The Modified Adomian Decomposition Method for. Solving Nonlinear Coupled Burger s Equations

Transcription

1 Nonlinear Analysis and Differential Equations, Vol. 3, 015, no. 3, HIKARI Ltd, The Modified Adomian Decomposition Method for Solving Nonlinear Coupled Burger s Equations M. Al-Mazmumy Department of Mathematics Science Faculty for Girls, King Abdulaziz University, Saudi Arabia H. Al-Malki Department of Mathematics Science Faculty for Girls, King Abdulaziz University, Saudi Arabia Copyright 015 M. Al-Mazmumy and H. Al-Malki. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, Adomian decomposition method (ADM) and some of its modification are considered to solve nonlinear coupled Bergur's equations. We consider two types of modification, the new modification of (ADM) and Laplace Adomian decomposition method (LADM). Finally the effectiveness of our methods is demonstrated by numerical experiments. Keywords: Adomian decomposition method (ADM), New modification method (MADM), Laplace Adomian decomposition method (LADM), coupled Burger s equations 1 Introduction Burgers equation is a fundamental partial differential equation from fluid mechanics. It occurs in various areas of applied mathematics, such as modeling of dynamics, heat conduction, and acoustic waves. It is named for Johannes Martinus Burgers ( ). It is very rare that a real life applications can be modeled by a single partial differential equation, usually it takes a system of coupled partial differential equations to yield a complete model. Due to its wide range of applicability some researchers have been interested in studying its solution using

2 11 M. Al-Mazmumy and H. Al-Malki various numerical techniques such as Variational iteration method (VIM) [1], Homotopy perturbation method (HPM) [5], Differential Transformation Method (DTM) [], Adomian Pade Technique[4]. In this paper we will apply the Adomian decomposition method (ADM) and some of its modifications to solve the system of coupled Burger s equations. The method was developed by George Adomian in the last twenty years ago [3, 7]. This method has many important applications especially in the fields of physics, chemistry, mechanics and other sciences. The main advantage of this method is that it can be used directly without using restrictive assumptions or linearization. Another important advantage is that this method is capable of greatly reducing the size of computation.this method will be successfully used to handle most types of partial differential equations that appear in several physical models and scientific models. The solution in this method is found as an infinite series whose terms are determined by a recursive relations using the Adomian polynomials which converges rapidly to accurate solution. This paper is organized as follows. In Section, the coupled Burger's equations are studied with the (ADM), new modification method (MADM) and (LADM). In Section 3, some applications are included to demonstrate the method. Finally, our results are given in Section 4 as conclusions. The coupled Burger s equations In this section, we describe the (ADM), (MADM) and (LADM) as it applies to coupled Burger's equations..1 Adomian decomposition method (ADM) The coupled Burger s equations is given by { u u t x v v t x u u + α (uv) = 0, x x (1) v v + +β (uv) = 0, x x In an operator form, eq (1) can be rewritten as { L tu L xx u ul x u + αl x uv = 0, L t v L xx v vl x v + βl x uv = 0, with initial condition u(x, 0) = f(x), v(x, 0) = g(x), Where L t = t so L 1 t t = dt 0 () (3) To illustrate (ADM) [3, 7], operating by L t 1 to the system () and using the initial conditions we have

3 The modified Adomian decomposition method 113 { u(x, t) = f(x) + L t 1 [u xx + uu x αl x uv], v(x, t) = g(x) + L t 1 [v xx + vv x βl x uv], (4) In (ADM) the unknown function u(x, t) and v(x, t) can be decomposed by infinite series of components u(x, t) = v(x, t) = n=0 u n (x, t), n=0 v n (x, t), (5) And the nonlinear terms by using the infinite series with Adomian's polynomials uu x = n=0 A n, vv x = n=0 B n, uv = n=0 C n, Where A n, B n, C n are Adomian's polynomials compute from the relations A 0 = u 0 u 0x, A 1 = u 0x u 1 + u 0 u 1x, A = u 0x u + u 1x u 1 + u x u 0, A 3 = u 0x u 3 + u 1x u + u x u 1 + u 3x u 0 B 0 = v 0 v 0x, B 1 = v 0x v 1 + v 0 v 1x, B = v 0x v + v 1x v 1 + v x v 0 B 3 = v 0x v 3 + v 1x v + v x v 1 + v 3x v 0, c 0 = u 0 v 0, c 1 = u 1 v 0 + u 0 v 1, c = u v 0 + u 1 v 1 + u 0 v, c 3 = u 3 v 0 + u v 1 + u 1 v + u 0 v 3 substituting (5) and (6) into (4) gives { n=0 u n(x, t) = f(x) + L 1 t [ n=0 u nxx (x, t) + n=0 A n αl x C n n=0 v n (x, t) = g(x) + L 1 t [ n=0 v nxx (x, t) + n=0 B n βl x C n then, the recursive relation is given below { u o = f(x), v 0 = g(x), { u k+1 = L t 1 [u kxx + A k αl x C k ], k 0, v k+1 = L t 1 [v kxx + B k βl x C k ], k 0, n=0 ], n=0 ], (6) (7) (8) (9)

4 114 M. Al-Mazmumy and H. Al-Malki having determined the component of u n, v n, n 0, the solution of coupled Burger's equations follows immediately. For numerical purposes we use: n φ n (t, x) = k=0 u k, n 0, (10) and n ψ n (t, x) = k=0 v k, n 0, (11). The new modification method (MADM) As we know above to solve a system of Burger equation by (ADM) we suggest that the zeros component u 0 and v 0 usually defined by u o = f, v 0 = g, but in the new modification,wazwaz [7,8] suggest that f, g be expressed in Taylor series { f = n=0 f n, g = n=0 g n, moreover,he suggest a new recursive relationship expressed in the form { u 0 = f 0, v 0 = g 0, { u k+1 = f k+1 + L t 1 [u kxx + A k αl x C k ], k 0 v k+1 = g k+1 + L t 1 [v kxx + B k βl x C k ], k 0 (1) (13) (14) having determined the component of u n, v n, n 0, the solution of coupled Burger's equations follows immediately..3 Laplace Adomian decomposition method (LADM) To illustrate (LADM) [6], applying Laplace transform with respect to variable t on both sides of Eq. () we have { sl[u(x, t)] u(x, 0) = L[u xx + uu x αl x uv], sl[v(x, t)] v(x, 0) = L[v xx + vv x βl x uv], using the initial condition { L[u(x, t)] = 1 s f(x) + 1 s L[u xx + uu x αl x uv], L[v(x, t)] = 1 s g(x) + 1 s L[v xx + vv x βl x uv], applying inverse Laplace transform we get { L 1 {L[u(x, t)]} = L 1 { 1 s f(x)} + L 1 { 1 s L[u xx + uu x αl x uv]}, L 1 {L[v(x, t)]} = L 1 { 1 s g(x)} + L 1 { 1 s L[v xx + vv x βl x uv]},

5 The modified Adomian decomposition method 115 There fore { u(x, t) = L 1 { 1 s f(x)} + L 1 { 1 s L[u xx + uu x αl x uv]}, v(x, t) = L 1 { 1 s g(x)} + L 1 { 1 s L[v xx + vv x βl x uv]}, (15) substituting (5) and (6) into (15) we have { n=0 u n(x, t) = f(x) + L 1 { 1 L[ u s n=0 nxx(x, t) + n=0 A n αl x n=0 C n ]}, n=0 v n (x, t) = g(x) +L 1 { 1 L[ v s n=0 nxx(x, t) + n=0 B n βl x n=0 C n ]}, then the recursive relation is given below: { u 0 = f(x), v 0 = g(x), { u k+1 = L 1 { 1 s L[u kxx + A k αl x C k ]}, k 0, v k+1 = L 1 { 1 s L[v kxx + B k βl x C k ]}, k 0, (16) (17) having determined the component of u n, v n, n 0 the solution of coupled Burger's equations follows immediately. 3 Applications In this section, two examples are provided to illustrate the methods, they are effectiveness and simplicity. Example 1: Consider the coupled Burger s equations with α = β = 1 { u u t x v v t x u u x + x (uv) = 0, (18) v v + + (uv) = 0, x x with initial conditions u(x, 0) = sin x, v(x, 0) = sin x, Solution: eq. (18) can be rewritten in an operator form, as { L tu L xx u ul x u + L x uv = 0, L t v L xx v vl x v + L x uv = 0,

6 116 M. Al-Mazmumy and H. Al-Malki First: by (ADM) Based on the above study and from the recursive relation (8) and (9) we obtain { u o = sin x, v 0 = sin x, { u k+1 = L t 1 [u kxx + A k L x C k ], k 0 v k+1 = L t 1 [v kxx + B k L x C k ], k 0 the first few component are u 1 = L t 1 [u 0xx + A 0 L x C 0 ] = L t 1 [ sin x + sin x cos x sin x cos x] = t sin x v 1 = L t 1 [v 0xx + B 0 L x C 0 ] = t sin x u = L t 1 [u 1xx + A 1 L x C 1 ] = L 1 t [t sin x 4t cos x sin x + 4t cos x sin x] = t sin x v = L 1 t [v 1xx + B 1 L x C 1 ] = t sin x u 3 = L t 1 [u xx + A L x C ]! = L 1 t [ t sin x + 4t sin x cos x 4t sin x cos x] = t3 sin x v 3 = L 1 t [v xx + B L x C ] = t3 sin x And so on. In the same manner the rest of components of the iteration were obtained. So the solution of u(x, t) and v(x, t) are t u(x, t) = sin x t sin x + { v(x, t) = sin x t sin x + t Therefore t u(x, t) = sin x [1 t +! { t3 + ] v(x, t) = sin x [1 t + t t3 + ]! The solution in a closed form is { u(x, t) = e t sin x. v(x, t) = e t sin x. t3 sin x sin x + t3 sin x sin x + Second: by the new modification (MADM) The Taylor expansion for sin x is given by sin x = x 1 x ! x5 1 7! x7 + Based on the above study and from the recursive relation for the new modification (13) and (14) we obtain!

7 The modified Adomian decomposition method 117 { u 0 = x v 0 = x { u k+1 = f k+1 + L t 1 [u kxx + A k L x C k ], k 0 v k+1 = g k+1 + L t 1 [v kxx + B k L x C k ], k 0 then, the first few component are u 1 = 1 x3 + L 1 t [u 0xx + A k L x C k ] = 1 x3 + L 1 t [0 + x x] = 1 x3 v 1 = 1 x3 + L 1 t [v 0xx + B k L x C k ] = 1 x3 u = 1 5! x5 + L 1 t [u 1xx + A 1 L x C 1 ] = 1 5! x5 + L t 1 [ x 1 3 x3 x 3 L x ( 1 3 x4 )] = 1 5! x5 + L t 1 [ x 4 3 x x] = 1 5! x5 xt v = 1 5! x5 + L t 1 [v 1xx + B 1 L x C 1 ] = 1 5! x5 xt u 3 = 1 7! x7 + L t 1 [u xx + A L x C k ] = 1 7! x7 + L 1 t [ x3 + (x5 x5 xt + + x5 tx) L 5! 1 4! x( x6 5! x t + x6 + x6 36 5! x t)] = 1 7! x7 + x3 t v 3 = 1 7! x7 + L 1 t [v xx + B L x C ] = 1 7! x7 + x3 And so on. In the same manner the rest of components of the iteration were obtained. So the solution of u(x, t) and v(x, t) are { u(x, t) = (x 1 x ! x5 ) t (x 1 x ! x5 ) + t! (x 1 x ! x5 ) + v(x, t) = (x 1 x ! x5 ) t (x 1 x ! x5 ) + t! (x 1 x ! x5 ) + Therefore t u(x, t) = sin x t sin x + sin x +! { v(x, t) = sin x t sin x + t sin x + The solution in a closed form is { u(x, t) = e t sin x, v(x, t) = e t sin x,! Third: by (LADM) Based on the above study and from the recursive relation (16) and (17) we obtained

8 118 M. Al-Mazmumy and H. Al-Malki { u 0 = sin x, v 0 = sin x, { u k+1 = L 1 { 1 s L[u kxx + A k L x C k ]}, k 0, v k+1 = L 1 { 1 s L[v kxx + B k L x C k ]}, k 0, The first few component are u 1 = L 1 { 1 s L[u 0xx + A 0 L x C 0 ]} = L 1 { 1 L[ sin x + sin x cos x sin x cos x]} s = L 1 { 1 L[ sin x]} = t sin x s v 1 = L 1 { 1 L[v s 0xx + B 0 L x C 0 ]} = t sin x u = L 1 { 1 s L[u 1xx + A 1 L x C 1 ]} = L 1 { 1 L[t sin x 4 sin x cos x + 4 sin x cos x]} s = L 1 { 1 L[t sin x]} = t sin x s v = L 1 { 1 L[v s 1xx + B 1 L x C 1 ]} = t sin x u 3 = L 1 { 1 L[u s xx + A L x C } = L 1 { 1 s L[ t sin x + 4 sin x cos x 4 sin x cos x]} = L 1 { 1 L t3 s [ t sin x]} = sin x v 3 = L 1 { 1 L[v s xx + B L x C } = t3 sin x The solution in a closed form is { u(x, t) = sin x e t. v(x, t) = sin x e t. and it is exactly the same as the results obtained in [6,7]. see Figure (1).

9 The modified Adomian decomposition method 119 Figure (1) Example : Consider the coupled Burger s equations with α = β = 5 { u u t x v v t x u u v v + 5 x x + 5 x x (uv) = 0, (uv) = 0, with initial condition u(x, 0) = v(x, 0) = λ (1 tanh 3 λx) where λ is an arbitrary constant.. Solution: In the follows, we apply (MADM), the Taylor expansion for λ(1 tanh 3 λx) is given by λ (1 tanh 3 λx) = λ 3 λ x λ4 x λ6 x λ8 x λ10 x 9 + 0(x 10 ) Based on the above study and from the recursive relation for the new modification (13) and (14) we obtain { u 0 = λ v 0 = λ then, we use the Maple Package to calculate the the first few component { u 1 = 3 λ x + L 1 t [u 0xx + A 0 5 L x(c 0 )] = 3 λ x v 1 = 3 λ x + L 1 t [v 0xx + B 0 L x (C 0 )] = 3 λ x

10 10 M. Al-Mazmumy and H. Al-Malki Similarly { u = 9 8 λ4 x λ3 t v = 9 8 λ4 x λ3 t { u 3 = λ6 x λ5 x t v 3 = λ6 x λ5 x t { u 4 = λ8 x λ6 xt λ7 x 4 t v 4 = λ8 x λ6 xt λ7 x 4 t In the same manner, the rest of components can be obtained using the recurrence relation (14). Substituting the quantities obtained in Eq. (10) and (11), the approximation solution in the series form of Example is φ 4 (t, x) = 4 u k = λ8 x λ7 x 4 t + ( x k=0 8 xt ) λ λ5 x t λ4 x λ3 t 3 λ x + λ ψ 4 (t, x) = 4 v k = λ8 x λ7 x 4 t + ( x k=0 8 xt ) λ λ5 x t λ4 x λ3 t 3 λ x + λ which is the same as the Taylor expansion of the exact solutions u(t, x) = v(t, x) = λ[1 tanh ( 3 λ(x 3λt)] And it is exactly the same as the results obtained in [1,].see Figure (). Figure ()

11 The modified Adomian decomposition method 11 4 Conclusions In this paper, the (ADM), the (MADM) and (LADM) are successfully applied to solve coupled Burger s equations. The results of example shows that the (MADM) is a powerful mathematical tool to solving coupled Burger s equations. It has shown that the method is reliable, efficient and requires fewer computations and the present scheme (LADM) gives better accuracy in comparison with the other method. It is also a promising method to solve other nonlinear equations. The solutions obtained are shown graphically. In our work, we use the Maple Package to calculate the series obtained from the method. References [1] M. A. Abdou, and A. A. Soliman, Variational iteration method for solving Burgers' and coupled Burgers' equations, J. Comput. Appl. Math. 181 (005), [] R. Abazari & A. Borhanifar, Numerical study of the solution of the Burgers and coupled Burgers equations by a differential transformation method, Computers and Mathematics with Applications, 59, (010) [3] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic, Dordrecht, [4] M. Dehghan, A. Hamidi, M. Shakourifar, The solution of coupled Burger s equations using Adomian Pade Technique, Appl. Math. 189 (007), [5] Khyati R. Desai and V. H. Pradhan, Solution of Burger s equation and coupled Burger s equations by Homotopy perturbation method", International Journal of Engineering Research and Applications (IJERA), (01), [6] M. Hussain and M. Khan, Modified Laplace Decomposition Method Pakistan, Applied Mathematical Sciences, 36 (010), [7] A. M. Wazwaz, Partial Differential Equations and Solitary Waves Theory, Higher Education Press, China, [8] A. M. Wazwaz, and S. M. El-Sayed, A new modification of the Adomian decomposition method for linear and nonlinear operator, Applied Mathematics and computation,1 (001),

12 1 M. Al-Mazmumy and H. Al-Malki Received: January 6, 015; Published: February 1, 015