Post on 30-Jan-2021
© 2020, IJSRMSS All Rights Reserved 45
International Journal of Scientific Research in __________________________ Research Paper . Mathematical and Statistical Sciences
Volume-7, Issue-3, pp.45-53, June (2020) E-ISSN: 2348-4519
Impact of Numerical Technique on the Solution of Boundary Value
Problem
Nisu Jain
Department of Mathematics, Govt. Mohindra College Patiala-147001, Punjab, India
Author’s Mail id: jainnisu@yahoo.in Tel.: +91-98768-62938
Available online at: www.isroset.org
Received: 28/May/2020, Accepted: 10/June/2020, Online: 30/June/2020
Abstract—A comparison of orthogonal collocation method (OCM) and orthogonal collocation on finite elements (OCFE)
is done to solve boundary value problem numerically and analytically. Both numerical techniques are applied on
dimensionless form of the model, then equations are discretized using numerical techniques. The results are obtained by
MATLAB ODE 15s system solver software. Comparison is shown both in tabulated and graphical form. Relative error is used to check the efficiency of the technique. 3-D graphs are used to specify solution variation for different values of
parameter.
Keywords— OCM; OCFE; Boundary Value Problems; Laplace Transformation
I. INTRODUCTION
Boundary value problems are used in science and
engineering for several years, yet they remain very active
and interesting research area because of their application
in various fields of science and engineering. There are
many types of differential equations such as diffusion
equation, advection-diffusion equation, Burgers-Huxley
equation, Burger-Fisher equation, Fisher-Kolmegnov equation, Fitzhogh-Nagumo equation, Kurmato-
Sivashinsky equation, Kawahara equation etc., which
describes the various physical problems.
Differential equations play a vital role in the field of
Mathematics, especially in modeling and simulation. The
differential equations with an additional set of constraints
are called boundary value problems.
Study of boundary value problems involves another
important step that is the solution of the boundary value
problem. Number of numerical techniques has been developed time to time to solve the different type of
boundary value problems. Laplace transform [1, 2, 3, 4],
Fourier transform [5, 6, 7], Homotopy perturbation method
[8, 4], Least square method [9, 10, 11], finite difference
method [12, 13, 14, 15], finite element method [16, 17, 18,
19, 20, 21], spline collocation [22, 23, 24, 25, 26, 27, 28,
29, 30, 31, 32, 33, 34, 35, 36 ], orthogonal collocation [37,
38, 39, 40 ,41], orthogonal collocation on finite elements
[42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55] etc.
Rate of convergence plays vital role for any numerical technique. Higher the rate of convergence implies better
will be the numerical technique. Laplace transform,
Fourier transform are from the family of analytic
techniques which are followed to solve the linear boundary
value problems. However, finite difference method and
collocation techniques are from the family of numerical
techniques. The main difference between analytic and
numerical techniques is that former solves the equation on
an abstract set whereas later discretize the problem on a
given set of points.
Collocation techniques are one of the weighted residual methods. In such type of techniques, the solution of the
given problem is replaced by an approximating function.
This approximating function is adjusted to the differential
equation and the boundary conditions as well. The residual
is set equal to zero at the collocation points. As the
collocation techniques have easy adaptability to the
computer codes and are simple than Galerkin method and
finite difference method, these techniques have gained
momentum in the solution of boundary value problems.
II. COLLOCATION POINTS Choice of the collocation points is an important and
sensitive part of orthogonal collocation method. In this
method, collocation points are taken to be the zeros of
orthogonal polynomials. Usually, the zeros of Jacobi
polynomial are taken as collocation points. Legendre and
Chebyshev polynomials are particular cases of Jacobi
polynomial.
The mapping of computational domain of the interval [-1,
1] to [0, 1] described below gives collocation points:
3
1
2 2
j
m j
x (1)
where xj is the jth collocation point in the interval [-1, 1].
http://www.isroset.org/
Int. J. Sci. Res. in Mathematical and Statistical Sciences Vol. 7, Issue.3, June 2020
© 2020, IJSRMSS All Rights Reserved 46
Interpolation points are taken roots of shifted Chebyshev
polynomial as:
1
)1(cos
m
jx j
; j = 1, 2,…, m+2 (2)
The discretization end points are fixed as 1 = 0 and m+2 = 1.
The zeros of Legendre polynomial are calculated from the
following recurrence relation:
)()2()()32()()1( 321 xPjxxPjxPj jjj
; j =2,…, m+1 (3)
Where P0(x) = 1 and P-1(x) = 0. In case of Legendre
polynomial, 0 and 1 are taken to be the boundary points.
III. ORTHOGONAL COLLOCATION ON FINITE
ELEMENTS
Due to orthogonal collocation method, slow convergence
of stiff system of boundary value problems, mainly in case
of orthogonal collocation method is conjectured with the
finite element method to combine the features of both the
methods.
Orthogonal collocation on finite elements was first
proposed by Patterson and Cresswell (1971) [58]. It was
further extended by Carey and Finlayson (1975) [56] to
solve effectiveness factor problem for large Thieles
modulus. Ma and Guiochon (1991) [59] have used orthogonal collocation on finite elements for calculation of
chromatographic elution band profiles for different
components. Arora et. al., (2009) [60] has checked the
asymptotic behavior of parabolic partial differential
equations using finite element collocation method.
In OCFE, the whole domain is divided into small sub
domains. The orthogonal collocation is applied within in
each sub-domain. The trial function and its first derivative
should be continuous at the boundaries of the elements.
In orthogonal collocation on finite elements, The global variable x varies in the ℓth element, where ℓ = 1, 2, 3,…,
ne. The node points are set at x1, x2, x3,…, xne+1 .
To apply the orthogonal collocation with in ℓth element, a
new variable is introduced in the ℓth element. The
variable is introduced in such a way that as x varies from
x to 1x , varies from 0 to 1 in the ℓth element such
that
1
x x
x x
.
Orthogonal collocation is applied on the variable within ℓth element, ℓ = 1,2,…,ne .To avoid the problem of double
calculation at the node points, the approximating function
and its first derivative are taken to be continuous at the
node points by using the principle of continuity.
1
1
xxyy
(4)
1
1
xxdx
dy
dx
dy
(5)
These conditions are also called the continuity conditions.
With the help of these conditions, the problem of double
calculation arising in the application of orthogonal
collocation on finite elements is overcome.
Solution of Boundary value Problems The mathematical equation involving diffusion-dispersion
phenomenon describing the pulp washing in packed beds
of finite length with adsorption isotherm is described by
Kukerja (1995) for washing zone is discussed below:
(6)
With adsorption isotherm
(7) equation (6) represent the mathematical models of
diffusion-dispersion of washing, is the time of the displacement, is the distance from the initial point of the displacing fluid, is the solute concentration, the average interstitial velocity co-efficient is , the
average interstitial velocity of the fluid is and the length the bed length is At the bed the boundary condition is:
(8)
)
And at the bed the initial condition
For
(10)
Conversion of the model into dimensionless form
Using the dimensionless variable change the relations in
dimensionless form by introducing the following:
The dimensionless time
, dimensionless
concentration of solute in liquor
, dimensionless
concentration of solute in fiber
, at the initial
stage of the experiment is equal to the ratio of total fluid
volume initial at the free value of the bed.
After dimensionless form, equation (6) becomes
(11)
(12)
(13)
And
(14)
These boundary conditions correspond to “radiation” at the
ends of a slab onto media at zero temperature. The general solution of equation (11) under the boundary condition for
an arbitrary temperature distribution is given by Carslaw
and Jaeger (1959) [61]. Result of the equation of the type
can be gotten by use of Laplace transform.
Taking the Laplace transform of then, we get:
Int. J. Sci. Res. in Mathematical and Statistical Sciences Vol. 7, Issue.3, June 2020
© 2020, IJSRMSS All Rights Reserved 47
̅̅ ̅ ̅̅ ̅ ∫
(15)
The equation (11) satisfies the initial condition (13) then one gets:
̅̅ ̅̅
̅̅ ̅
(16)
Where
(17)
Equations (12) and (13) of boundary conditions can be
written as:
̅̅ ̅ ̅̅ ̅
(
) ̅̅ ̅
The solution of equation (15) is:
̅̅ ̅
[
] [
]
[
] [
]
[
] [
]
[
] [
]
[
] [
]
(
)
(
)
(
)
(
)
(
)
(
)
The inverse transform of equation becomes
[
(
√
√
)
(
√
√
)
]
[
(
√
√
)
√
√
]
[
(
√
√
)
√
√
]
[
(
√
√
)
√
√
]
[
(
√
√
)
√
√
]
Int. J. Sci. Res. in Mathematical and Statistical Sciences Vol. 7, Issue.3, June 2020
© 2020, IJSRMSS All Rights Reserved 48
[
(
√
√
)
(
√
√
)
]
[
(
√
√
)
(
√
√
)
]
[
(
√
√
)
(
√
√
)
]
[
(
√
√
)
(
√
√
)
]
[
(
)
(
)
]
[
(
)
(
)
]
[
(
)
(
)
]
[
(
)
(
)
]
[
(
)
(
)
]
[
(
)
(
)
]
[
(
)
(
)
]
[
(
)
(
)
]
[
(
)
(
)
]
Now,
[ [
(
)
] [
(
)
]]
[
[
(
)
]
[
(
)
]]
[
[
(
)
]
[
(
)
]]
Int. J. Sci. Res. in Mathematical and Statistical Sciences Vol. 7, Issue.3, June 2020
© 2020, IJSRMSS All Rights Reserved 49
[
[
(
)
]
[
(
)
]]
[
[
(
)
]
[
(
)
]]
[
[
(
)
] [
(
)
]]
[
[
(
)
]
[
(
)
]]
[
[
(
)
]
[
(
)
]]
[
[
(
)
]
[
(
)
]]
For the solution of use some direct result of inverse transform through the complex integral transform are available in Carslaw et. al. (1959) [61]. The above
boundary value problem with suitable boundary and initial
condition solved by OCM and OCFE collocation
techniques. The complete analysis of description of
mathematical model for different values of peclet numbers
are presented both in Tabular and Graphical form.
The above boundary value problem with suitable boundary
and initial condition solved by OCM and OCFE
collocation techniques. The complete analysis of
description of mathematical model for different values of
peclet numbers is presented both in Tabular and Graphical form.
From table (1), the behavior of concentration has been
described for Peclet Number 4, 6, 8 and 10. It is clear from
tabulated values that OCFE gives better results for Pe=10
as comparison to Pe=4. From tables (2) and (3) it is clear
that as the value of Peclet number increase it goes in
converging stage in less time. From, figure (1) it is clear
that for Pe=10 solution profile converges more rapidly
than less value of peclet number. As the value of Peclet
number increases, solution profile going to converging in less time. From figure(2) and figure(3) it is clear that
solution profiles goes in converging stage more rapidly for
large value of peclet number that is for Pe=80, even in less
than t=1 as comparison to small values of peclet number,
which shows the efficiency of numerical technique. The
performance of the solute concentration with respect to the
dimensionless time and distance has been exposed in the
3-D graphs for different values of Peclet Number. The
comparison of OCM and OCFE is done in terms of their
relative errors. The comparison of OCM and OCFE in
terms of relative error for Pe=2 is given in table (4) and
shown graphically in figure (4). It can be seen easily that for Pe=2 OCM gives more relative error as comparison
with OCFE. For Pe=10 the comparison of OCM and
OCFE in terms of relative error is given table (5) and
graphically presented in figure (5).
Fig. 1: Behaviour of solution profiles at different Peclet numbers
(Pe)
Fig. 2: Behaviour of solution profiles at different Peclet numbers
(Pe)
Int. J. Sci. Res. in Mathematical and Statistical Sciences Vol. 7, Issue.3, June 2020
© 2020, IJSRMSS All Rights Reserved 50
Fig. 3: Behaviour of solution profiles at different Peclet numbers
(Pe)
Fig. 4: Comparison of relative error for OCM and OCFE at
Pe=10
Fig. 5: Comparison of relative error for OCM and OCFE at
Pe=20
Fig. 6: 3D behaviour of solute concentration profiles for Pe=5
Fig. 7: 3D behaviour of solute concentration profiles for Pe=10
Fig. 8: 3D behaviour of solute concentration profiles for Pe=20
Fig. 9: 3D behaviour of solute concentration profiles for Pe=40
Table 1: Exit solute concentration (c) for different values of peclet number (pe)
time t pe=4 pe=6 pe=8 pe=10
0.00E+00 1.04E+00 1.04E+00 1.04E+00 1.04E+00
2.00E-01 4.89E-01 5.54E-01 6.01E-01 6.39E-01
4.00E-01 2.49E-01 2.63E-01 2.70E-01 2.73E-01
6.00E-01 1.44E-01 1.39E-01 1.31E-01 1.22E-01
8.00E-01 8.80E-02 7.84E-02 6.72E-02 5.71E-02
1.00E+00 5.48E-02 4.58E-02 3.59E-02 2.78E-02
1.20E+00 3.44E-02 2.72E-02 1.97E-02 1.39E-02
1.40E+00 2.16E-02 1.64E-02 1.09E-02 7.06E-03
1.60E+00 1.36E-02 9.87E-03 6.14E-03 3.64E-03
1.80E+00 8.58E-03 5.96E-03 3.46E-03 1.89E-03
2.00E+00 5.40E-03 3.61E-03 1.96E-03 9.87E-04
2.20E+00 3.40E-03 2.19E-03 1.11E-03 5.17E-04
2.40E+00 2.14E-03 1.32E-03 6.30E-04 2.72E-04
2.60E+00 1.35E-03 8.02E-04 3.58E-04 1.43E-04
2.80E+00 8.51E-04 4.86E-04 2.03E-04 7.52E-05
3.00E+00 5.36E-04 2.95E-04 1.16E-04 3.96E-05
Int. J. Sci. Res. in Mathematical and Statistical Sciences Vol. 7, Issue.3, June 2020
© 2020, IJSRMSS All Rights Reserved 51
Table 2: Exit solute concentration (c) for different values of
peclet number (pe) time t pe=15 pe=20 pe=25 pe=30
0.00E+00 1.04E+00 1.04E+00 1.04E+00 1.04E+00
2.00E-01 7.10E-01 7.59E-01 7.97E-01 8.26E-01
4.00E-01 2.76E-01 2.74E-01 2.70E-01 2.66E-01
6.00E-01 1.01E-01 8.36E-02 6.92E-02 5.74E-02
8.00E-01 3.79E-02 2.52E-02 1.67E-02 1.09E-02
1.00E+00 1.46E-02 7.67E-03 3.93E-03 1.80E-03
1.20E+00 5.75E-03 2.37E-03 8.63E-04 5.37E-05
1.40E+00 2.31E-03 7.42E-04 1.56E-04 -1.66E-04
1.60E+00 9.45E-04 2.34E-04 1.43E-05 -1.09E-04
1.80E+00 3.91E-04 7.48E-05 -5.29E-06 -4.85E-05
2.00E+00 1.63E-04 2.41E-05 -4.25E-06 -1.80E-05
2.20E+00 6.85E-05 7.86E-06 -1.99E-06 -5.99E-06
2.40E+00 2.89E-05 2.58E-06 -7.83E-07 -1.86E-06
2.60E+00 1.23E-05 8.54E-07 -2.82E-07 -5.45E-07
2.80E+00 5.22E-06 2.85E-07 -9.58E-08 -1.48E-07
3.00E+00 2.22E-06 9.59E-08 -3.13E-08 -3.54E-08
Table 3: Exit solute concentration (c) for different values of
peclet number (pe) time t pe=40 pe=50 pe=65 pe=80
0.00E+00 1.04E+00 1.04E+00 1.04E+00 1.04E+00
2.00E-01 8.70E-01 9.00E-01 9.32E-01 9.54E-01
4.00E-01 2.56E-01 2.47E-01 2.35E-01 2.24E-01
6.00E-01 3.95E-02 2.72E-02 1.52E-02 7.78E-03
8.00E-01 4.40E-03 1.34E-03 -4.43E-04 -9.41E-04
1.00E+00 -3.60E-04 -1.46E-03 -2.54E-03 -3.37E-03
1.20E+00 -1.01E-03 -1.90E-03 -2.93E-03 -3.47E-03
1.40E+00 -7.71E-04 -1.49E-03 -2.58E-03 -3.46E-03
1.60E+00 -4.00E-04 -8.18E-04 -1.64E-03 -2.61E-03
1.80E+00 -1.56E-04 -3.11E-04 -5.85E-04 -8.24E-04
2.00E+00 -4.92E-05 -8.36E-05 -1.07E-04 -7.39E-05
2.20E+00 -1.35E-05 -1.82E-05 -1.17E-05 9.85E-06
2.40E+00 -3.39E-06 -4.17E-06 -5.90E-06 -1.15E-05
2.60E+00 -6.79E-07 -5.45E-07 -1.79E-06 -7.86E-06
2.80E+00 -1.28E-08 5.66E-07 2.56E-06 5.89E-06
3.00E+00 9.30E-08 5.85E-07 2.85E-06 7.78E-06
Table 4 : Comparison of OCM and OCFE for pe=2 time t EXACT OCM ERROR OCFE ERROR
0.00E+00 1.04E+00 1.08E+00 4.00E-02 1.04E+00 0.00E-02
2.00E-01 3.22E-01 8.92E-01 5.70E-01 3.90E-01 6.80E-02
4.00E-01 1.64E-01 5.75E-01 4.11E-01 2.12E-01 4.80E-02
6.00E-01 8.55E-02 3.48E-01 2.63E-01 1.25E-01 3.95E-02
8.00E-01 4.47E-02 2.09E-01 1.64E-01 7.49E-02 3.02E-02
1.00E+00 2.34E-02 1.25E-01 1.02E-01 4.49E-02 2.15E-02
1.20E+00 1.22E-02 7.51E-02 6.29E-02 2.69E-02 1.47E-02
1.40E+00 6.38E-03 4.50E-02 3.86E-02 1.61E-02 9.72E-03
1.60E+00 3.33E-03 2.70E-02 2.37E-02 9.67E-03 6.34E-03
1.80E+00 1.74E-03 1.62E-02 1.45E-02 5.80E-03 4.06E-03
2.00E+00 9.11E-04 9.69E-03 8.78E-03 3.48E-03 2.57E-03
2.20E+00 4.76E-04 5.81E-03 5.33E-03 2.08E-03 1.60E-03
2.40E+00 2.49E-04 3.48E-03 3.23E-03 1.25E-03 1.00E-03
2.60E+00 1.30E-04 2.09E-03 1.96E-03 7.49E-04 6.19E-04
2.80E+00 6.80E-05 1.25E-03 1.18E-03 4.49E-04 3.81E-04
3.00E+00 3.56E-05 7.50E-04 7.14E-04 2.69E-04 2.33E-04
Table 5 : Comparison of OCM and OCFE for pe=10
time t EXACT OCM ERROR OCFE ERROR
0.00E+00 1.04E+00 1.08E+00 4.00E-02 1.04E+00 0.00E-02
2.00E-01 6.31E-01 1.01E+00 3.79E-01 6.39E-01 8.00E-03
4.00E-01 2.73E-01 9.69E-01 6.96E-01 2.73E-01 0.00E-02
6.00E-01 1.24E-01 7.61E-01 6.37E-01 1.22E-01 2.00E-03
8.00E-01 5.95E-02 5.25E-01 4.66E-01 5.71E-02 2.40E-03
1.00E+00 2.96E-02 3.30E-01 3.00E-01 2.78E-02 1.80E-03
1.20E+00 1.52E-02 1.92E-01 1.77E-01 1.39E-02 1.30E-03
1.40E+00 7.89E-03 1.03E-01 9.51E-02 7.06E-03 8.30E-04
1.60E+00 4.16E-03 5.01E-02 4.59E-02 3.64E-03 5.20E-04
1.80E+00 2.21E-03 2.13E-02 1.91E-02 1.89E-03 3.20E-04
2.00E+00 1.18E-03 6.94E-03 5.76E-03 9.87E-04 1.93E-04
2.20E+00 6.29E-04 5.65E-04 6.40E-05 5.17E-04 1.12E-04
2.40E+00 3.37E-04 -1.70E-03 2.04E-03 2.72E-04 6.50E-05
2.60E+00 1.81E-04 -2.09E-03 2.27E-03 1.43E-04 3.80E-05
2.80E+00 9.71E-05 -1.76E-03 1.86E-03 7.52E-05 2.19E-05
3.00E+00 5.21E-05 -1.26E-03 1.31E-03 3.96E-05 1.25E-05
Table 6 : Comparison of OCM and OCFE for pe=20
time t EXACT OCM ERROR OCFE ERROR
0.00E+00 1.04E+00 1.08E+00 4.00E-02 1.04E+00 0.00E-02
2.00E-01 8.70E-01 9.85E-01 1.15E-01 8.70E-01 0.00E-02
4.00E-01 2.56E-01 1.06E+00 8.04E-01 2.56E-01 0.00E-02
6.00E-01 3.95E-02 9.48E-01 9.09E-01 3.95E-02 0.00E-02
8.00E-01 4.40E-03 7.05E-01 7.01E-01 4.40E-03 0.00E-02
1.00E+00 -3.60E-04 4.37E-01 4.37E-01 -3.60E-04 0.00E-02
1.20E+00 -1.01E-03 2.19E-01 2.20E-01 -1.01E-03 0.00E-02
1.40E+00 -7.71E-04 7.41E-02 7.49E-02 -7.71E-04 0.00E-02
1.60E+00 -4.00E-04 -3.05E-03 2.65E-03 -4.00E-04 0.00E-02
1.80E+00 -1.56E-04 -3.27E-02 3.25E-02 -1.56E-04 0.00E-02
2.00E+00 -4.92E-05 -3.55E-02 3.55E-02 -4.92E-05 0.00E-02
2.20E+00 -1.35E-05 -2.67E-02 2.67E-02 -1.35E-05 0.00E-02
2.40E+00 -3.39E-06 -1.58E-02 1.58E-02 -3.39E-06 0.00E-02
2.60E+00 -6.79E-07 -7.00E-03 7.00E-03 -6.79E-07 0.00E-02
2.80E+00 -1.28E-08 -1.52E-03 1.52E-03 -1.28E-08 0.00E-02
3.00E+00 9.30E-08 1.11E-03 1.11E-03 9.30E-08 0.00E-02
V. CONCLUSION
The model s solved using both the numerical technique
OCM and OCFE . The efficiency of model and numerical
technique is checked for different values of peclet number. It has been concluded that solution profiles rapidly
converges for large values of peclet number as comparison
to smaller one. Comparison has been presented both in
tabular and graphical form. This has been concluded that
relative error for OCFE is very less as comparison with
OCM. Thus OCFE is time friendly and more efficient
numerical technique as comparison to OCM.
Conflict of Interest: I declare that I have no conflict of
interests.
REFERENCES
[1] Cuomo S, Amore L D and Murli A (2007) Error analysis of a collocation
method for numerically inverting a Laplace transform in case of real samples.
Journal of Computational and Applied Mathematics 210:149-158.
[2] Davis GB (1985) A Laplace transform technique for the analytical solution of a diffusion-convection equation over a
finite domain. Applied Mathematical Modelling 9:69-71.
[3] Liao HT and Shiau CY (2000) Analytical solution to an axial dispersion model for the fixed-bed adsorber. AIChE Journal
46(6):1168-1176.
[4] Zheng Y and Gu T (1996) Analytical solution to a model for the
startup period of fixed-bed reactors. Chemical Engineering
Science 51(15):3773-3779.
[5] Jaiswal DK, Kumar A and Yadav RR (2011) Analytical solution to the one-dimensional advection-diffusion equation
with temporally dependent coefficients. Journal of Water
Resource and Protection 3:76-84.
Int. J. Sci. Res. in Mathematical and Statistical Sciences Vol. 7, Issue.3, June 2020
© 2020, IJSRMSS All Rights Reserved 52
[6] Kumar A, Jaiswal DK and Yadav RR (2012) Analytical solutions of one-dimensional temporally dependent advection-
diffusion equation along longitudinal semi-infinite
homogeneous porous domain for uniform flow. IOSR Journal
of Mathematics 2:1-11.
[7] Nurmuhammad A, Muhammad M, Mori M and Sugihara (2005) Double exponential transformation in the Sinc-collocation method for a boundary
value problem with fourth-order ordinary differential equation. Journal of
Computational and Applied Mathematics 182(1):32-50.
[8] He JH (2003) Homotopy perturbation method: a new non linear analytical technique. Applied Mathematics and Computation
135:73-79.
[9] Jia X, Zeng F and Gu Y (2013) Semi-analytical solutions to one-dimensional advection-diffusion equations with variable
diffusion coefficient and variable flow velocity. Applied
Mathematics and Computation 221:268-281.
[10] Rasmuson A and Nerettnieks I (1980) Exact solution of a model for diffusion in particles and longitudinal dispersion in packed
beds. AIChE Journal 26(4):686-690.
[11] Solsvik J and Jakobsen HA (2012) Effects of Jacobi polynomials on the numerical solution of the pellet equation
using the orthogonal collocation, Galerkin, tau, and least
squares methods. Computers and Chemical Engineering 39:1-
21.
[12] Caglar H, Caglar N and Elfaituri K (2006) B-spline interpolation compared with finite difference, finite element and
finite volume methods which applied to two-point boundary
value problems. Applied Mathematics and Computation
175:72-79.
[13] Fletcher CA J (1983) A comparison of finite element and finite difference solutions of the one-and two-dimensional Burgers
equations. Journal of Computational Physics 51:159-188.
[14] Gurarslan G, Karahan H, Alkaya D, Sari M, and Yasar M, (2013) Numerical Solution of Advection-Diffusion Equation
Using a Sixth-Order Compact Finite Difference Method.
Mathematical Problems in Engineering
http://dx.doi.org/10.1155/2013/ 672936.
[15] Zhao J. (2013) Compact finite difference methods for high order integro-differential equations. Applied Mathematics and
Computation 221:66-78.
[16] Caldwell J, Wanless P and Cook A E (1981) A finite element approach to Burgers equation. Applied Mathematical Modelling
5:189-193.
[17] Chen H and Jiang Z (2004) A characteristics mixed finite element method for Burgers equation. Journal of Applied
Mathematics and Computation 15:29-51.
[18] Luo Z, Li H and Sun P (2013) A reduced-order Crank-Nicolson finite volume element formulation based on POD method for
parabolic equations. Applied Mathematics and Computation
219:5887-5900.
[19] Onah SE (2002) Asymptotic behaviour of the Galerkin and the finite element collocation methods for a parabolic equation.
Applied Mathematics and Computation 127: 207-213.
[20] Robalo RJ, Almeida RM, Coimbra MC, and Rodrigues AE (2013) A Matlab implementation of the moving finite element
method for simulation of pulp washing problems. International
Conference on Approximation Methods and Numerical
Modelling in Environment and Natural Resources 13: 1-11.
[21] Sharma D, Jiwari R. and Kumar S (2012) Numerical solution of two point boundary value problems using Galerkin-Finite
element method. International Journal of Nonlinear Science
13:204-210.
[22] Ali A (2009) Chebyshev collocation spectral method for solving the RLW equation.International Journal of Nonlinear Science 7(2):131-142.
[23] Bialecki B (2003) A fast solver for the orthogonal spline collocation solution of
the biharmonic Dirichelt problem on rectangles. Journal of Computer Physics
191:601-621.
[24] Bialecki B and Fairweather G (2001) Orthogonal spline collocation methods for partial differential equations. Journal of Computational and Applied Mathematics
128(1-2):55-82.
[25] Botella O (2002). On a collocation B-spline method for the solution of the Navier-Stokes equations. Computers and Fluids 31:397-420.
[26] Danumjaya P and Nandakumarab A K (2006) Orthogonal cubic spline collocation method for the Cahn-Hilliard equation. Applied Mathematics and
Computation 182:1316-1329.
[27] Goh J, Majid AA and Ismail AIM (2012) Cubic B-Spline collocation method for one-dimensional heat and advection-
diffusion equations. Journal of Applied Mathematics 8:1-8.
[28] Houstis EN, Christara CC and Rice JR (1986) Quadratic-Spline Collocation methods for two point boundary value problems.
Computer Science Technical Reports 503:1-22.
[29] Johnson R W (2005) Higher order B-spline collocation at the Greville abscissa. Applied Numerical Mathematics 52:63-75.
[30] Mittal A (2016) Numerical Solution of Burgers equation using
efficient computational Technique using cubic B-splines.
International Journal of Theoretical & Applied Sciences
8(1):55-59.
[31] Mittal R C and Arora G (2010) Efficient numerical solution of fisher’s equation by using B-spline method. International Journal of Computer Mathematics
87(13):3039-3051.
[32] Morinishi Y, Tamano S and Nakabayashi K (2003) A DNS algorithm using B-spline collocation method for compressible turbulent channel flow. Computers
and Fluids 32(5):751-776.
[33] Pedas A and Tamme E (2006) Spline collocation method for integro-differential equations with weakly singular kernels. Journal of Computational and Applied
Mathematics 197:253-269.
[34] Rashidinia J, Mohammadi R and Jalilian R (2008) Cubic spline method for two-point boundary value problems. International
Journal of Engineering Science 19: 39-43.
[35] Sun W, Wu J and Zhang X (2007) Non conforming spline collocation methods in irregular domains. Numerical methods for partial differential equations
23(6):1509-1529.
[36] Woo G And Kim S (2001) Preconditioning C1- Quadratic spline collocation
method of elliptic equations by finite difference method. Bulletin of the Korean
Mathematical Society 38(1):17-27.
[37] Arden S, Wilde SNC and Langdon S (2007) A collocation method for high frequency scattering by convex. Journal of Computational and Applied
Mathematics 1-15.
[38] Feiz M (1997) A 1-D multigroup diffusion equation nodal
model using the orthogonal collocation method. Annals of
Nuclear Energy 24(3):187-196.
[39] Soliman MA (1998). Studies on the method of orthogonal collocation IV. Laguerre and Hermite orthogonal collocation method. Journal of King Saud
University Engineering Science 12:1-14.
[40] Villadsen JV and Stewart WE (1967). Solution of Boundary Value Problems by orthogonal collocation. Chemical
Engineering Science 22:1483-1501.
[41] Wu LY, Chung LL, Wu CH, Huang HH and Lin K W (2011). Vibrations of nonlocal Timoshenko beams using orthogonal collocation method. Procedia
Engineering 14:2394-2402.
[42] Arora S and Kaur I (2015) Numerical solution of heat conduction problems using orthogonal collocation on finite
elements. Journal of the Nigerian Mathematical Society 34:286-
302.
[43] Arora S and Kaur I (2016) An efficient scheme for numerical solution of Burgers equation using quintic Hermite interpolating
polynomials. Arabian Journal of Mathematics 5:23-34.
[44] Arora S and Potucek F (2012) Verification of mathematical model for displacement washing of kraft pulp fibres. Indian
Journal of Chemical Technology 19:140-148.
[45] Arora S, Dhaliwal SS and Kukreja VK (2005) Solution of two point boundary value problems using orthogonal collocation on
finite elements. Applied Mathematics and Computation 171:
358-370.
[46] Arora S, Dhaliwal SS and Kukreja V K (2006) A
computationally efficient technique for solving two point
boundary value problems in porous media. Applied
Mathematics and Computation 183: 1170-1180.
[47] Arora S, Dhaliwal SS and Kukreja VK (2006a). Application of orthogonal collocation on finite elements for solving non-linear
Int. J. Sci. Res. in Mathematical and Statistical Sciences Vol. 7, Issue.3, June 2020
© 2020, IJSRMSS All Rights Reserved 53
boundary value problems. Applied Mathematics and
Computation 183: 516-523.
[48] Arora S, Kaur I and Potucek F (2015a) Modelling of displacement washing of pulp fibers using the Hermite
collocation method. Brazilian Journal of Chemical Engineering
32(2):563-575.
[49] Arora S, Kaur I, Kumar H and Kukreja VK (2015b). A robust technique of cubic hermite collocation for solution of two phase
non linear model. Journal of King Saud University-Engineering
Sciences 1-7.
[50] Barrozo M A S, Henrique H M, Sartori D J M and Freire JT (2006). The use of the orthogonal collocation method on the study of the drying kinetics of soybean
seeds. Journal of Stored Products Research 42(3): 348-356.
[51] Chang PW and Finlayson BA (1977). Orthogonal collocation on finite elements
for elliptic equations. Advances in Computer Methods for Partial Differential
Equations-II 79-86.
[52] Ganaie IA, Arora S and Kukreja VK (2013). Modelling and simulation of a packed bed of pulp fibers using mixed
collocation method. International Journal of Differential
Equations 2013:1-7.
[53] Lang AW and Sloan DM (2002). Hermite collocation solution of near singular problems using numerical coordinate transformations based on adaptively.
Journal of Computational and Applied Mathematics 140:499-520.
[54] Mittal A, Ganaie IA, Kukreja VK, Parumasur N and Singh P (2013). Solution of diffusion-dispersion models using a
computationally efficient technique of orthogonal collocation
on finite elements with cubic Hermite as basis. Computers and
Chemical Engineering 58: 203-210.
[55] Mittal A, Ganaie IA, Parumasur N, Singh P and Kukreja VK (2013). Numerical solution of diffusion-dispersion models
using orthogonal collocation on finite elements with Hermite
basis for Robin conditions. Applied Mathematical Sciences 7
(34):1663-1673.
[56] Carey GF and Finlayson BA (1975). Orthogonal Collocation on finite elements. Chemical Engineering Science 30:587-596.
[57] Liu F & Bhatia SK (2001). Application of Petrov-Galerkin methods to transient boundary value problems in chemical
engineering: Adsorption with steep gradients in bidisperse
solids. Chemical Engineering and Science 56:3727-3735.
[58] Paterson WR and Cresswell DL (1971). A simple method for the calculation of effectiveness factors. Chemical Engineering
Science 26:605-616.
[59] Ma Z and Guiochon G (1991). Application of orthogonal collocation on finite elements in the simulation of non-linear
chromatography. Computers and Chemical Engineering
15(6):415-426.
[60] Arora S and Potucek F (2009). Modelling of displacement
washing of pulp: Comparison between model and experimental
data. Cellulose Chemistry Technology 43(7-8):307-315.
[61] Carslaw H S and Jaeger J C (1959). Conduction of heat in solids, Oxford University Press London.
AUTHORS PROFILE
Dr. Nisu Jain Ph.D. Mathematics from Punjabi University,
Patiala in 2019. She is currently working as Assistant
Professor in Department of mathematics, Govt. Mohindra
College, Patiala since September 2019. His main research
work focus on Differential equations. She has 9 years of
teaching experience.
.