Application of the Homotopy Perturbation Method in Steady State Brusselator Model

Advances in Chemical Science Volume 3 Issue 3, November 2014

www.seipub.org/sepacs

Application of the Homotopy Perturbation Method in Steady State Brusselator Model V.Ananthaswamy1, P. Jeyabarathi2

*

Department of Mathematics, The Madura College (Autonomous), Maduri, Tamil Nadu, India

1

Department of Mathematics, K.L.N. Engineering College, Pottapalayam, Tamil Nadu, India

2 1

ananthu9777@rediffmail.com; 2 bsc.jeya@yahoo.com

Abstract In this research paper the analytical expression for the concentrations of the reaction-diffusion equations in steady state Brusselator model is discussed. The approximate analytical solution of the steady state non-linear boundary value problem is derived by using the Homotopy perturbation method. The Homotopy-perturbation method has the advantage of being more concise for analytical and numerical functions. This method can be easily extended to find the solution of the other strongly non-linear initial and boundary value problems in engineering and chemical sciences.

Keywords Two-point Boundary Value Problem; Non-linear ReactionDiffusion Equation; Steady State Brusselator Model; Homotopy Perturbation Method

Introduction Systems of ordinary differential equations (ODEs) arise in mathematical models throughout science and engineering. When an explicit condition (or conditions) that a solution must satisfy is specified at one value of the independent variable, usually its lower bound, this is referred to as an initial value problem (IVP). When the conditions to be satisfied occur at more than one value of the independent variable, this is referred to as a boundary value problem (BVP). If there are two values of the independent variable at which conditions are specified, then this is a two-point boundary value problem (TPBVP). TPBVPs occur in a wide variety of problems, including the modeling of chemical reactions, heat transfer, and diffusion. They are also of interest in optimal control problems. For a typical TPBVP, when expressed as system of first-order ODEs, there are one or more unknown initial states. Solving the TPBVP is equivalent to finding these initial states.

In other cases, the initial states are known, but there are unknown model parameters, and solving the TPBVP requires finding values for these parameters.

There may also be a combination of unknown initial states and parameters that must be determined to solve theTPBVP. A TPBVP may have no solution, a single solution, or multiple solutions. There are many techniques available for the numerical solution of TPBVPs for ODEs [Heath (2002), Chen et. al (2005 & 2006) ].

Reaction-diffusion (RD) systems arise frequently in the study of chemical and biological phenomena and are naturally modeled by partial differential equations.

Finding accurate and efficient methods for solving nonlinear system of PDEs has long been an active research undertaking. In this analysis, we study the reaction-diffusion Brusselator model which models a chemical reaction diffusion process by homotopy perturbation method. Wazwaz (2000) used the Adomian decomposition method to handle the reaction-diffusion Brusselator model. In recent years, much attention has been given to the study of the Homotopy-perturbation method (HPM) for solving a wide range of problems whose mathematical models yield differential equation or system of differential equations.

Mathematical Formulations of the Problem The steady state Brusselator model is a reactiondiffusion system in fixed, bounded and one dimensional medium. The reaction scheme of the system is [Lin et. al (2008)]

A X 2 X Y 3 X B  X Y  D X E The system is open, allowing the concentrations of the reactants and to be kept at a constant level throughout the reaction/diffusion medium. In the event of all forward rate constants set to one and all reverse rate constants set to zero, the steady-state

31

www.seipub.org/sepacs

Advances in Chemical Science Volume 3 Issue 3, November 2014

concentration profiles for X and Y are described are by

DX

2 X  ( B  1) X  A  X 2Y z 2

(1)

 2Y  X 2Y  B X z 2

(2)

DY

Where A , B , X and Y are real and denote the concentrations of the corresponding chemical species, DX and DY are the diffusion coefficients for X and Y respectively. The spatial coordinates in the reaction/diffusion medium is the independent variable z 0, L . The corresponding boundary

conditions are

X (0)  X ( L)  A and Y (0)  Y ( L)  B / A

(3)

Solution of the Boundary Value Problem Using the Homotopy Perturbation Method Linear and non-linear phenomena are of fundamental importance in various fields of science and engineering. Most models of real – life problems are still very difficult to solve. Therefore, approximate analytical solutions such as Homotopy perturbation method (HPM) [Ghori et. al (2007), Ozis et. al(2007), Li et. al(2007), Mousa et. al(2008), He (1999 and 2003), Ariel (2010), Ananthaswamy et. al(2012) and Subha et. al(2014) ] were introduced. This method is the most effective and convenient ones for both linear and nonlinear equations. Perturbation method is based on assuming a small parameter. The majority of nonlinear problems, especially those having strong nonlinearity, have no small parameters at all and the approximate solutions obtained by the perturbation methods, in most cases, are valid only for small values of the small parameter. Generally, the perturbation solutions are uniformly valid as long as a scientific system parameter is small. However, we cannot rely fully on the approximations, because there is no criterion on which the small parameter should exist.

Thus, it is essential to check the validity of the approximations numerically and/or experimentally.

To overcome these difficulties, HPM have been proposed recently.

Recently, many authors have applied the Homotopy perturbation method (HPM) to solve the nonlinear boundary value problems in physics and engineering sciences [Ghori et. al (2007), Ozis et. al(2007), Li et.

32

al(2007), Mousa et. al(2008)]. Recently this method is also used to solve some of the non-linear problem in physical sciences [He (1999 and 2003)]. This method is a combination of Homotopy in topology and classic perturbation techniques. Ji-Huan He used to solve the Lighthill equation (1999), the Diffusion equation (2003) and the Blasius equation [He (2003) and Ariel (2010)].

The HPM is unique in its applicability, accuracy and efficiency. The HPM uses the imbedding parameter p as a small parameter, and only a few iterations are needed to search for an asymptotic solution.

Appendix A: Basic Concept of the Homotopy Perturbation Method [He (1999 and 2003)] To explain this method, let us consider the following function:

Do (u)  f (r )  0,

r 

(A.1)

with the boundary conditions of Bo (u ,

u )  0, n

r 

(A.2)

where Do is a general differential operator, Bo is a boundary operator,

f (r ) is a known analytical

function and  is the boundary of the domain  . In general, the operator Do can be divided into a linear part L and a non-linear part N . The eqn.(A.1) can therefore be written as

L(u)  N (u)  f (r )  0

(A.3)

By the Homotopy technique, we construct Homotopy v(r, p) :  [0,1]   that satisfies

a

H (v, p)  (1  p)[ L(v)  L(u0 )]  p[ Do (v)  f (r )]  0 (A.4) H (v, p)  L(v)  L(u0 )  pL(u0 )  p[ N (v)  f (r )]  0 (A.5) where p  [0, 1] is an embedding parameter, and u0 is an initial approximation of the eqn.(A.1) that satisfies the boundary conditions. From the eqns. (A.4) and (A.5), we have

H (v,0)  L(v)  L(u0 )  0

(A.6)

H (v,1)  Do (v)  f (r )  0

(A.7)

When p=0, the eqns.(A.4) and (A.5) become linear equations. When p =1, they become non-linear equations. The process of changing p from zero to unity is that of L(v)  L(u0 )  0 to Do (v)  f (r )  0 . We first use the embedding parameter p as a small

Advances in Chemical Science Volume 3 Issue 3, November 2014

parameter and assume that the solutions of the eqns.

(A.4) and (A.5) can be written as a power series in p :

v  v0  pv1  p 2 v2  ...

(A.8)

Setting p  1 results in the approximate solution of the eqn. (A.1):

u  lim v  v0  v1  v2  ...

(A.9)

p 1

This is the basic idea of the HPM. Appendix B: Solution of the Non-linear Eqns.

(1)-(3) by Using the Homotopy Perturbation Method

www.seipub.org/sepacs

 d 2 (Y0  pY1  p 2Y2  .....)   dz 2  

1  p  

 d 2 (Y0  pY1  p 2Y2  .....)    2 dz     ( X  pX  p 2 X  ....) 2   0 1 2      (Y0  pY1  p 2Y2  .....)   p    0 2    B ( X 0  pX 1  p X 2  ....)      DY             

(B.6)

Comparing the coefficients of like powers of p in the eqns.(B.5) and (B.6) we get

In this Appendix, we indicate how the eqns. (4) and (5) is derived in this paper. To find the solution of the eqns.(1)-(3) ,we construct the Homotopy as follows:

d2X

1  p  

2



( B  1) X   DX 

 dz  d 2 X  ( B  1) X  A  X 2Y  p 2  DX   dZ

p0 :

d 2 X0 dz

2

 ( B  1) X 0   0 DX  

p0 : (B.1)

   0  

p1 :

 d 2Y   d 2Y ( X 2Y  BX )   p   2   0 2 DY  dz   dz 

1  p  

(B.2)

The analytical solution of the eqns. (B.1) and (B.2) are

d 2 X1 dz

2

d 2Y0 dz 2

0

(B.8)

 ( B  1) X 1   A  X 02 Y0     DX  DX  

p1 :

(B.7)

 BX 0  X 02 Y0   DY dz 

d 2Y1 2

   0 

   0 

(B.10)

The initial approximations are as follows:

X  X 0  pX1  p 2 X 2  ..........

(B.3)

X 0  0   X 0 ( L)  A and Y0  0   Y0 ( L)  B / A

Y  Y0  pY1  p 2Y2  ..........

(B.4)

X i  0   X i ( L)  0 and

Substituting the eqns.(B.3) and (B.4) eqns.(B.1) and (B.2) respectively we get

into the

Y0  0   Y0 ( L)  B / A, i  1, 2,3.....

(B.11) (B.12)

Solving the eqns.(B.7)-(B.10) and using the boundary condition eqn.(B.11) and (B.12), we obtain the following results:

 d 2 ( X 0  pX 1  p 2 X 2  ....)    2 dz   1  p   ( B  1)( X 0  pX 1  p 2 X 2  ....)    DX    d 2 ( X 0  pX 1  p 2 X 2  ...)    2 dz     ( B  1)( X  pX  p 2 X  ....)  A    0 1 2      ( X 0  pX 1  p 2 X 2  ....) 2  p    0 2   (Y0  pY1  p Y2  ...)     DX             

(B.9)

  A k L kz kL k z X0    (1  e ) e  (1  e ) e  (B.13)  2sinh(kL) 

Y0  (B.5)

B A

(B.14)

 A   k1 k L kz kL k z X1     (1  e ) e  (1  e ) e    2 2sinh( kL )    k DX   (1  e  k L ) 2 e 2 k z  (1  ek L ) 2 e 2 k z   AB  kL k L   6 (1  e ) (1  e )   2 2 12 k DX sinh (k L)   

  

       (B.15)

33

www.seipub.org/sepacs

 AB Y1   2  DY k

Advances in Chemical Science Volume 3 Issue 3, November 2014



kL 2 k L 2   A B (1  e )  (1  e )    16 DY k 2 sinh 2 (k L)  

 A B (1  e k L ) (1  e  k L ) ( L z  z 2 )     4 DY sinh 2 (k L)    k L k z k L  k  A B (1  e ) e  (1  e ) e z      2 DY sinh 2 (k L)    k L 2 2 k z k L 2  2  A B (1  e ) e  (1  e ) e k z   16 k 2 DY sinh 2 (k L) 



   





k

(B.16)



( B  1) DX

 (1  e k L ) 2  (1  ek L ) 2  AB  6 (1  ek L ) (1  e  k L )    k1  12 k 2 DX sinh 2 (k L)

(6)

(7)

Results and Discussions

   

respectively.

Figure 1 and 2 shows the dimensionless concentration X versus the dimensionless distance z. From the Fig. 1 (a)- (g), it is clear that, when the dimensionless L constant increases, the corresponding concentration X ( z ) decreases in some fixed values of

According to the HPM, we can conclude that

the dimensionless parameters DX , DY , A and B. From

where k and k1 are defined by the eqns. (6) and (7)

X  lim X ( z)  X 0  X1

(B.17)

Fig.2 (a)-(b), it is noted that when the diffusion coefficient increases, the corresponding DX

Y  lim Y ( z )  Y0  Y1

(B.18)

concentrations X ( z ) also increases in some fixed

p 1

p 1

After putting the eqns. (B.13) and (B.15) into an eqn.(B.17), and (B.14) and (B.16) into an eqn. (B.18), we obtain the solution in the text eqns.(4) and (5).

values of the dimensionless parameters and for various ranges of the dimensionless constant L . (a)

The approximate analytical solution of eqns.(1)-(3) using the Homotopy perturbation method is given by

 k A  k L kz kL k z X ( z)   1  (1  e ) e  (1  e ) e 2sinh( kL )    A    2   k DX 



 (4)

  (1  e k L ) 2 e2 k z  (1  ek L ) 2 e2 k z    AB    6 (1  ek L ) (1  e k L )      2 2 12 k DX sinh (k L)       B  AB Y ( z )    A  DY k 2



kL 2 k L 2   A B (1  e )  (1  e )    16 DY k 2 sinh 2 (k L)  

 A B (1  e k L ) (1  e  k L ) ( L z  z 2 )     4 DY sinh 2 (k L)    k L k z k L  k  A B (1  e ) e  (1  e ) e z      2 DY sinh 2 (k L)    k L 2 2 k z k L 2  2  A B (1  e ) e  (1  e ) e k z   16 k 2 DY sinh 2 (k L) 





where

34



(b)

   

(5)

   

Advances in Chemical Science Volume 3 Issue 3, November 2014

(c)

www.seipub.org/sepacs

(g)

(d) FIGURE 1: DIMENSIONLESS CONCENTRATION X VERSUS THE DIMENSIONLESS DISTANCE z . THE CONCENTRATION X ( z ) WERE COMPUTED USING THE EQN. (4) IN SOME FIXED VALUES OF THE DIMENSIONLESS PARAMETERS DX , DY , A and B , WHEN (A) L  0.05 , (B) L  0.10 , (C) L  0.15, (D) L  0.20, (E) L  0.25, (F) L  0.30 AND (G) L  0.35 . (a)

(e)

(b)

(f)

FIGURE 2: DIMENSIONLESS CONCENTRATION X VERSUS THE DIMENSIONLESS DISTANCE z . THE CONCENTRATION X ( z ) WERE COMPUTED USING THE EQN. (4) IN SOME FIXED VALUES OF THE DIMENSIONLESS PARAMETERS DY , A and B AND VARIOUS VALUES OF THE DIFFUSION COEFFICIENT DX WHEN (A) L  0.10 AND (B) L  0.35 .

35

www.seipub.org/sepacs

Advances in Chemical Science Volume 3 Issue 3, November 2014

(a)

(b)

(e)

(f)

(c)

(g)

(d)

FIGURE 3: DIMENSIONLESS CONCENTRATION Y VERSUS THE DIMENSIONLESS DISTANCE z . THE CONCENTRATION Y ( z ) WERE COMPUTED USING THE EQN. (4) IN SOME FIXED VALUES OF THE DIMENSIONLESS PARAMETERS DX , DY , A and B , WHEN (A) L  0.05 , (B) L  0.10 , (C) L  0.15, (D) L  0.20, (E) L  0.25 , (F) L  0.30 AND (G) L  0.35 .

Figure 3 and 4 shows the dimensionless concentration Y versus the dimensionless distance z. From the Fig. 2 (a)- (g), it is inferred that when the dimensionless L constant increases, the corresponding concentration Y ( z ) also increases in some fixed values of the dimensionless parameters DX , DY , A and B. From Fig.4 (a)-(b), it is observed that when the

36

Advances in Chemical Science Volume 3 Issue 3, November 2014

diffusion coefficient DX increases, the corresponding concentrations Y ( z ) also increases in some fixed values of the dimensionless parameters and for various ranges of the dimensionless constant L . (a)

www.seipub.org/sepacs

Homotopy-perturbation method is a promising tool for linear and non-linear system of ordinary and partial differential equations.

ACKNOWLEDGEMENT

The authors are also thankful to Shri. S. Natanagopal, the Secretary, The Madura College Board, Madurai, Dr. R. Murali, The Principal, The Madura College (Autonomous) and Mr. S. Muthukumar, Head of the Department of Mathematics, The Madura College (Autonomous), Madurai, Tamil Nadu, India for their constant encouragement.

Appendix C: Nomenclature Symbo l

(b)

Meaning

A

Dimensionless constant

B

Dimensionless constant Steady state concentration profile for the chemical specie A Steady state concentration profile for the chemical specie B

X

Y z

Dimensionless distance

L

Dimensionless constant Diffussion coefficient for the concentration of the chemical specie A Diffussion coefficient for the concentration of the chemical specie B

DX DY REFERENCES FIGURE 4: DIMENSIONLESS CONCENTRATION X VERSUS THE DIMENSIONLESS DISTANCE z . THE CONCENTRATION X ( z ) WERE COMPUTED USING THE EQN. (4) IN SOME FIXED VALUES OF THE DIMENSIONLESS PARAMETERS DY , A and B AND VARIOUS VALUES OF THE DIFFUSION COEFFICIENT DX WHEN (A) L  0.10 AND (B) L  0.35 .

Conclusion This investigation is carried out using the Homotopy perturbation method (HPM) to approximate the solutions of well-known steady state Brusselator reaction-diffusion model. The HPM is very simple and straightforward. HPM avoids the difficulties arising in finding the Adomian polynomials and transformation formulas.Very recently, the HPM has been widely used to solve many problems in engineering and sciences because of it's reliability and the reduction in the size of computations. It is shown that the

Ananthaswamy V.,

Rajendran L., “Analytical solution

of non-isothermal diffusion- reaction processes

and

effectiveness factors, Article ID 487240, 2012, 1-14 (2012).

Ananthaswamy V., Rajendran L., “Analytical solution of two-point non-linear boundary value problems in a porous catalyst particles”, International Journal of Mathematical Archive. 3 (3), 810-821 (2012).

Ananthaswamy V., Rajendran L., “Analytical solutions of

some

value

two-point

problems”,

non-linear

Applied

elliptic

Mathematics.

boundary 3,

1044-

1058 (2012).

Ariel P.D., “Alternative approaches to construction of Homotopyperturbation algorithms, Nonlinear. Sci. Letts.

A., 1, 43-52 (2010).

Chen Y., “Dynamic systems optimization. Ph.D. thesis.

37

www.seipub.org/sepacs

Advances in Chemical Science Volume 3 Issue 3, November 2014

reaction-diffusion Brusselator model”, Appl. Math.

Los Angeles”, CA: University of California, (2006).

Chen

Y.,

Manousiouthakis

V.,

“Identification

of

all

solutions of TPBV problems”,Presented at AIChE Annual Meeting. Cincinnati, OH, Paper #11e (2005).

Ghori

Q.

K.,

A.

M.,

“Application of Homotopy perturbation method

to

squeezing

Ahmed

flow

M.,

of

a

and

Siddiqui

Newtonian

fluid”,

Int.

J.

Nonlinear Sci. Numer. Simulat. 8, 179-184 (2007).

He J. H., “A simple perturbation approach to Blasius equation”,

Appl.

Math.

Comput.

140,

217-222

(2003).

He

J. H., “Homotopy perturbation method: a new nonlinear

analytical

technique”,

Appl.

Math.

Comput. 135, 73-79 (2003).

He

J. H., “Homotopy perturbation technique”, Comp Meth.Appl. Mech. Eng. 178, 257-262 (1999).[10]

Heath M. T., “Scientific computing: An introductory survey” (2nd ed.). New York, NY:

McGraw-Hill

(2002).

Li S.J., Liu Y. X., “An Improved approach to nonlinear dynamical system identification using PID neural networks”, Int. J. Nonlinear Sci. Numer. Simulat. 7 , 177182 (2006).

Lin Y., Enzer J. A., solutions

of

and Stadtherr M. A., “Enclosing all

two-point

boundary

value

problems

for ODE”, Computers and Chemical Engineering.

32, 1714–1725 (2008).

Mousa M. M., Ragab S. F., Nturforsch Z., “Application of the Homotopy perturbation method to linear and nonlinear

Schrödinger

equations“,.

Zeitschrift

für

Naturforschung. 63, 140-144 (2008).

Ozis T., Yildirim A., “A Comparative study of He’s Homotopy

perturbation

method

for

determining

frequency-amplitude relation of a nonlinear oscillator with

discontinuities”, Int. J. Nonlinear Sci. Numer.

Simulat. 8, 243-248 (2007).

Subha

M.,

Ananthaswamy

V.,

and

Rajendran

L.,

“Analytical solution of non-linear boundary value problem for the electrohydrodynamic flow equation”, International

Journal

of

Automation

and

Control

Engineering. 3(2), 48-56 (2014).

Wazwaz A. M., “The decomposition method applied to systems of partial differential equations and to the

38

Comput. 110, 251-264 (2010).

Author Introduction Dr. Vembu Ananthaswamy received his M.Sc. Mathematics degree from The Madura College (Autonomous), Madurai-625011, Tamil Nadu, India during the year 2000. He has received his M.Phil degree in Mathematics from Madurai Kamaraj University, Madurai, Tamil Nadu, India during the year 2002. He has received his Ph.D., degree (Under the guidance of Dr. L. Rajendran, Assistant Professor, Department of Mathematics, The Madura College, Tamil Nadu, India) from Madurai Kamaraj University, Madurai, Tamil Nadu, India, during the year October 2013. He has more than 14 years and 6 months of teaching experiences for Engineering Colleges, Arts & Science Colleges and Deemed University.

He has 3 years of research experiences. At present he is working as Assistant Professor, Department of Mathematics, The Madura College (Autonomous), Madurai-625 011, Tamil Nadu, India from 2008 onwards. His present research interest includes: Mathematical modeling based on differential equations and asymptotic approximations, Analysis of system of non-linear reaction diffusion equations in physical, chemical and biological sciences, Numerical Analysis, Mathematical Biology, Mathematical and Computational Modeling, Mathematical Modeling for Ecological systems. He has participated more than 20 State level/National/International seminars and workshops and presented 7 research papers in National and International Conferences.

He has published more than 30 research articles in peer-reviewed National and International Journals and communicated 5 research articles in reputed National and International Journals.

Presently he is Reviewer/Editorial Board Member/Advisory Board Member in 67 reputed National and International Journals. He has reviewed more than 17 research articles in reputed International Journals. Currently he has been doing one ongoing minor research project sanctioned by UGC in the duration of 18 months.

Ms. Ponraj Jeyabarathi received her M.Sc., in Mathematics from Lady Doak College (Autonomous), Madurai during the year 2011. She has received her M.Phil., degree in Mathematics from The Madura College (Autonomous), Maudrai, Tamil Nadu, India during the year 2012. Her M.Phil., thesis title is “Analytical expression of a Steady – State Concentration of the Species Enzyme-Containing Polymer Modified Electrode”. She has more than 2 years and 6 months of teaching experiences in Arts & Science and

Advances in Chemical Science Volume 3 Issue 3, November 2014

Professional Colleges. Currently, she is working as Assistant Professor in the Department of Mathematics, K.L.N. Engineering College, Pottappalayam, Madurai, Tamil Nadu, India from the year 2013 to till date. She is going to register her Ph.D., degree under the guidance of Dr. V. Ananthaswamy, Assistant Professor, Department of

www.seipub.org/sepacs

Mathematics, The Madura College (Autonomous), Madurai, Tamil Nadu, India. Her current research area includes Mathematical Modeling in Biological sciences, Asymptotic methods, perturbation techniques. She has participated in National and International Conferences.

39