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Chapter 1

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Chapter 1

INTRODUCTION

1.1 History of Fractional Calculus

Fractional Calculus deals with the study of so-called fractional order integral and derivative

operators over real or complex domains, and their applications. The definitions of the frac-

tional derivative (which generalizes the ordinary differential operator) are diverse. Questions

such as ‘what does the 1/3 or√

2 derivative of a function mean?’, and ‘How can these Frac-

tional operators be applied, losing as they do so many fundamental properties with respect

to ordinary derivatives?’ have been the center of attention of many important scientists over

the last two centuries. A rigorous, encyclopedic study of Fractional Calculus was written by

Samko, Kilbas and Marichev in 1993 [101].

In his discovery of calculus, Leibnitz introduced the symbol for the nth derivative dnydxn ≡

Dny, where n is a non-negative integer. Can the meaning of derivative of integral order

be extended to have meaning when n is any number - fractional, irrotational or complex?

L’Hospital asked Leibnitz about the possibility that n be a fraction. What if n = 12?

Lacroix (1816) discussed the following results in his 700 pages long book on calculus:[72]

Dnxm =m!

(n−m)!xm−n, D ≡ d

dx,

where n(≥ 0) and m are integers.

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For fractional number α and β,

Dαxβ =Γ(β + 1)

Γ(β − α+ 1)xβ−x

D12x =

Γ(2)

Γ(32)x

12 =

2√πx

12 (Lacroix, 1816)

Riemann-Liouville’s definition (1847) of integraion of arbitrary order α is

cD−αx f(x) =

1

Γ(2)

∫ x

c

(x− t)α−1f(t)dt, α ≥ 0,

where α is fractional, irrotational or complex number.

For a complex function f(z) of a complex variable z = x+ iy,

cDαy f(z) = g(z)

cDαxf(x) = g(x)

Abel (1820) considered the problem of tantochronous motion which involved fractional

integral of order 12. The Abel integral equation also involves integral of fractional order

0 < α < 1.

Euler’s integral formula in real analysis and Cauchy’s integral formula in complex analy-

sis also involve the ideas of fractional integral of arbitrary order α and fractional derivatives

of arbitrary order α.

The Euler integral formula for the hypergeometric function 2F1(a, b, c;x) can be expressed

as a fractional integral of order (c− b).


A fractional derivative is nothing more than an operator which generalizes the ordinary

derivative, such that if the derivative is represented by the operator Dα then, when α = 1,

it lets us get back to the ordinary differential operator D. Much more attention has been

given to fractional calculus and its applications for the last three decades. In recent years

considerable interest in fractional calculus has been stimulated by the applications that this

calculus finds in numerical analysis and different areas of physics and engineering, possibly

including fractal phenomena.

During time fractional calculus was built on formal foundations by many famous mathe-

maticians, Laplace (1812), Fourier (1822), Abel (1823- 1826), Liouville (1832-1837), Riemann

(1847), Grunwald (1867-1872), Letnikov (1868- 1872), Heaviside (1892-1912), Weyl (1917),

Erdelyi (1939-1965) and many others like Hardy (1917), Hardy and Littlewood (1925, 1928,

1932), Kober (1940), and Kuttner (1953) examined some rather special, but natural, proper-

ties of differintegrals of functions belonging to Lebesgue and Lipschitz classes. Erdelyi (1939,

1940, 1954) and Osler (1970a) have given definitions of differintegrals with respect to arbi-

trary functions, and Post (1930) used difference quotients to define generalized differentiation

for operators f(D), where D denotes differentiation and f is a suitably restricted function.

Riesz (1949) has developed a theory of fractional integration for functions of more than one

variable. Erdelyi (1964, 1965) has applied the fractional calculus to integral equations and

Higgins (1967) has used fractional integral operators to solve differential equations. Other

applications include those to rheology (Scott Blair et al., 1947; Shermergor, 1966; Scott

Blair, 1947, 1950a,b; Scott Blair and Caffyn, 1949; Graham et al, 1961), to electrochemistry

(Belavin et al., 1964; Oldham, 1969a; Oldham and Spanier, 1970; Grenness and Oldham,

1972), and to general transport problems (Oldham, 1973b; Oldham and Spanier, 1972)and

many others (cf.Gorenflo and Mainardi [56]).


1.2 Definitions of Fractional Calculus

Some definitions of fractional calculus are as follows:

1. L. Euler (1730)

Euler generalized the formula

dnxm

dxn= m(m− 1)...(m− n+ 1)xm−n

by using of the following property of Gamma function,

Γ(m+ 1) = m(m− 1)...(m− n+ 1)Γ(m− n+ 1)

to obtain

dnxm

dxn=

Γ(m+ 1)

Γ(m− n+ 1)xm−n

Gamma function is defined as follows.

Γ(z) =

∫ ∞

0

e−ttz−1dt, Re(z) > 0

2. J. B. J. Fourier (1820-1822):

By means of integral representation

f(x) =1

2π

∫ ∞

−∞f(z)dz

∫ ∞

−∞cos(px− pz)dp

he wrote

dnf(x)

dxn=

1

2π

∫ ∞

−∞f(z)dz

∫ ∞

−∞cos(px− pz + n

π

2)dp

3. N. H. Abel (1823- 1826):

Abel considered the integral representation∫ x

0s′(η)dη

(x−η)α = ψ(x) for arbitrary α and then

wrote

s(x) =1

Γ(1− α)

d−αψ(x)

dx−α


4. J. Lioville (1832 - 1855):

I. In his first definition, according to exponential representation of a function

f(x) =∞∑

n=0

cneanx, he generalized the formula dmf(x)

dxv = ameax as

dvf(x)

dxv=

∞∑n=0

cneanx

II. Second type of his definition was Fractional Integral∫ µ

Φ(x)dxµ =1

(−1)µΓ(µ)

∫ ∞

0

Φ(x+ α)αµ−1dα∫ µ

Φ(x)dxµ =1

Γ(µ)

∫ ∞

0

Φ(x− α)αµ−1dα

By substituting of τ = x+ α and τ = x− α in the above formulas respectively, he obtained∫ µ

Φ(x)dxµ =1

(−1)µΓ(µ)

∫ ∞

x

(τ − x)µ−1Φ(τ)dτ∫ µ

Φ(x)dxµ =1

Γ(µ)

∫ x

−∞(x− τ)µ−1Φ(τ)dτ.

III. Third definition, includes Fractional Derivative,

dnF (x)

dxµ=

(−1)µ

hµ

(F (x)

µ

1F (x+ h) + ...+

µ(µ− 1)

1.2F (x+ 2h)− ...

)dnF (x)

dxµ=

1

hµ

(F (x)

µ

1F (x− h) + ...+

µ(µ− 1)

1.2F (x− 2h)− ...

)5. G. F. B. Riemann (1847 - 1876):

His definition of Fractional Integral is

D−vf(x) =1

Γ(v)

∫ x

c

(x− t)v−1f(t)dt+ ψ(t)

6. N. Ya. Sonin (1869), A. V. Letnikov (1872), H. Laurent (1884):

They considered to the Cauchy Integral formula

f (n)(z) =n!

2πi

∫c

f(t)

(t− z)n+1dt

and substituted n by v to obtain

Dvf(z) =Γ(v + 1)

2πi

∫ x+

c

f(t)

(t− z)v+1dt


7. Riemann-Liouvill definition:

The popular definition of fractional calculus is this which shows joining of two previous

definitions.

aDαt f(t) = 1

Γ(n−α)

(ddt

)n ∫ t

af(τ)dτ

(t−τ)α−n+1

(n− 1 ≤ α < n)

8. Grunwald-Letnikove:

This is another joined definition which is sometimes useful.

aDαt f(t) = lim

h→0h−α

[ t−ah ]∑

j=0

(−1)j

α

j

f(t− jh)

9. M. Caputo (1967):

The second popular definition is

Ca D

αt f(t) =

1

Γ(α− n)

∫ t

a

f (n)(τ)dτ

(t− τ)α+1−n, (n− 1 ≤ α < n)

10. K. S. Miller, B. Ross (1993):

They used differential operator D as

Dαf(t) = Dα1Dα2 ...Dαnf(t) = (α1, α2...αn)

which Dαi is Riemann-Liouvill or Caputo definitions.

Fractional calculus is one of the most intensively developing areas of the mathematical

analysis as a result of its increasing range of applications. Operators for fractional differen-

tiation and integration have been used in various fields such as: fluid flow, viscoelasticity,

control theory of dynamical systems, electrical networks, probability and statistics, dynami-

cal processes in porous structures, electrochemistry of corrosion, optics and signal processing,

rheology, hydraulics of dams, potential fields, diffusion problems and waves in liquids and

gases,etc.

The use of half-order derivatives and integrals leads to a formulation of certain electro-

chemical problems which is more economical and useful than the classical approach in terms

of Fick’s law of diffusion(cf. Crank [43]). The main advantage of the fractional calculus is


that the fractional derivatives provide an excellent instrument for the description of mem-

ory and hereditary properties of various materials and processes. Now a days it includes

irrational and complex orders, whereas originally it considered only orders in the real line.

In special treaties the mathematical aspects and applications of the fractional calculus are

extensively discussed. In particular, this discipline involves the notion and methods of solv-

ing of differential equations involving fractional derivatives of the unknown function (called

fractional differential equations).

1.3 Riemann-Liouville Fractional Integral

In this section, we discuss the various aspects of the Riemann-Liouville fractional integral.

We begin with a formal definition ( see Definition 1.3.1 below).

Let X be a positive number and let f be continuous on [0, X]. Then if ν ≥ 1,∫ t

0

(t− ξ)ν−1f(ξ)dξ (1.3.1)

exists as a Riemann integral for all t ∈ [0, X]. Of course, (1.3.1) will exist under more general

conditions. For example, if f is continuous on (0, X] and behaves like tλ for −1 < λ < 0

in a neighbourhood of the origin and if 0 < Re ν < 1, then (1.3.1) exists as an improper

Riemann integral. The following definition, however, is sufficiently broad for our purposes.

Definition 1.3.1. Let Re ν > 0 and let f be piecewise continuous on J ′ = (0,∞) and

integrable on any finite subinterval of J = [0,∞). Then for t > 0 we call

0D−νt f(t) =

1

Γ(ν)

∫ t

0

(t− ξ)ν−1f(ξ)dξ (1.3.2)

the Riemann-Liouville fractional integral of f of order ν.

Let us discuss this definition. As we have observed above, (1.3.2) is an improper integral

if 0 < Re ν < 1. We require f to be piecewise continuous only on J ′ = (0,∞) (the interval J

excluding the origin) to accommodate functions that behave like ln t or tµ (for −1 < µ < 0)

in a neighbourhood of the origin. We shall denote by C the class of functions described in

Definition 1.3.1. [One readily may generalize C to include, for example, such functions as

f(ξ) = |ξ − a|λ, λ > −1, 0 < a < t. We seldom shall have occasion to do so].


For example, if f(t) = tµ with µ > −1, then

0D−νt tµ =

Γ(µ+ 1)

Γ(µ+ ν + 1)tµ+ν , t > 0 (1.3.3)

[since (1.3.2) is now essentially the beta function]. Because µ+Re ν may be negative, we see

from this example why we must include the caveat t > 0 in our definition of the fractional

integral. [Of course, if µ ≥ 0, then (1.3.3) is continuous on J ]. To avoid minor mathematical

complications not related to the fractional calculus, and with little loss of generality, we

shall, as a practical matter, assume that ν is real. Occasionally, we indicate that certain

formulas are valid for Re ν > 0 rather than just for ν > 0.

If we write (1.3.2) as the Stieltjes integral

0D−νt f(t) =

1

Γ(ν + 1)

∫ t

0

f(ξ)dα(ξ),

where

α(ξ) = −(t− ξ)ν (1.3.4)

is a (continuous) monotonic increasing function of ξ on [0, t], then if f is continuous on [0, t],

the first mean value theorem for integrals implies that∫ t

0

f(ξ)dα(ξ) = f(x)tν

for some x ∈ [0, t]. Hence

limt→0

0D−νt f(t) = 0. (1.3.5)

If f is not continuous (but still of class C), then (1.3.5) need not be true. In fact, we see

from (1.3.3) with ν > 0, µ > −1, that

limt→0

0D−νt tµ =

0, µ+ ν > 0

Γ(µ+ 1), µ+ ν = 0

∞, µ+ ν < 0.

Furthermore, we also conclude from (1.3.3) that even the continuity of f at the origin does not

guarantee, the differentiability of 0D−νt f(t) at t=0. ( For example, let µ > 0 and µ+ ν < 1).


1.4 Caputo’s Fractional Derivative

Suppose that α > 0, t > a, and α, a, t ∈ R. The operator of fractional calculus

Dα∗ f(t) =

1

Γ(n−α)

∫ t

af (n)(τ)

(t−τ)α+1−ndτ, n− 1 < α < n ∈ N,

dn

dtnf(t), α = n ∈ N,

is called the Caputo fractional derivative or Caputo differential operator of fractional calculus

of order α: This operator is introduced by the Italian mathematician Caputo in 1967 [34].

For simplicity we consider the case a = 0 in our thesis.

The definition

aDpt f(t) =

1

Γ(k − p)

dk

dtk

∫ t

0

(t− τ)k−p−1f(τ)dτ, (k − 1 ≤ p < k) (1.4.1)

of the fractional differentiation of the Riemann-Liouville type played an important role in

the development of the theory of fractional derivatives and integrals and for its applications

in pure mathematics ( solution of integer-order differential equations, definitions of new

function classes, summation of series, etc.,).

However, the demands of modern technology require a certain revision of the well-

established pure mathematical approach. There have appeared a number of works, especially

in the theory of viscoelasticity and in hereditary solid mechanics, where fractional derivatives

are used for a better description of material properties. Mathematical modelling based on

enhanced rheological models naturally leads to differential equations of fractional order and

to the necessity of the formulation of initial conditions to such equations.

Applied problems require definitions of fractional derivatives allowing the utilization of

physically interpretable initial conditions, which contain f(a), f ′(a), etc.

Unfortunately, the Riemann-Liouville approach leads to initial conditions containing the

limit values of the Riemann-Liouville fractional derivatives at the lower terminal t = a, for


example

limt→a

aDα−1t f(t) = b1,

limt→a

aDα−2t f(t) = b2, (1.4.2)

...

limt→a

aDα−nt f(t) = bn,

where bk, k = 1, 2, . . . , n are given constants.

In spite of the fact that initial value problems with such initial conditions can be suc-

cessfully solved mathematically, their solutions are practically useless, because there is no

known physical interpretation for such types of initial conditions.

Here we observe a conflict between the well-established and polished mathematical theory

and practical needs.

A certain solution to this conflict was proposed by M. Caputo first in his paper [34]

and two years later in his book [35], and recently (in Banach spaces) by El-Sayed [50, 51].

Caputo’s definition can be written as

Ca D

αt f(t) =

1

Γ(α− n)

∫ t

a

f (n)(τ)dτ

(t− τ)α+1−n, (n− 1 < α < n). (1.4.3)

Under natural conditions on the function f(t), for α→ n the Caputo derivative becomes

a conventional n-th derivative of the function f(t). Indeed, let us assume that 0 ≤ n− 1 <

α < n and that the function f(t) has n+1 continuous bounded derivatives in [a, T ] for every

T > a. Then

limα→n

Ca D

αt f(t) = lim

α→n

{f (n)(a)(t− a)n−α

Γ(n− α+ 1)+

1

Γ(n− α+ 1)

∫ t

a

(t− τ)n−αf (n+1)(τ)dτ

}= f (n)(a) +

∫ t

a

f (n+1)(τ)dτ = f (n)(t), n = 1, 2, . . . .

This says that, similarly to the Riemann-Liouville approach, the Caputo approach also

provides an interpolation between integer-order derivatives.

The main advantage of Caputo’s approach is that the initial conditions for fractional

differential equations with Caputo derivatives take on the same form as for integer-order


differential equations, i.e. contain the limit values of integer-order derivatives of unknown

functions at the lower terminal t = a.

To underline the difference in the form of the initial conditions which must accompany

fractional differential equations in terms of the Riemann-Liouville and the Caputo deriva-

tives, let us recall the corresponding Laplace transform formulas for the case a = 0.

The formula for the Laplace transform of the Riemann-Liouville fractional derivative is

∫ ∞

0

e−pt{ 0Dαt f(t)}dt = pαF (p)−

n−1∑k=0

pk0D

α−k−1t f(t)|t=0, (n− 1 ≤ α < n), (1.4.4)

whereas Caputo’s formula, first obtained in [34], for the Laplace transform of the Caputo

derivative is∫ ∞

0

e−pt{ C0 D

αt f(t)}dt = pαF (p)−

n−1∑k=0

pα−k−1f (k)(0), (n− 1 < α ≤ n). (1.4.5)

We see that the Laplace transform of the Riemann-Liouville fractional derivative allows

utilization of initial conditions of the type (1.4.4), which can cause problems with their

physical interpretation. On the contrary, the Laplace transform of the Caputo derivative

allows utilization of initial values of classical integer-order derivatives with known physical

interpretations.

The Laplace transform method is frequently used for solving applied problems. To choose

the appropriate Laplace transform formula, it is very important to understand which type

of definition of fractional derivative must be used.

Another difference between the Riemann-Liouville definition (1.4.1) and the Caputo def-

inition (1.4.3) is that the Caputo derivative of a constant is 0, whereas in the cases of a finite

value of the lower terminal a the Riemann-Liouville fractional derivative of a constant C is

not equal to 0, but

0Dαt C =

Ct−α

Γ(1− α). (1.4.6)

Putting a = −∞ in both definitions and requiring reasonable behaviour of f(t) and its

derivatives for t→ −∞, we arrive at the same formula

−∞Dαt f(t) = −∞

CDαt f(t) =

1

Γ(n− α)

∫ t

−∞

f (n)(τ)dτ

(t− τ)α+1−n, (n− 1 < α < n), (1.4.7)


which shows that for the study of steady-state dynamical processes the Riemann-Liouville

definition and the Caputo definition must give the same results.

There is also another difference between the Riemann-Liouville and the Caputo ap-

proaches, which we would like to mention here and which seems to be important for ap-

plications. Namely, for the Caputo derivative we have

Ca D

αt

(Ca D

mt f(t)

)= C

a Dα+mt f(t), (m = 0, 1, . . . ; n− 1 < α < n) (1.4.8)

while for the Riemann-Liouville derivative

aDmt ( aD

αt f(t)) = aD

α+mt f(t), (m = 0, 1, . . . ; n− 1 < α < n). (1.4.9)

The interchange of the differentiation operators in formulas (1.4.8) and (1.4.9) is allowed

under different conditions:

Ca D

αt

(Ca D

mt f(t)

)=C

a Dmt

(Ca D

αt f(t)

)=C

a Dα+mt f(t), (1.4.10)

f (s)(0) = 0, s = n, n+ 1, . . . ,m, (m = 0, 1, 2, . . . ;n− 1 < α < n)

aDmt ( aD

αt f(t)) = aD

αt ( aD

mt f(t)) = aD

α+mt f(t), (1.4.11)

f (s)(0) = 0, s = 0, 1, . . . ,m, (m = 0, 1, 2, . . . ;n− 1 < α < n).

We see that contrary to the Riemann-Liouville approach, in the case of the Caputo

derivative there are no restrictions on the values f (s)(0), (s = 0, 1, . . . , n− 1).

1.5 Fractional Differential Equations

A fractional differential equation is an equation which contains fractional derivatives;

a fractional integral equation is an integral equation containing fractional integrals. The

Laplace transform and its convolution integral are found to be useful in the study of fractional

differential equations.

Now, let us see, how fractional differential equation is applied in differential problems.

We analyse the most simple differential equations of fractional order.


The simple fractional relaxation and oscillation equations

The classical phenomena of relaxation and oscillations in their simplest form are known

to be governed by linear ordinary differential equations, of order one and two respectively,

that hereafter we recall with the corresponding solutions. Let us denote by u = u(t) the

field variable and by q(t) a given continuous function, with t ≥ 0. The relaxation differential

equation reads

u′(t) = −u(t) + q(t), (1.5.1)

whose solution, under the initial condition u(0+) = c0, is

u(t) = c0e−t +

∫ t

0

q(t− τ)e−τdτ (1.5.2)

The oscillation differential equation reads

u′′(t) = −u(t) + q(t), (1.5.3)

whose solution, under the initial conditions u(0+) = c0 and u′(0+) = c1 , is

u(t) = c0 cos t+ c1 sin t+

∫ t

0

q(t− τ) sin τdτ (1.5.4)

From the point of view of the fractional calculus a natural generalization of equations (1.5.1)

and (1.5.3) is obtained by replacing the ordinary derivative with a fractional one of order α.

In order to preserve the type of initial conditions required in the classical phenomena, we

agree to replace the first and second derivative in (1.5.1) and (1.5.3) with a Caputo fractional

derivative of order α with 0 < α < 1 and 1 < α < 2, respectively. We agree to refer to

the corresponding equations as the simple fractional relaxation equation and the simple

fractional oscillation equation. Generally speaking, we consider the following differential

equation of fractional order α > 0,

Dα∗ u(t) = Dα

(u(t)

m−1∑k=0

tk

k!u(k)(0+)

)= −u(t) + q(t), t > 0 (1.5.5)

Here m is a positive integer uniquely defined by m − 1 < α ≤ m, which provides the

number of the prescribed initial values u(k)(0+) = ck, k = 0, 1, 2, ...,m − 1. Implicit in the


form of (1.5.5) is our desire to obtain solutions u(t) for which the u(k)(t) are continuous for

t ≥ 0, k = 0, 1, ...,m − 1. In particular, the cases of fractional relaxation and fractional

oscillation are obtained for m = 1 and m = 2 , respectively We note that when α is the

integer m the equation (1.5.5) reduces to an ordinary differential equation whose solution can

be expressed in terms of m linearly independent solutions of the homogeneous equation and

of one particular solution of the inhomogeneous equation. We summarize the well-known

result as follows

u(t) =m−1∑k=0

ckuk(t) +

∫ t

0

q(t− τ)uδ(τ)dτ (1.5.6)

uk(t) = Jku0(t), u(h)k (0+) = δkh, h, k = 0, 1, ...,m− 1, (1.5.7)

uδ(t) = −u′0(t) (1.5.8)

Thus, the m functions uk(t) represent the fundamental solutions of the differential equation

of order m, namely those linearly independent solutions of the homogeneous equation which

satisfy the initial conditions in (1.5.7). The function uδ(t), with which the free term q(t)

appears convoluted, represents the so called impulse response solution, namely the particular

solution of the inhomogeneous equation with all ck ≡ 0, k = 0, 1, ...,m− 1, and with q(t) =

δ(t). In the cases of ordinary relaxation and oscillation we recognize that u0(t) = e−t = uδ(t)

and u0(t) = cost, u1(t) = Ju0(t) = sint = cos(t− π2) = uδ(t), where J is the integral operator

and Jα is fractional integral of order α respectively.

1.5.1 Applications

In this section we review some applications of fractional differential equations.

1. Identification of biochemical reactors

The fractional differential equations have not been yet used for modeling or identification

of biochemical reactors using actual experimental data. But also the analysis of different

parameter estimation approaches used with fractional differential equations. The different

approaches considered for parameter estimation step involved in the identification task were:

(a) solving a nonlinear set of algebraic equations, obtained from the derivatives of the ob-

jective function with respect to the parameters being estimated.


(b) solving a multivariable nonlinear deterministic optimization problem.

(c) solving a multivariable nonlinear heuristic optimization problem, based on genetic algo-

rithms.

These techniques were used not only for fractional identification, but also for first and sec-

ond integer order models. After parameter estimation, the identified models are analyzed by

standard statistical tests (Otto, 1999) in order to quantify and evaluate the quality of the

achieved data fitness, see [65].

2. Model arterial viscoelasticity

Some models based on fractional order differential equations were presented to describe cell

and tissue biomechanics (Djordjevic et al. 2003; Koeller, 1984; Suki et al. 1994). These

equations derive into fractional viscoelastic concepts. Briefly, if a spring represents a zero

order element and a dashpot a first order element, a new component called spring-pot can

be conceived with an intermediate order 0 < α < 1. The mechanical response can interpo-

late between pure elastic and viscous behaviors. Both temporal relaxation and frequency

responses of a spring-pot follow power-law functions that seem to be naturally adapted to

fit arterial requirements. In [45] modify an Standard Linear Solid (SLS) model, replacing

a dashpot with a spring-pot of order 0 < α < 1 defined using fractional derivatives, to de-

scribe arterial viscoelasticity in vitro. Uniaxial stress-relaxation was registered during 1-hour

in human arteries at 2 stress levels and the parameters of the proposed model were adjusted.

And, an estimation of the frequency response in arteries was presented and discussed.

3. Schrdinger equation

The Schrdinger equation control the dynamical behaviour of quantum particles. In [1], Adda

and Cresson, considered to study of α-differential equations and discussed a fundamental

problem concerning the Schrdinger equation in the framework of Nottales scale relativity

theory.


1.6 Motivation

1.6.1 Examples

The objective of this thesis is to study the existence of results of various types of fractional

differential and integrodifferential equations and impulsive fractional differential equations.

We shall here list out some examples that motivate the study of existence of fractional

dynamical systems and impulsive differential equations.

Example 1.6.1(a)

Tautochrone Problem

It may be important to point out that the first application of fractional calculus was made

by Abel(1802-1829) in the solution of an integral equation that arises in the formulation

of the tautochronous problem. This problem deals with the determination of the shape of

a frictionless plane curve through the origin in a vertical plane along which a particle of

mass m can fall in a time that is independent of the starting position. If the sliding time is

constant T, then the Abel integral equation is√2gT =

∫ η

0

(η − y)12f ′(y)dy,

where g is the acceleration due to gravity, (ξ, η)is the initial position and s=f(y) is the

equation of the sliding curve. It turns out that this equation is equivalent to the fractional

integral equation

T√

2g = Γ(1

2)D

−12

η f ′(η)

Indeed, Heaviside gave an interpretation of√p = D

12 so that D

12t 1 = 1√

πt

Example 1.6.1(b)

Applications in the diffusion equation

It is well known that, in the classical case, the diffusion equation is given by

∂u

∂t= b2

∂2u

∂x2(1.6.1)

A general solution is given for a fractional diffusion equation defined in a bounded space

domain. The fractional time derivative is described in the Caputo sense. The Caputo frac-

tional derivative is considered here because it allows for the standard inclusion of traditional


initial and boundary conditions in the formulation, whereas models based on other fractional

derivatives may require the values of the fractional derivative terms at the initial time. Keep-

ing this definition in mind, the fractional diffusion equation (1.6.1) can be written as

∂αu

∂tα= b2

∂2u

∂x2(1.6.2)

where alpha is a parameter describing the order of the fractional derivative, b denotes a

constant coefficient with dimension (Time)(Length)−α/2, x and t are the space and time

variables, and u = u(x; t) is the field defined in the space domain [0, L], considering the

following boundary conditions

u(0, t) = u(L; t) = 0; t > 0,

u(x, 0) = f(x), 0 < x < L,

ut(x, 0) = 0; 0 < x < L, (1.6.3)

for 1 < α ≤ 2. The last boundary condition is assumed to ensure the continuous dependence

of the solution on the parameter α in the transition from α = 1− to α = 1+ [2]. Taking the

finite sine transform of Eq. (1.6.2), integrating the second term of the resulting equation by

parts, and applying the boundary conditions, we obtain

dαu

dtα+ (ban)2u = 0 (1.6.4)

and

u = u(n, t) =

∫ L

0

u(x, t)sin(anx)dx, (1.6.5)

is the finite sine transform of u(x, t), where a = π/L, and n is a wave number, Eq. (1.6.4)

may be called a diffusion-wave equation in a wave number domain. Taking the finite sine

transform of Eq. (1.6.3), we obtain

u(n, 0) =

∫ L

0

f(x)u(x, t)sin(anx)dx, (1.6.6)

and ut(n, 0) = 0 for 1 < α ≥ 2. Taking the Laplace transform of Eq. (1.6.2) and using the

initial conditions and the properties of the Caputo derivative, we obtain

Un,s =sα−1u(n, 0)

sα + (ban)2(1.6.7)


where s is the Laplace transform parameter, and U(n, s) is the Laplace transform of u(n, t).

Taking first the inverse Laplace transform of Eq. (1.6.7) and then the inverse finite sine

transform of the resulting equation, we obtain

u(x, t) =2

L

∞∑n=1

Eα(−b2a2n2tα)sin(anx)×∫ L

0

f(r)sin(anr)dr, (1.6.8)

where Eα is the Mittag-Leffler function. The Mittag-Leffler function has several interesting

properties.

In particular, we have E1(−z) = e−z and E2(−z2) = cos(z). Using these identities, the

solutions of Eq. (1.6.2) for α = 1 and 2 are given as

u(x, t) =2

L

∞∑n=1

e−(ban)2tsin(anx)×∫ L

0


and

u(x, t) =2

L

∞∑n=1

cos(bant)sin(anx)×∫ L

0


Equations (1.6.9) and (1.6.10) represent the diffusion and the wave solutions. These are

special cases of the solution (Eq. (1.6.8)) of the fractional diffusion-wave equation.

Example 1.6.1(c)

Fluid Flow and the design of a Weir Notch

A weir notch is an opening in a dam (weir) that allows water to spill over the dam, (see

Fig. 1), where we have indicated a cross section of the dam and a partial front view. (The

sketch is not to scale.) Our problem is to design the shape of the opening such that the rate

of flow of water through the notch (say, in cubic feet per second) is a specified function of

the height of the opening.

Starting from physical principles we derive the equation for determining the shape of

the notch. It turns out to be an integral equation of the Riemann- Liouville type. After

formulating the problem, we shall, of course, solve it.

Let the x-axis denote the direction of flow, the z-axis the vertical direction, and the y-axis

the transverse direction along the face of the dam.


Fig. 1

See Fig. 2, where we have drawn an enlarged view of a portion of Fig. 1. The axes are

oriented as indicated, and h is the height of the notch.

The solid square at point I and the solid square at point II are supposed to indicate

the same element of fluid as it moves from point I [with coordinates (x0, y0, z0)] to point

II [with coordinates (0, y0, z0)] along the same “tube of flow.” Then by Bernoulli’s theorem

from hydrodynamics

PI

ρ+ gz0 +

1

2V 2

1 =PII

ρ+ gz0 +

1

2V 2

II , (1.6.11)

where ρ is the density of water, g the acceleration of gravity, and PI and VI are the pressure

and velocity at point I while PII and VII are the corresponding quantities at point II. If we

assume that point I is far enough upstream, VI is negligible and we may write (1.6.11) as

PI − PII =1

2ρV 2

II (1.6.12)


Figure 2

Now

PI = (atmospheric pressure) + (the pressure exerted by a column of water of height

h− z0) and since point II is in the plane of the notch ( the shaded area of Fig. 2)(b)

PII = atmospheric pressure.

Thus PI − PII is a constant (namely, ρg) times (h− z0) and (1.6.12) implies that

VII =√

2g(h− z0)12 . (1.6.13)

Referring to Fig. 3, we see that the element of area dA (the shaded region in Fig. 3) is

dA = 2|y|dz (1.6.14)

where we have assumed that the shape of the notch is symmetrical about the z-axis. Now

|y| is some function of z, say

|y| = f(z), (1.6.15)

and we may write (1.6.14) as

dA = 2f(z)dz.


Figure 3

Thus the incremental rate of flow of water through the area dA is

dQ = V dA,

where V is the velocity of flow at height z, and from (1.6.13)

dQ = 2√

2g(h− z)12f(z)dz.

The total flow of water through the notch is thus

Q =

∫ h

0

dQ(z) = 2√

2g

∫ h

0

(h− z)12f(z)dz. (1.6.16)

Equation (1.6.16) is the desired integral equation for the determination of f when Q is given.

In the notation of the fractional calculus we may write it as

Q(h) =√

2gπD−32 f(h). (1.6.17)

Therefore

f(h) =1√2gπ

D32Q(h),

which is the desired solution.


Example 1.6.1(d)

Electric transmission lines

During the last decades of the nineteenth century, Heaviside successfully developed his oper-

ational calculus without rigorous mathematical arguments. In 1892 he introduced the idea

of fractional derivatives in his study of electric transmission lines. Based on the symbolic op-

erator form solution of heat equation due to Gregory(1846), Heaviside introduced the letter

p for the differential operator ddt

and gave the solution of the diffusion equation

∂2u

∂x2= a2p

for the temperature distribution u(x, t) in the symbolic form

u(x, t) = Aexp(ax√p) +Bexp(−ax√p)

in which p ≡ ddx

was treated as constant, where a, A and B are also constant.

Example 1.6.1(e)[102]

Ultrasonic wave propagation in human cancellous bone

Fractional calculus is used to describe the viscous interactions between fluid and solid struc-

ture. Reflection and transmission scattering operators are derived for a slab of cancellous

bone in the elastic frame using Blots theory. Experimental results are compared with the-

oretical predictions for slow and fast waves transmitted through human cancellous bone

samples

Example 1.6.1(f)[103]

Application of fractional calculus in the theory of viscoelasticity

The advantage of the method of fractional derivatives in theory of viscoelasticity is that

it affords possibilities for obtaining constitutive equations for elastic complex modulus of

viscoelastic materials with only few experimentally determined parameters. Also the frac-

tional derivative method has been used in studies of the complex moduli and impedances

for various models of viscoelastic substances.


Example 1.6.1(g)[86]

Fractional differentiation for edge detection

In image processing, edge detection often makes use of integer-order differentiation operators,

especially order 1 used by the gradient and order 2 by the Laplacian. This paper demon-

strates how introducing an edge detector based on non-integer (fractional) differentiation

can improve the criterion of thin detection, or detection selectivity in the case of parabolic

luminance transitions, and the criterion of immunity to noise, which can be interpreted in

term of robustness to noise in general.

Example 1.6.1(h)[71]

Application of Fractional Calculus to Fluid Mechanics

Application of fractional calculus to the solution of time-dependent, viscous-diffusion fluid

mechanics problems are presented. Together with the Laplace transform method, the appli-

cation of fractional calculus to the classical transient viscous-diffusion equation in a semi-

infinite space is shown to yield explicit analytical (fractional) solutions for the shearstress

and fluid speed anywhere in the domain. Comparing the fractional results for boundary

shear-stress and fluid speed to the existing analytical results for the first and second Stokes

problems, the fractional methodology is validated and shown to be much simpler and more

powerful than existing techniques.


1.7 Nonlocal Cauchy Problem

Nonlocal Cauchy problem, namely the Cauchy problem for a differential equation with

a nonlocal initial condition u(t0) + g(t1, ..., tp, u) = u0 (here 0 ≤ t0 < t1 < ... < tp ≤ t0 + a

and g is a given function) is one of the important topics in the study of the analysis theory.

Interest in such a problem stems mainly from the better effect of the nonlocal initial condition

than the usual one in treating physical problems. Actually the nonlocal initial condition

u(t0) + g(t1, ..., tp, u) = u0 models many interesting natural phenomena with which the

normal initial condition u(0) = u0 may not fit in. For instance, the function g(t1, ..., tp, u)

may be given by

g(t1, ..., tp, u) =

p∑i=1

Ciu(ti),

Ci, i = 1, 2, ..., p are constants. In this case we are permitted to have the measurements

at t = 0, t1, ..., tp, rather than just at t = 0. Thus more information is available. More

specially, letting g(t1, ..., tp, u) = −u(tp) and u0 = 0 yields a periodic problem and letting

g(t1, ..., tp, u) = −u(t0) + u(tp) gives a backward problem. The study of existence of so-

lutions to evolution equations with a nonlocal condition in Banach space was initiated by

Byszewski [33]. In Byszewski and Lakshmikantham [32] and the references therein, one can

find other information about the importance of nonlocal initial conditions in applications.

There have been many papers concerning this topic [23, 36, 40, 87, 88, 93, 108]. The diffusion

phenomenon of a small amount of gas in a transparent tube by using the formula

g(u) =

p∑j=0

cju(tj)

where cj, j = 0, 1, ..., p are given constants and 0 < t0 < t1 < ........ < tp < a. In this case

the above equation allows the additional measurement at tj, j = 0, 1, ..., p.


1.8 Impulsive Differential Equations

The theory of impulsive differential equations is an important branch of differential equa-

tions. Many evolution processes in applied sciences are represented by differential equations.

However, the situation is quite different in various physical phenomena that have an abrupt

change in their states such as mechanical systems with impact, biological systems involving

thresholds, population dynamics, chemical technology, industrial robotics, medicine and so

on. In the mathematical simulation of such phenomena, the duration of these abrupt changes

are often negligible compared to the total duration of the processes and it is reasonable to

assume that the processes change its state instantaneously, that is in the form of impulses.

Adequate mathematical model of such processes with impulsive effects are impulsive differ-

ential equations.

The theory of impulsive differential equations has wide applications in many real world

phenomena, in particular mechanical and biological systems where discontinuities arise in

their state. There has a significant development in impulsive theory especially in the area

of impulsive differential equations with fixed moments; see for instance the monographs by

Bainov and Simeonov [22], [75] and the references therein. Here we list some examples

Example 1.8.1

Growth of fish population [94]

A fish breeding pond maintained scientifically is an example of a system involving im-

pulsive behaviour. Here the natural growth of the fish population is disturbed by making

catches at certain time intervals and by adding fresh breed. The impulses are given at times

t1, t2, .... This problem can be described by:

Dx(t) = αx(t)Du(t), (1.8.1)

x(t0) = x0, (1.8.2)

where D is the distributional derivative, u is a right-continuous function of bounded varia-

tions and α ∈ R. Assume that u is of the form

u(t) = t+∞∑

k=1

akHk(t)


where

Hk(t) =

0 if t < tk,

1 if t ≥ tk,

where ak are real numbers. Generally a right continuous function of bounded variation

contains an absolutely continuous part and a singular part. Note that the discontinuities of

u are isolated. Further

Du = 1 +∞∑

k=1

akHk(tk),

where Hk(tk) is the Dirac measure concentrated at tk. It has been shown subsequently that

the unique solution of this equation is given by

x(t) =eα(t−t0)

Πk−1i=1 (1− ai)

x0, ai 6= 1, tk−1 ≤ t < tk. (1.8.3)

Clearly, if ai = 0 for i = 1, 2, ..... in (1.8.3) then the equations (1.8.1)-(1.8.2) reduces to the

equation:

x′(t) = αx(t), α ∈ R,

x(t0) = x0,

and solution (1.8.3) reduces to

x(t) = eα(t−t0)x0, t ≥ t0.

By consideration several varieties of fish growing in one pond, the model of growth given

in (1.8.1) can be generalized to a system of equations.


Example 1.8.2

Drug distribution in the human body [75]

Consider a simple two compartmental model for drug distribution in the human body

which is proposed by Kruger-Thiemer. It is assumed that the drug, which is administered

orally, is first dissolved into the gastro-intestinal tract. The drug is then absorbed into the

so-called apparent volume of distribution (a lumped compartment which accounts for blood,

muscle, tissue, etc.), and finally is eliminated from the system by the kidneys. Let x(t) and

y(t) denote the amounts of drug at time t in the gastro-intestinal tract and apparent volume

of distribution respectively, and let k1 and k2 be relevant rate constants. The dynamical

system of this model is then

x′ = −k1x, y′ = −k2y + k1x. (1.8.4)

We now postulate that at the moments of time 0 < t1 < t2 < ... < tN < T, the drug is

ingested in amounts δ0, δ1, ..., δN . So that we havex(t+i ) = x(t−i ) + δi, i = 1, 2, . . . , N,

y(t+i ) = y(t−i ),

x(0) = δ0, y(0) = 0.

(1.8.5)

To achieve the desired therapeutic effect, it is required that the amount of drug in the

apparent volume of distribution never goes below a constant level or plateau during the

time interval. Finally, take the biological cost function f(δ) = 12

N∑i=0

δ2i both to minimize

side effects and the cost of the drug. The problem is to find infδ≥0

f(δ) subject to (1.8.4) and

(1.8.5). Clearly (1.8.4) and (1.8.5) represent a simple impulsive differential system.


1.9 Existence

In fact fractional differential equations are considered as an alternative model to nonlinear

differential equations [31]. It draws a great application in nonlinear oscillations of earth-

quakes [59], many physical phenomena such as seepage flow in porous media and in fluid

dynamic traffic models. Fractional derivative can eliminate the deficiency of continum traffic

flow. The most important advantage of using fractional differential equations in these and

other applications [63] is their nonlocal property. It is well known that the integer order dif-

ferential operator is a local operator but the fractional order differential operator is non-local.

This means that the next state of a system depends not only upon its current state but also

upon all of its historical states. This is probably the most relevant feature for making this

fractional tool useful from an applied standpoint and interesting from a mathematical stand-

point and in turn led to the sustained study of theory of fractional differential equations [68].

The existence of solutions of abstract differential equations is investigated in [24] whereas the

existence of solutions of fractional differential equations by using fixed point techniques have

been discussed by several authors [47],[67],[66],[119],[54],[117],[64],[46],[3],[58]. The problem

of existence of solutions of evolution equations with nonlocal conditions was initiated by

Byszewski[33] and subsequently studied by several authors for different kinds of problems.

Hernandez et al. [62] discussed the recent developments in the theory of abstract fractional

differential equations in which the resolvent operator plays an important role in proving their

existence results.

The present study mainly deals with the fixed point approach of proving existence the-

orems for fractional integrodifferential and impulsive differential equations in Banach spaces.

Various types of fractional integrodifferential equations and impulsive fractional differen-

tial equations have been studied by the following authors.


In [25] Krishnan Balachandran, Juan J. Trujillo studied the nonlocal Cauchy problem for

nonlinear fractional integrodifferential equations in Banach spaces

cDqu(t) = A(t)u(t) + f(t), 0 ≤ t ≤ T,

u(0) = u0

where 0 < q < 1, A(t) is a bounded linear operator on a Banach space X, u0 ∈ X and

f : J → X is continuous.

In [24] K. Balachandran, J.Y. Park discussed the Nonlocal Cauchy problem for abstract

fractional semilinear evolution equations

dqu(t)

dtq= A(t)u(t), 0 ≤ t ≤ T

u(0) = u0

where 0 < q < 1 and A(t) is a bounded linear operator on a Banach Space X and u0 ∈ X.

In [38] Y.K. Chang, V. Kavitha, M. Mallika Arjunan, proved the Existence and unique-

ness of mild solutions to a semilinear integrodifferential equation of fractional order

Dqx(t) + Ax(t) = f

(t, x(t),

∫ t

0

e(t, s, x(s))ds

), t ∈ J = [0, a]

x(0) + g(x) = x0

where 0 < q < 1, −A is the infinitesimal generator of a noncompact,analytic semigroup

T (t), t ≤≤ 0 on a Banach Space (X, | · |), e, f, g are functions.

In [116] Xinwei Su studied the Solutions to boundary value problem of fractional order

on unbounded domains in a Banach space

Dα0+u(t) = f(t, u(t)), t ∈ J := [0,∞),

u(0) = 0, Dα−10+ u(∞) = u∞

where 1 < α ≤ 2, f ∈ C(J × E,E), u∞ ∈ E,Dα0+ and Dα−1

0+ are the Riemann-Liouville

fractional derivatives and Dα−10+ u(∞) = lim

t→∞Dα−1

0+ u(∞)u(t)


In [109] G.Wang, B.Ahmad and L.Zhang studied Impulsive anti-periodic boundary value

problem for nonlinear differential equations of fractional order of the form

cDα = f(t, u(t)), 2 < α ≤ 3, t ∈ J ′,

∆u(tk) = Qk(u(tk)), k = 1, 2, ...., p

∆u′(tk) = Ik(u(tk)), k = 1, 2, ..., p

∆u′′(tk) = I∗k(u(tk)), k = 1, 2, ..., p

u(0) = −u(1), u′(0) = −u′(1), u

′′(0) = −u′′(1)

where cDα is the Caputo fractional derivative.

In [115], Wei Lin discussed both the local and global existence of solutions to the equationDαt x(t) = f(t, x(t)), t ∈ [0, T ]

xk(t0) = x0(k), k = 0, 1, 2, ......, n− 1

in a finite dimensional space. The results are obtained via construction and the contraction

mapping principle. In particular, feed-back control of chaotic fractional differential equation

is theoretically investigated and the fractional Lorenz system as a numerical example is

further provided to verify the analytical result.

G. M. Mophou et al [88], studied the Cauchy problem with nonlocal conditionsDqx(t) = Ax(t) + tnf(t, x(t), Bx(t)), t ∈ [0, T ], n ∈ Z+

x(0) + g(x) = x0,

in general Banach space X with 0 < q < 1 and A is the infinitesimal generator of a C0

semigroup of bounded linear operator. By means of the Krasnoselskii’s theorem, existence

of solutions was also obtained.

In [10], B. Ahmad proved the existence of solutions to the equationcDqx(t) = f(t, x(t), ), t ∈ [0, T ],

x(0) = −x(T ), x′(0) = −x′(T ), x”(0) = −x”(T )


The results are obtained via construction and the contraction mapping principle and Kras-

noselskii’s fixed point theorem.

In [29] M. Benchohra et al. studied the existence and uniqueness of solutions for the

initial value problems (IVP for short), for fractional order differential equations

cDαy(t) = f(t, y), t ∈ J = [0, T ], t 6= tk, (1.9.1)

∆y∣∣t=tk

= Ik(y(t−k )), (1.9.2)

y(0) = y0, (1.9.3)

where k = 1, . . . ,m, 0 < α ≤ 1, cDα is the Caputo fractional derivative, f : J × R → R

is a given function, Ik : R → R, and y0 ∈ R, 0 = t0 < t1 < · · · < tm < tm+1 = T ,

∆y|t=tk = y(t+k )− y(t−k ), y(t+k ) = limh→0+ y(tk + h) and y(t−k ) = limh→0− y(tk + h) represent

the right and left limits of y(t) at t = tk.

In [20] Atmania et al. obtained some results regarding local existence and uniqueness for

some fractional integrodifferential problem with a finite number of impulses of the form:

Dαu(t) = f(t, u(t)); t ∈ [t0, t0 + τ ], t 6= tk, k = 1, . . . ,m; (1.9.4)

with the initial condition

Dα−1u(t0) = u0; (t− t0)1−αu(t)

∣∣t=t0

=u0

Γ(α); (1.9.5)

subject to the impulsive conditions

Dα−1(u(t+k )− u(t−k )) = Ik(t); t = tk, k = 1, . . . ,m;

(t− tk)1−αu(t)

∣∣t=tk

=Ik(tk)

Γ(α), k = 1, . . . ,m.

(1.9.6)

In [37] Y.Chen, Z.lv studied the initial value problems of fractional differential equations

by using the Leggett-Williams fixed point theorem.

Dαn0+u(t)−

n−1∑j=1

aj(t)Dαj

0+u(t) = f(t, u(t)), 0 ≤ t ≤ 1,

u(0) = u′(0) = 0,

(1.9.7)


where 0 < α1 < α2 < · · · < αn−1 < αn − 1 < 1 < αn < 2, n ≥ 2, n ∈ Z, an ∈ R,

f : [0, 1] × [0,+∞) → [0,+∞) is continuous and aj : [0, 1] → (0,+∞) (j = 1, 2, . . . , n − 1)

are continuously differentiable.

In [98] M. Rchid Sidi Ammi et al. obtained the existence and uniqueness of solutions the

functional integro-differential fractional equation

dα

dtα

[ x(t)

f(t, x(t))

]= g(t, xt,

∫ t

0

k(s, xs)ds)

a.e., t ∈ I, (1.9.8)

subject to

x(t) = φ(t), t ∈ I0, (1.9.9)

where dα/dtα denotes the Riemann-Liouville derivative of order α, 0 < α < 1, and xt : I0 →

C is the continuous function defined by xt(θ) = x(t+ θ) for all θ ∈ I0, under suitable mixed

Lipschitz and other conditions on the nonlinearities f and g.

In [105] Syed Abbs discuss the existence and uniqueness of solution to fractional order

ordinary and delay differential equations of the form

dαx(t)

dtα= g(t, x(t)), t ∈ [0, T ]

x(0) = x0, 0 < α < 1,

(1.9.10)

anddαx(t)

dtα= f(t, x(t), x(t− τ)), t ∈ [0, T ]

x(t) = φ(t), t ∈ [−τ, 0] 0 < α < 1,

(1.9.11)

under suitable conditions on g, f and φ. We assume that g satisfies Lipschitz condition with

Lipschitz constant Lg and f(t, x, y) can be written as f1(t, x) + f2(t, x, y), where both f1, f2

are Lipschitz continuous with Lipschitz constants Lf1 and Lf2 respectively.

Motivated by the above works, in this thesis we study the existence results of fractional

integrodifferential and differential equations with impulsive conditions.

1.10 Methods

In recent years, existence and uniqueness results for various types of fractional differential

and integrodifferential equations are proved by using valuable tools such as semigroup theory,


Laplace transform method, probability density function, α-resolvent operator and fixed point

methods. In this thesis we prove the existence theorems by the applications of semigroup

theory and α-resolvent operator and fixed point techniques.

1.10.1 Semigroup Theory

Theory of semigroups of bounded linear operators developed quite rapidly since the dis-

covery of the generation theorem by Hille and Yosida in 1948. By now, it is an extensive

mathematical subject with substantial applications to many fields of analysis. The theory

of semigroups of bounded linear operators is closely related to the solution of differential

and fractional equations in Banach spaces [57]. For more details about semigroup theory,

we refer Pazy [95] and [52].

1.10.2 Resolvent Operator

Let A be a closed and linear operator with domain D(A) defined on a Banach space X and

α > 0. Let ρ(A) be the resolvent set of A. We call A the generator of an α−resolvent

family if there exists ω ≥ 0 and a strongly continuous function Sα : R+ → L(X) such that

{λα : Reλ > ω} ⊂ ρ(A) and

(λαI − A)−1x =

∫ ∞

0

e−λtSα(t)xdt, Reλ > ω, x ∈ X.

In this case, Sα(t) is called the α−resolvent family generated by A.[19]

The concept of solution operator is closely related to the concept of a resolvent family

(see [97, Chapter 1]). For more details on α-resolvent family and solution operators, we refer

to [81, 97] and the references therein.

1.10.3 Fixed Point Method

To prove the existence results, the modern method known as fixed point techniques are

widely used. Fixed point method is used to prove the existence theorems for fractional dif-

ferential and integrodifferential equations and impulsive problem. The Banach fixed point

theorem is an important source of existence and uniqueness theorem in different branch of


analysis. Recently, Leray-Schauder Alternative, Krasnoselskii, Schauder and nonlinear alter-

native of Granas and Frigon, Sadovskii fixed point theorems are used to prove the existence

of solutions for various types of fractional differential and integrodifferential equations.

In this thesis we use the Krasnoselskii, Schauder and Sadovskii fixed point techniques to

prove our existence results.

In Chapter II, we deal with the existence and uniqueness of the solutions for fractional

integrodifferential equations with nonlocal initial and integral conditions in a Banach space.

The results are established by the application of the contraction mapping principle and the

Krasnoselskii’s fixed point theorem and an application is also given.

In Chapter III, the existence and uniqueness theorem for the nonlinear fractional mixed

Volterra-Fredholm integrodifferential equation with nonlocal initial condition are studied.

We point out that such a kind of initial conditions with non-local restrictions could play an

interesting role in the applications of the mentioned model. The results obtained are used

in the example.

In Chapter IV, we are mainly focused upon the existence and uniqueness of mild solutions

to a semilinear fractional integro-differential equations with nonlocal conditions in Banach

spaces. Based upon suitable fixed point thereoms combined with the study of an α-resolvent

family, the main results are established under the Krasnoselskii-Krein-type conditions.

In Chapter V, the existence of solutions for fractional evolution equations with boundary

conditions in Banach spaces is proved. The results are obtained by using fractional calculus

and the fixed point theorems.

In Chapter VI, we establish some sufficient conditions for the existence of solutions for a

class of initial value problem of impulsive fractional differential equations and anti periodic

boundary value problem of an impulsive integrodifferential equations. An example is also

provided.


1.11 Contributions of the author

In the light of the above the author has obtained some significant generalizations on the

following topics:

(1) Existence results for some fractional integrodifferential systems in Banach spaces

(2) Existence of solutions to fractional mixed integrodifferential equations with nonlocal

initial condition

(3) Existence of solutions for nonlocal semilinear fractional integro-differential equation with

Krasnoselskii-Krein-type conditions

(4) Existence of solutions for fractional semilinear evolution boundary value problem in

Banach Spaces

(5) Existence results for impulsive fractional dynamical systems with integral and anti-

periodic boundary conditions.

The rest of the thesis contains a detailed account of the above topics.


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