Impact of Numerical Technique on the Solution of Boundary Value … · 2020. 7. 6. · equation,...

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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: [email protected] 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].

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Int. J. Sci. Res. in Mathematical and Statistical Sciences Vol. 7, Issue.3, June 2020


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:



̅̅ ̅ ̅̅ ̅ ∫

(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

[

(

√

√

)

(

√

√

)

]

[

(

√

√

)

√

√

]

[

(

√

√

)

√

√

]

[

(

√

√

)

√

√

]

[

(

√

√

)

√

√

]



[

(

√

√

)

(

√

√

)

]

[

(

√

√

)

(

√

√

)

]

[

(

√

√

)

(

√

√

)

]

[

(

√

√

)

(

√

√

)

]

[

(

)

(

)

]

[

(

)

(

)

]

[

(

)

(

)

]

[

(

)

(

)

]

[

(

)

(

)

]

[

(

)

(

)

]

[

(

)

(

)

]

[

(

)

(

)

]

[

(

)

(

)

]

Now,

[ [

(

)

] [

(

)

]]

[

[

(

)

]

[

(

)

]]

[

[

(

)

]

[

(

)

]]



[

[

(

)

]

[

(

)

]]

[

[

(

)

]

[

(

)

]]

[

[

(

)

] [

(

)

]]

[

[

(

)

]

[

(

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[

[

(

)

]

[

(

)

]]

[

[

(

)

]

[

(

)

]]

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)


(Pe)




(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




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



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.



[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



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.

.

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