza, M. Engelsberg, M. De, N. Do, N. Miranda, and
M. A. Silva. Concentration-dependent diffusivity and anomalous diffusion: A magnetic resonance
76
imaging study of water ingress in porous zeolite. Physical Review E, 73:011204, 2006. doi:
10.1103/PhysRevE.73.011204.
[87] J. Crank. THE MATHEMATICS OF DIFFUSION. Clarendon Press, Oxford, 1975.
[88] M. Zayernouri and G. E. Karniadakis. Fractional spectral collocation methods for linear and
nonlinear variable order FPDEs. Journal of Computational Physics, 293:312–338, 2015. ISSN
00219991. doi: 10.1016/j.jcp.2014.12.001.
[89] C. Soon, C. Coimbra, and M. Kobayashi. The variable viscoelasticity oscillator. Annalen der
Physik, 14(6):378–389, jun 2005. ISSN 0003-3804. doi: 10.1002/andp.200410140. URL http:
//doi.wiley.com/10.1002/andp.200410140.
[90] P. Zhuang, F. Liu, V. Anh, and I. Turner. Numerical Methods for the Variable-Order Fractional
Advection-Diffusion Equation with a Nonlinear Source Term. SIAM Journal on Numerical Analysis,
47(3):1760–1781, jan 2009. ISSN 0036-1429. doi: 10.1137/080730597. URL http://epubs.
siam.org/doi/abs/10.1137/080730597.
[91] R. Lin, F. Liu, V. Anh, and I. Turner. Stability and convergence of a new explicit finite-difference
approximation for the variable-order nonlinear fractional diffusion equation. Applied Mathematics
and Computation, 212(2):435–445, 2009. ISSN 00963003. doi: 10.1016/j.amc.2009.02.047.
[92] C.-M. Chen, F. Liu, V. Anh, and I. Turner. Numerical Schemes with High Spatial Accuracy for a
Variable-Order Anomalous Subdiffusion Equation. SIAM Journal on Scientific Computing, 32(4):
1740–1760, jan 2010. ISSN 1064-8275. doi: 10.1137/090771715. URL http://epubs.siam.
org/doi/abs/10.1137/090771715.
[93] C.-M. Chen, F. Liu, I. Turner, and V. Anh. Numerical methods with fourth-order spatial accuracy
for variable-order nonlinear Stokes’ first problem for a heated generalized second grade fluid.
Computers & Mathematics with Applications, 62(3):971–986, 2011. ISSN 08981221. doi: 10.
1016/j.camwa.2011.03.065.
[94] C.-M. Chen, F. Liu, I. Turner, V. Anh, and Y. Chen. Numerical approximation for a variable-
order nonlinear reaction–subdiffusion equation. Numerical Algorithms, 63(2):265–290, jun 2013.
ISSN 1017-1398. doi: 10.1007/s11075-012-9622-6. URL http://link.springer.com/10.1007/
s11075-012-9622-6.
[95] C.-M. Chen, F. Liu, V. Anh, and I. Turner. Numerical methods for solving a two-dimensional
variable-order anomalous subdiffusion equation. Mathematics of Computation, 81(277):345–366,
jan 2012. ISSN 0025-5718. doi: 10.1090/S0025-5718-2011-02447-6. URL http://www.ams.
org/jourcgi/jour-getitem?pii=S0025-5718-2011-02447-6.
[96] S. Chen, F. Liu, and K. Burrage. Numerical simulation of a new two-dimensional variable-order
fractional percolation equation in non-homogeneous porous media. Computers & Mathematics
with Applications, 67(9):1673–1681, 2014. ISSN 08981221. doi: 10.1016/j.camwa.2014.03.003.
77
[97] H. Sun, W. Chen, C. Li, and Y. Chen. Finite Difference Schemes for Variable-Order Time Fractional
Diffusion Equation. International Journal of Bifurcation and Chaos, 22(04):1250085, apr 2012.
ISSN 0218-1274. doi: 10.1142/S021812741250085X. URL http://www.worldscientific.com/
doi/abs/10.1142/S021812741250085X.
[98] S. Shen, F. Liu, J. Chen, I. Turner, and V. Anh. Numerical techniques for the variable order
time fractional diffusion equation. Applied Mathematics and Computation, 218(22):10861–10870,
2012. ISSN 00963003. doi: 10.1016/j.amc.2012.04.047.
[99] S. Shen, F. Liu, V. Anh, I. Turner, and J. Chen. A characteristic difference method for the variable-
order fractional advection-diffusion equation. Journal of Applied Mathematics and Computing,
42(1-2):371–386, jul 2013. ISSN 1598-5865. doi: 10.1007/s12190-012-0642-0. URL http:
//link.springer.com/10.1007/s12190-012-0642-0.
[100] H. Zhang, F. Liu, M. S. Phanikumar, and M. M. Meerschaert. A novel numerical method for the time
variable fractional order mobile–immobile advection–dispersion model. Computers & Mathematics
with Applications, 66(5):693–701, 2013. ISSN 08981221. doi: 10.1016/j.camwa.2013.01.031.
[101] X. Zhao, Z.-z. Sun, and G. E. Karniadakis. Second-order approximations for variable order frac-
tional derivatives: Algorithms and applications. Journal of Computational Physics, 293:184–200,
2015. ISSN 00219991. doi: 10.1016/j.jcp.2014.08.015.
[102] L. E. S. Ramirez and C. F. M. Coimbra. On the selection and meaning of variable order oper-
ators for dynamic modeling. International Journal of Differential Equations, 2010, 2010. ISSN
16879643. doi: 10.1155/2010/846107.
[103] H. Zhang, F. Liu, M. S. Phanikumar, and M. M. Meerschaert. A novel numerical method for
the time variable fractional order mobile-immobile advection-dispersion model. Computers and
Mathematics with Applications, 66(5):693–701, 2013. ISSN 08981221. doi: 10.1016/j.camwa.
2013.01.031. URL http://dx.doi.org/10.1016/j.camwa.2013.01.031.
[104] C. Li and F. Zeng. Numerical Methods Fractional Calculus for Fractional Calculus. CRC Press,
2015. ISBN 9781482253801.
[105] C. Lubich. Discretized Fractional Calculus. SIAM Journal on Mathematical Analysis, 17(3):704–
719, may 1986. ISSN 0036-1410. doi: 10.1137/0517050. URL http://epubs.siam.org/doi/
abs/10.1137/0517050.
[106] Z. Li, L. Liu, S. Dehghan, Y. Chen, and D. Xue. A review and evaluation of numerical tools
for fractional calculus and fractional order control. arXiv:1511.07521 [cs], 7179(March), 2015.
ISSN 13665820. doi: 10.1080/00207179.2015.1124290. URL http://arxiv.org/abs/1511.
07521%5Cnhttp://www.arxiv.org/pdf/1511.07521.pdf.
[107] C. Tadjeran, M. M. Meerschaert, and H.-P. Scheffler. A second-order accurate numerical approx-
imation for the fractional diffusion equation. Journal of Computational Physics, 213(1):205–213,
78
2006. ISSN 00219991. doi: 10.1016/j.jcp.2005.08.008. URL http://www.sciencedirect.com/
science/article/pii/S0021999105003773.
[108] D. Ingman and J. Suzdalnitsky. Control of damping oscillations by fractional differential op-
erator with time-dependent order. Computer Methods in Applied Mechanics and Engineer-
ing, 193(52):5585–5595, dec 2004. ISSN 00457825. doi: 10.1016/j.cma.2004.06.029. URL
http://linkinghub.elsevier.com/retrieve/pii/S0045782504002944.
[109] D. Valerio and J. Sa da Costa. Variable-order fractional derivatives and their numerical approxi-
mations. Signal Processing, 91(3):470–483, 2011. ISSN 01651684. doi: 10.1016/j.sigpro.2010.
04.006.
[110] O. Santos and et al. A comparative survey of finite difference methods for the solution of time,
space and time-space fractional diffusion equations. paper in submission, October 2016.
[111] O. Santos and et al. Influence of time, space and solution dependent variable order in the be-
haviour of subdiffusive systems. paper in submission, October 2016.
79
80
Appendix A
Schemes for fractional diffusion
equations in matrix form
A.1 Time Fractional Diffusion Equations
A.1.1 First Order Weighted Average Scheme
Equation (2.4.9) can be given in matrix form by
(E − µ(1− θ)S)un+1 = un + µ
n∑k=0
[θω(1−γ)k + (1− θ)ω(1−γ)
k ]Sun−k
+ µθ
n∑k=0
ω(1−γ)k Xun−kb + µ(1− θ)
n+1∑k=0
ω(1−γ)k Xun+1−k
b + τθfn + τ(1− θ)fn+1 (A.1.1)
where µ =Kγτ
γ
h2, ω(1−γ)
k is given by equation (2.2.3), un = [uni , un2 , · · · , unM−1]T , unb = [un0 , u
nM ]T ,
fn = [fni , fn2 , · · · , fnM−1]T , E = IM−1 and
S =
−2 1 0 · · · 0 0
1 −2 1 · · · 0 0
0 1 −2 · · · 0 0...
......
. . ....
...
0 0 0 · · · −2 1
0 0 0 · · · 1 −2
(M−1)×(M−1)
X =
1 0
0 0...
...
0 0
0 1
(M−1)×2
A.1.2 Second Order Finite Difference Scheme
Equation (2.4.25) can be written in matrix form as
A.1
[E − rυ(γ)0 S]un+1 = un + rυ(γ)0 Xun+1
b
+ r
n−1∑k=0
(ω(γ)k − ω(γ)
k−1 + υ(γ)k+1 − υ
(γ)k )[Sun−k +Xun−kb ] + r(ω(γ)
n − ω(γ)n−1[Su0 −Xu0b ] +
τ
2(fn + fn+1]
(A.1.2)
with r , ω(γ)k and υ
(γ)k defined as in equations (2.4.20), (2.4.21) and (2.4.22), respectively. Additionally
un = [uni , un2 , · · · , unM−1]T , unb = [un0 , u
nM ]T , fn = [fni , f
n2 , · · · , fnM−1]T , E = IM−1 and
S =
−2 1 0 · · · 0 0
1 −2 1 · · · 0 0
0 1 −2 · · · 0 0...
......
. . ....
...
0 0 0 · · · −2 1
0 0 0 · · · 1 −2
(M−1)×(M−1)
X =
1 0
0 0...
...
0 0
0 1
(M−1)×2
A.1.3 Third Order Finite Difference Scheme
Equation (2.4.41) can be written in matrix form as
(E +h2
12S)u1 = −h
2
12Xu1b =
2Kγτγ
Γ(γ + 3)(Su1 +Xu1b) + (E +
h2
12S)(F 1 +XF 1
b) (A.1.3)
and equation (2.4.40) as
gγ0 (E +h2
12S)un = −
n∑k=1
gγk (E +h2
12S)un−k − (
h2
12)
n∑k=0
gγkXun−kb +
Kγτγ
h2(Sun + unb ) + τγ(E +
h2
12S)fn +
h2τγ
12Xfnb (A.1.4)
where gγk is defined as in equation (2.2.17), un = [uni , un2 , · · · , unM−1]T , unb = [un0 , u
nM ]T , fn = [fni , f
n2 , · · · , fnM−1]T ,
fnb
= [fn0 , fnM ]T ,Fn = [Fni , F
n2 , · · · , FnM−1]T ,Fnb = [Fn0 , F
nM ]T , E = IM−1 and
S =
−2 1 0 · · · 0 0
1 −2 1 · · · 0 0
0 1 −2 · · · 0 0...
......
. . ....
...
0 0 0 · · · −2 1
0 0 0 · · · 1 −2
(M−1)×(M−1)
X =
1 0
0 0...
...
0 0
0 1
(M−1)×2
A.2
A.2 Space Fractional Diffusion Equations
A.2.1 First Order Finite Difference Scheme
Equation (2.5.6) can be written in matrix form as
(E − µ1A− µ2AT )un+1 = un + τfn+1 + µ1
ω(α)2 0...
...
ω(α)M ω
(α)0
un+1b + µ2
ω(α)0 ω
(α)M
......
0 ω(α)2
un+1b (A.2.1)
where µ1 =K1τ
hα, µ2 =
K2τ
hα, ω(1−γ)
k is given by equation (2.2.3), un = [uni , un2 , · · · , unM−1]T , unb =
[un0 , unM ]T , fn = [fni , f
n2 , · · · , fnM−1]T , E = IM−1 and
A =
ω(α)1 ω
(α)0 0 · · · 0 0
ω(α)2 ω
(α)1 ω
(α)0 · · · 0 0
ω(α)3 ω
(α)2 ω
(α)1 · · · 0 0
......
.... . .
......
ω(α)M−2 ω
(α)M−3 ω
(α)M−4 · · · ω
(α)1 ω
(α)0
ω(α)M−1 ω
(α)M−2 ω
(α)M−3 · · · ω
(α)2 ω
(α)1
(M−1)×(M−1)
A.2.2 Second Order Finite Difference Scheme
Equation (2.5.9) can be written in matrix form as
[E − τ
2hα(K1A+K2A
T )]un+1 = [E +τ
2hα(K1A+K2A
T )]un +τ
2(fn+1 + fn) +Hn (A.2.2)
where un = [uni , un2 , · · · , unM−1]T , fn = [fni , f
n2 , · · · , fnM−1]T , E = IM−1. Furthermore
A =
g(α)1 g
(α)0 0 · · · 0 0
g(α)2 g
(α)1 g
(α)0 · · · 0 0
g(α)3 g
(α)2 g
(α)1 · · · 0 0
......
.... . .
......
g(α)M−2 g
(α)M−3 g
(α)M−4 · · · g
(α)1 g
(α)0
g(α)M−1 g
(α)M−2 g
(α)M−3 · · · g
(α)2 g
(α)1
(M−1)×(M−1)
A.3
with g(α)k defined in equation (2.2.36) and
H =τ
2hα
K1g(α)2 +K2g
(α)0
K1g(α)3
...
K1g(α)M−1
K1g(α)M
(un0 + un+1
0 ) +τ
2hα
K2g(α)M
K2g(α)M−1...
K2g(α)3
K2g(α)2 +K1g
(α)0
(unM + un+1
M )
A.2.3 Fourth Order Finite Difference Scheme
Equation (2.5.15) can be written in matrix form as
(E+cαS−µ1A−µ2AT )un+1 = (E+cαS+µ1A+µ2A
T )un+(τ
2)(E+cαS)(fn+1+fn)+(
τ
2)(cαX)(fn+1
b+fn
b)
(A.2.3)
where µ1 =K1τ
2hα,µ2 =
K2τ
2hα, cα is defined in equation (2.2.41), un = [uni , u
n2 , · · · , unM−1]T , fn =
[fni , fn2 , · · · , fnM−1]T , fn
b= [fn0 , f
nM ]T , E = IM−1 and
S =
−2 1 0 · · · 0 0
1 −2 1 · · · 0 0
0 1 −2 · · · 0 0...
......
. . ....
...
0 0 0 · · · −2 1
0 0 0 · · · 1 −2
(M−1)×(M−1)
X =
1 0
0 0...
...
0 0
0 1
(M−1)×2
A =
g(α)1 g
(α)0 0 · · · 0 0
g(α)2 g
(α)1 g
(α)0 · · · 0 0
g(α)3 g
(α)2 g
(α)1 · · · 0 0
......
.... . .
......
g(α)M−2 g
(α)M−3 g
(α)M−4 · · · g
(α)1 g
(α)0
g(α)M−1 g
(α)M−2 g
(α)M−3 · · · g
(α)2 g
(α)1
(M−1)×(M−1)
where the coefficients g(α)k were defined in equation (2.2.46).
A.4
A.3 Time-space Fractional Diffusion Equations
A.3.1 First Order Finite Difference Scheme
Equation (2.6.8) can by given in matrix form as
(b0E − µ1A− µ2AT )un = b0u
n−1 −n−1∑k=1
(bn−k−1 − bn−k)uk + bn−1u0
+ µ1
ω(α)2 0...
...
ω(α)M ω
(α)0
unb + µ2
ω(α)0 ω
(α)M
......
0 ω(α)2
unb + τγfn (A.3.1)
where µ1 =K1τ
γ
hα, µ2 =
K2τγ
hα, ω(1−γ)
k is given by equation (2.2.3), bk is given by (2.2.6), un =
[uni , un2 , · · · , unM−1]T , unb = [un0 , u
nM ]T , fn = [fni , f
n2 , · · · , fnM−1]T , E = IM−1 and
A =
ω(α)1 ω
(α)0 0 · · · 0 0
ω(α)2 ω
(α)1 ω
(α)0 · · · 0 0
ω(α)3 ω
(α)2 ω
(α)1 · · · 0 0
......
.... . .
......
ω(α)M−2 ω
(α)M−3 ω
(α)M−4 · · · ω
(α)1 ω
(α)0
ω(α)M−1 ω
(α)M−2 ω
(α)M−3 · · · ω
(α)2 ω
(α)1
(M−1)×(M−1)
A.3.2 Second Order Finite Difference Scheme
Equation (2.6.15) can by given in matrix form as
(b0E − µ1A− µ2AT )un = b0u
n−1 −n−1∑k=1
(bn−k−1 − bn−k)uk + bn−1u0
+ µ1
g(α)2 0...
...
g(α)M g
(α)0
unb + µ2
g(α)0 g
(α)M
......
0 g(α)2
unb + τγfn (A.3.2)
where µ1 =K1τ
γ
hα, µ2 =
K2τγ
hα, g(α)k is defined in equation (2.2.36), bk is given by (2.2.6), un =
[uni , un2 , · · · , unM−1]T , unb = [un0 , u
nM ]T , fn = [fni , f
n2 , · · · , fnM−1]T , E = IM−1 and
A =
g(α)1 g
(α)0 0 · · · 0 0
g(α)2 g
(α)1 g
(α)0 · · · 0 0
g(α)3 g
(α)2 g
(α)1 · · · 0 0
......
.... . .
......
g(α)M−2 g
(α)M−3 g
(α)M−4 · · · g
(α)1 g
(α)0
g(α)M−1 g
(α)M−2 g
(α)M−3 · · · g
(α)2 g
(α)1
(M−1)×(M−1)
A.5
A.3.3 Fourth Order Finite Difference Scheme
Equation (2.6.24) can be written in matrix form as
(E + cαS −K1µA−K2µAT )un = (E + cαS)
n−1∑k=0
(bn−k−1 − bn−k)uk
+ (E + cαS)bn−1u0 − cαXunb + cαX
n−1∑k=0
(bn−k−1 − bn−k)ukb
+ cαbn−1Xu0b + µ(E + cαS)fn + µcαXf
n
b
+K1µ
g(α)2 0...
...
g(α)M g
(α)0
unb +K2µ
g(α)0 g
(α)M
......
0 g(α)2
unb (A.3.3)
where µ = τγΓ(2 − γ), cα is defined in equation (2.2.41), bk is given by (2.2.6), the coefficients g(α)k
were defined in equation (2.2.46), un = [uni , un2 , · · · , unM−1]T , unb = [un0 , u
nM ]T , fn = [fni , f
n2 , · · · , fnM−1]T ,
fnb
= [fn0 , fnM ]T , E = IM−1 and
S =
−2 1 0 · · · 0 0
1 −2 1 · · · 0 0
0 1 −2 · · · 0 0...
......
. . ....
...
0 0 0 · · · −2 1
0 0 0 · · · 1 −2
(M−1)×(M−1)
X =
1 0
0 0...
...
0 0
0 1
(M−1)×2
A =
g(α)1 g
(α)0 0 · · · 0 0
g(α)2 g
(α)1 g
(α)0 · · · 0 0
g(α)3 g
(α)2 g
(α)1 · · · 0 0
......
.... . .
......
g(α)M−2 g
(α)M−3 g
(α)M−4 · · · g
(α)1 g
(α)0
g(α)M−1 g
(α)M−2 g
(α)M−3 · · · g
(α)2 g
(α)1
(M−1)×(M−1)
A.6