An Approximation to the Solution of the Brusselator System by Adomian Decomposition Method and Comparing the Results with Runge-Kutta Method

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1 Int. J. Contemp. Mat. Sciences, Vol. 2, 27, no. 2, An Approximation to te Solution of te Brusselator System by Adomian Decomposition Metod and Comparing te Results wit Runge-Kutta Metod J. Biazar and Z. Ayati Department of Matematic, Faculty of Science University of Guilan, P.O. Box 1914, Rast, Iran Abstract Adomian decomposition metod, as a convenience device as been used to solve many functional equations so far. In tis manuscript, we consider a system of nonlinear ordinary differential equations, wic governs on general reaction in biocemistry as a teoretical problem of concentration kinetics. Tese system, wic is known as Brusselator system as been solved by applying Adomian decomposition metod and te results are compared wit te results of runge kutta metod. Keywords: decomposition metod, Brusselator system, runge kutta metod 1 Introduction Brusselator system as te following general form: x t = a + x2 y (b +1)x, y t = bx x2 y. (1) were a> and a> are constant(google searc). It is one of te simplest but most fundamental models wic displays biological and cemical oscillations. Let ave te following initial conditions: x() = N 1 y() = N 2 (2)

2 984 J. Biazar and Z. Ayati 2 Solution of te system (1) by te Adomian decomposition metod: Let s consider te system (1) in te operator form: L t x = a + x 2 y (b +1)x, L t y = bx x 2 y. (3) were L t = t By applying te inverse operatorl 1 t system (3) we derive: = (.)dt to eac equation in te x(t) =N 1 + at + (x2 y (b +1)x)dt y(t) =N 2 + (bx x2 y)dt. Adomian decomposition metod, well addressed in [1, 2] considers x and y as te summations of two series, say (4) x =Σ n= x n, y =Σ n= y n (5) And te nonlinear term x 2 y is represented as: x 2 y =Σ n= A n(x,x 1,...,x n,y,y 1,...,y n ). (6) Were A n s are called Adomian polynomials and sould be determined. Adomian polynomials ave been calculated by using an alternate algoritm for calculating Adomian polynomial [3]. Terefore we ave: A = x 2 y A 1 =2x y x 1 + x 2 y 1 A 2 = x 2 1 y +2x y 1 x 1 +2x y x 2 + x 2 y 2 A 3 =2x 1 y x 2 + x 2 1y 1 +2x y 2 x 1 +2x y 1 x 2 +2x y x 3 + x 2 y 3 A 4 = x 2 2 y +2x 1 y 1 x 2 +2x 1 y x 3 +x 2 1 y 2+2x y 3 x 1 +2x y 2 x 2 +2x y 1 x 3 +2x y x 4 +x 2 y 4 Substituting (5) and (6) into (4) leads to te following system, Σ n= x n = N 1 + at +Σ n= A ndt (b + 1)Σ n= x ndt Σ n= y n = N 2 + bσ n= x ndt Σ n= A ndt. (7)

3 Approximate solutions of Brusselator system 985 From wic we defined x = N 1 + at, y = N 2 x n+1 = A ndt (b +1) x ndt n =, 1, 2,... (8) y n+1 = b x ndt A ndt n =, 1, 2,... Knowing Adomian polynomial, few first terms of te series (5), are calculated using te sceme (9). x 1 = 1 3 a2 N 2 t 3 + an 1 N 2 t 2 abt2 2 at2 2 + N 2 1 N 2t N 1 bt N 1 t y 1 = 1 3 a2 N 2 t 3 an 1 N 2 t 2 + abt2 2 N 2 1 N 2t + N 1 bt x 2 = 1 18 a4 N 2 t a3 N 2 2 t a3 N 1 N 2 t a3 bt a2 N 2 t a2 bn 2 t a2 N 1 N 2 2 t a2 N 2 1 N 2 t a2 bn 1 t abt ab2 t at3 4 3 an 1N 2 t an 2 1 N 2 2 t abn 2 1 t3 an 3 1 N 2t abn 1N 2 t 3 + N 3 1 N 2 2 t2 3 2 N 2 1 N 2t b2 N 1 t 2 + bn 1 t bn 3 1 t2 1 2 N 4 1 N 2t N 1t bn 2 1 N 2t 2 y 2 = 1 18 a4 N 2 t a3 N 2 2 t a3 N 1 N 2 t a3 bt a2 N 2 t a2 bn 2 t a2 N 1 N 2 2 t a2 N 2 1 N 2 t a2 bn 1 t abt3 1 6 ab2 t abn 1N 2 t an 2 1 N 2 2 t3 5 6 abn 2 1 t3 + an 3 1 N 2t 3 + an 1 N 2 t 3 N 3 1 N 2 2 t bn 2 1 N 2t b2 N 1 t bn 1t 2 + N 2 1 N 2t bn 3 1 t N 4 1 N 2t 2. We can determine te components x,x 1,x 2,... and y,y 1,y 2,... as many as is necessary to enance te desired accuracy for te approximations. So te approximations x (n) =Σ n 1 i= x i and y (n) =Σ n 1 i= y i, determine n-terms approximation to te solutions.

4 986 J. Biazar and Z. Ayati 3 Solution of te system (1) by te runge kutta metod To solve te Brusselator system, (1), by Runge-Kutta metod, let s define te following functions: f(t, x, y) =a + x 2 (t)y(t) (b +1)x(t) g(t, x, y) =bx(t) x 2 (t)y(t) Runge-Kutta metod computes te approximations by te following iterative equations x n+1 = x n + 6 (m 1 +2m 2 +2m 3 + m 4 ) y n+1 = y n + 6 (n 1 +2n 2 +2n 3 + n 4 ) Were m 1 = f(t n,x n,y n ), n 1 = g(t n,x n,y n ) m 2 = f(t n + 2,x n + m 1 2,y n + n 1 n 2 = g(t n + 2,x n + m 1 2,y n + n 1 m 3 = f(t n + 2,x n + m 2 2,y n + n 2 n 3 = g(t n + 2,x n + m 2 2,y n + n 2 m 4 = f(t n +, x n + m 3, y n + n 3 ) n 4 = g(t n +, x n + m 3, y n + n 3 ) 4 Numerical results To illustrate Adomian decomposition metod and comparing te results of tis metod wit te results of Runge-Kutta metod, ere are two examples. Example1 Consider te following system: x t = 2x + x2 y y t = x x2 y.

5 Approximate solutions of Brusselator system 987 were x() = 1, y() = 1 By using Adomian decomposition metod we ave x n+1 = y n+1 = x =1, y =1 A n dt 2 x n dt n =, 1, 2,... x n dt A n dt n =, 1, 2,... Ten-terms approximations to te solutions by using tis sceme, te following are computed. x =1 t t3 3 8 t t t t t t9 y = t2 1 2 t t4 3 4 t t t t t9 For some specified values of, te results of Adomian metod and Runge-Kutta metod are compared in table 1. table1 A comparison between te solutions of Ex. 1 by Adomian metod and Runge Kutta metod t A.D.M Runge-Kutta metod t x(t) y(t) x(t) y(t) Example2 Consider system (1), by te following coefficients: N 1 =2, N 2 =3, a =.2, b =.1 By using (8), five-terms approximation to te solutions would be as: x =2+9.82t t t t t t t t t 9

6 988 J. Biazar and Z. Ayati t t t 12 y =3 11.8t t t t t t t t t t t t 12 In table 2 te results of Adomian decomposition metod and Runge-Kutta metod, for some value of t, are presented for comparison. table2 A comparison between te solutions of Ex. 2 by Adomian metod and Runge Kutta metod t A.D.M Runge-Kutta metod t x(t) y(t) x(t) y(t) Conclusion Te main goal of tis work as been to derive an approximation for te solutions Brusselator system. We ave acieved tis goal by applying Adomian decomposition metod. Te results of applying A.D.M ave been compared by te results of Runge- Kutta metod for different coefficients of te system. we ave derived different agreements for te results of two metods. As te results in te tables sow in te first example we ave acieved better agreement. Te computations are done using Maple 9. 6 Reference [1] G. Adomian, Solving Frontier problem of Pysics: Te Decomposition Metod, Kluwer Academic press,1994

7 Approximate solutions of Brusselator system 989 [2] G. Adomian, Nonlinear Stocastic Systems and Applications to Pysics, Kluwer, [3] J.Biazar, E. Babolian, A. Nouri, R. Islam, An alternate algoritm for computing Adomian Decomposition metod in special cases, Applied Matematics and Computation 38 (2-3) (23) Received: November 15, 26