Adomian Decomposition Method (ADM) for Nonlinear Wave-like Equations with Variable Coefficient

Transcription

1 Applied Mathematical Sciences, Vol. 4, 1, no. 49, Adomian Decomposition Method (ADM) for Nonlinear Wave-like Equations with Variable Coefficient Mohammad Ghoreishi School of Mathematical Sciences, Universiti Sains Malaysia, 118 USM, Pulau Pinang A. I. B. Md. Ismail School of Mathematical Sciences, Universiti Sains Malaysia, 118 USM, Pulau Pinang N. H. M. Ali School of Mathematical Sciences, Universiti Sains Malaysia, 118 USM, Pulau Pinang Abstract The Adomian Decomposition Method (ADM) has been widely applied in solving partial differential equations which represent various phenomena in engineering and physics. This method is easy to program and can provide analytical solutions to the problems by just utilizing the initial conditions. Partial differential equations with nonlinear components play a crucial role in representing applications in mathematical physics. In this paper, nonlinear wave-like equations with variable coefficients are solved using ADM and the results are compared with known exact solutions. We shall show that ADM is able to solve this type of equations effectively and accurately without any need for discretization, perturbation, transformation and linearization. Keywords: Adomian decomposition, wave-like equation, variable coefficient

2 43 M. Ghoreishi, A.I.B.Md. Ismail and N.H.M. Ali 1 Introduction Over the last two decades or so, the Adomian Decomposition Method (ADM) has proven successful in deriving analytical solutions of linear and nonlinear differential equations by providing solutions in terms of convergent power series. This method does not require discretization of the variables and thus does not face the problem of computational rounding off errors. The theoretical convergence analysis of the series solution of ADM has been carried out amongst others, by Cherrault [6], Adomian [3], Cherrault et al. [7], Chrysos et al. [8] and Ngarhasta et al. [11]. In recent years, there have been great development in the application of ADM in solving partial differential equations with variable coefficients. Achouri and Omrani [1], for example, applied the ADM to obtain numerical solutions for the damped generalized regularized long-wave (DGRLW) equation with variable coefficients where they focus on the nonlinear DGRLW equations of the following forms u t (ϕ(x, t)u x t ) au x x + u x + u p u x = (1) In addition, they compared the ADM with the modified decomposition method (MDM) to solve the homogeneous and inhomogeneous DGRLW equations with variable coefficients. They obtain a high degree of accuracy in the numerical results performed and in most cases, the n-term approximation is accurate for quite low values of n. Soufyane and Boulmaf [1] have applied ADM to obtain analytical solutions to the linear and nonlinear parabolic equation with variable physical parameters in time and space of the following form u t = f (x, y, t)u x x + g(x, y, t)u y y + h(x, y, t)n(u) () The ADM was tested on several examples using a symbolic MATLAB Toolbox and the results were shown to confirm the accurateness and efficiency of this method in solving these parabolic equations. Linear heat-like and wave-like equations with variable coefficients have also been solved by Wazwaz and Gorguis [13] using the ADM. The method was shown to be capable of reducing the computational efforts compared to standard methods while still exhibiting a high degree of accuracy of the numerical solution. It is observed however that there are only a limited number of studies being researched on the use of ADM for solving nonlinear wave-like equations with variable coefficients. The aim of this paper is to obtain the analytical solution of this type of equations using ADM. The paper is organized as follows: In section, a description of the ADM applied to nonlinear wave-like equation is presented. In section 3, we demonstrate the applicability of ADM in solving examples of some nonlinear wave-like equations

3 ADM for nonlinear wave-like equation 433 and compare the solutions with known exact solutions. The conclusion is given in section 4. ADM for nonlinear wave-like equation In this section we describe the application of Adomian Decomposition Method for nonlinear wave-like equation with initial condition based on the analysis by previous researchers. Consider the wave equation k+m u t t = F 1i j (X, t, u) x k i x m j i,j=1 F i j (u xi, u x j ) + G 1i (X, t, u) p i=1 x p i G i (u xi )+ (3) H(X, t, u) + S(X, t) with the initial conditions u(x, ) = a (X ) u t (X, ) = a 1 (X ). (4) Here X = (x 1, x,..., x n ) and F 1i j, G 1i are nonlinear functions of X, t and u. F i j, G i are nonlinear functions of derivatives of x i and x j whilst H, S are nonlinear functions. k, m, p are integers. Eq (3) is first written in the operator form k+m L t t u = F 1i j (X, t) x k i x m j i, j=1 F i j (L xi u, L x j u) + G 1i (X, t) p i=1 H(X, t, u) + S(X, t). x p i G i (L xi u)+ (5) where the following notations shall be used L t t (.) = (.), L t xi x j (.) = (.) x i x j and L xi (.) = (.). The inverse operator L 1(.) = (.) d t d t is assumed x i t t to exist and easily obtained. This inverse operator is then applied in Eq (5) to obtain u(x, t) = u(x, ) + tu t (X, ) + L 1 [ k+m F t t 1i j (X, t, u) x k i,j=1 i x F m i j (L xi u, L x j u)+ j (6) G 1i (X, t, u) p x p G i (L xi u) + H(X, t, u))] + L 1 (S(X, t)). t t i i=1

4 434 M. Ghoreishi, A.I.B.Md. Ismail and N.H.M. Ali The unknown solution u is assumed to be given by an infinite series of the form [,3] u(x, t) = u k (X, t). (7) k= According to the ADM, we consider Eq. (.4) and represent the nonlinear expressions Nu = Mu = k+m F 1i j (X, t, u) x k i x F m i j (L xi u, L x j u) j G 1i (X, t, u) p x p G i (L xi u) i i, j=1 i=1 Ku = H(X, t, u) by the infinite series of polynomials given A n, B n, C n : Nu = A n, Mu = B n, Ku = A n, B n and C n are called Adomian s polynomials which can be calculated using the methods described in [,14,15] and are given by 1 d n A n = B n = C n = n! dλ (N( n λ k u k )) for n =, 1,.... There exists other general formulas for Adomian polynomials and details can be found in [4,5,14]. The remaining component u n (x, t) for n 1, is computed as follows with u (x, t) := k= λ= a (X ) + ta 1 (X ) S(X, t) = C n a (X ) + ta 1 (X ) + L 1 (S(X, t)) S(X, t) t t u n+1 (X, t) = L 1 t t (A n + B n + C n ). Thus we can calculate the terms of the solution (in infinite series form) term by term until we obtain the required accuracy with the inclusion of more terms leading to greater accuracy. Further details on ADM for wave equation can be found in [1] and the references therein. (8)

5 ADM for nonlinear wave-like equation Illustrative examples In this section, the ADM described earlier will be demonstrated on four illustrative examples and we compare the approximate analytical solution obtained for our nonlinear wave-like problems with known exact solutions. We define E m (X, t) to be the absolute error between the exact solution u(x, t) and m term approximate solution ϕ m (X, t)(= m 1 k= u k(x, t)) as follows : E m (X, t) = u(x, t) ϕ m (X, t). We used Mathematica for our calculations. Example 1: Let us consider the dimensional nonlinear wave-like equation with variable coefficients u t t = x y (u x xu y y ) x y (x yu xu y ) u (9) The initial conditions are u(x, y, ) = e x y and u t (x, y, ) = e x y. It can be verified that the exact solution is u(x, y, t) = e x y (sin t + cos t). Using the ADM, we have Thus L t t (u) = x y (L x xul y y u) u(x, y, t) = u(x, y, )+tu t (x, y, )+L 1 t t We assume u(x, y, t) = u(x, y, ) + tu t (x, y, ) + L 1 t t where Nu = u x x u y y = x y (x y L x L y u) u x y (L x xul y y u) x y Mu x y Nu A n, Mu = x yu x u y = Using (8) we obtain Adomian s polynomials as follows x y (x y L x L y u) u B n L 1 t t (u) A = (u ) x x (u ) y y A 1 = (u 1 ) x x (u ) y y + (u ) x x (u 1 ) y y A = (u ) x x (u ) y y + (u 1 ) x x (u 1 ) y y + (u ) x x (u ) y y

6 436 M. Ghoreishi, A.I.B.Md. Ismail and N.H.M. Ali and B = x y(u ) x (u ) y B 1 = x y(u 1 ) x (u ) y + x y(u ) x (u 1 ) y B = x y(u ) x (u ) y + x y(u 1 ) x (u 1 ) y + x y(u ) x (u ) y and so on. Thus we obtain the the first few components as follows u 1 (x, y, t) = u (x, y, t) = u (x, y, t) = u(x, y, ) + tu t (x, y, ) = e x y (1 + t) x y A x y B u x y A 1 x y B 1 u 1 and so on. The solution of the problem using ADM is as follows u(x, y, t) = d td t = e x y ( t t3 6 ) d td t = e x y ( t4 4 + t5 1 ) u n (x, y, t) = e x y 1 + t t t3 6 + t4 4 + t5 1 It is clear that the series converges to the exact solution. Table 1 shows the absolute errors E 6 when x, y and t vary from 1 to 9. From the table, it can be seen that for all test points, the ADM provides a very rapid and accurate approximation for the nonlinear wave-like equation. Clearly it can be observed that the ADM generates very accurate results for example 1. Table 1: Absolute error E 6 for variable x,y and t t/x,y Figure 1 and shows absolute error between the solution obtained by ADM with four and eight terms, respectively, and exact solution at x = y = 5. The larger the value of m, the more accurate the solution become.

7 ADM for nonlinear wave-like equation e t Figure 1: Error between the solution obtained using ADM with four terms and the exact solution at x = y = 5 e t Figure : Error between the solution obtained using ADM with eight terms and the exact solution at x = y = 5

8 438 M. Ghoreishi, A.I.B.Md. Ismail and N.H.M. Ali Example : Let us consider the nonlinear partial differential equation u t t = u x (u xu x x u x x x ) + u x x (u3 ) x x 18u5 + u, < x < 1 t > with initial conditions u(x, ) = e x and u t (x, ) = e x. It can be verified that the exact solution is u(x, t) = e x+t. Using the ADM, u(x, t) = u(x, )+tu t (x, )+L 1 t t We assume u x (L xul x x ul x x x u) + L x x (L x xu 3 ) 18u 5 + u. u(x, t) = u(x, ) + tu t (x, ) + L 1 t t (Nu + Mu 18Ku) + L x x(u) where Nu = u x (u xu x x u x x x ) = Mu = u x x (u3 ) = x x Ku = u 5 = C n. B n A n We can use the general form (8) and obtain Adomian s polynomials as follows A = u x ((u ) x (u ) x x (u ) x x x ) A 1 = u u 1 x ((u ) x (u ) x x (u ) x x x )+u x [(u 1) x (u ) x x (u ) x x x +(u ) x (u 1 ) x x (u ) x x x + (u ) x (u ) x x (u 1 ) x x x ], B = (u ) x x ((u ) 3 ) x x B 1 = (u ) x (u 1 ) x x ((u ) 3 )+(u x x ) x x (3(u ) (u x x 1) x x ) and C = u 5, C 1 = 5u 4 u 1

9 ADM for nonlinear wave-like equation 439 and so on. Thus, we have and u 1 (x, t) = u (x, t) = k= u (x, t) = u(x, ) + tu t (x, ) = (1 + t)e x. (A + B 18C + u )d td t = ( t + t3 6 )ex (A 1 + B 1 18C 1 + u 1 )d td t = ( t4 4 + t5 1 )ex and so on. The solution in series form is given by u(x, t) = u k (x, t) = e x 1 + t + t + t3 6 + t4 4 + t Table shows the absolute error E 6 for various values x and t. Again, the computed solutions are compared with the exact solution. It can be observed that the ADM gives very accurate approximations where the absolute errors become smaller as more terms are used in the computations. Table : Absolute error E 6 for variable x and t t/x Figure 3 and 4 shows error between solution obtained using ADM with four and eight terms, respectively, and exact solution at x = 5. It can clearly be seen that the ADM is very accurate for example. Example 3 : Finally we solve the following one dimensional nonlinear wave-like equation u t t = x x (u xu x x ) x (u x x ) u, x 1 t

10 44 M. Ghoreishi, A.I.B.Md. Ismail and N.H.M. Ali 8 e t Figure 3: Error between the solution obtained using ADM with four terms and the exact solution at x = 5 e t Figure 4: Error between the solution obtained using ADM with eight terms and exact solution at x = 5

11 ADM for nonlinear wave-like equation 441 subject to the initial conditions u(x, ) = and u t (x, ) = x and boundary conditions u(, t) = and u(1, t) = sin t. It can be verified that the exact solution is u(x, t) = x sin t. We apply the ADM and obtain We assume where and u(x, t) = u(x, ) + tu t (x, ) + L 1 t t u(x, t) = u(x, ) + tu t (x, ) + L 1 t t Nu = u x u x x = x x (u xu x x ) x (u x x ) u. x x (Nu) x (Mu) u A n, Mu = (u x x ) = n=1 Using (8) we obtain Adomian s polynomials as follows A = (u ) x (u ) x x A 1 = (u ) x (u 1 ) x x + (u 1 ) x (u ) x x A = (u ) x (u ) x x + (u 1 ) x (u 1 ) x x + (u ) x (u ) x x B n. B = (u ) x x b 1 = (u ) x x (u 1 ) x x B = (u 1 ) x x + (u ) x x (u ) x x n=1 and so on. Starting with u (x, t) = u(x, ) + tu t (x, ) = t x and using We can obtain u n+1 (x, t) = u 1 (x, t) = u (x, t) = u 3 (x, t) = n=1 x x (A n) x (B n ) u n d td t x x (A ) x (B ) u d td t = t3 3! x x x (A 1) x (B 1 ) u 1 d td t = t5 5! x x x (A ) x (B ) u d td t = t7 and so on. Using (.5) we have u(x, t) = u n (x, t) = x t t3 3! + t5 5! t7 7! + 7! x

12 44 M. Ghoreishi, A.I.B.Md. Ismail and N.H.M. Ali which converges to the exact solution. Table 3 shows the absolute error E 6 for various values x and t. Amongst the three examples presented, ADM seems to be able to generate the most accurate approximations for this example. It can be seen that the absolute errors are the least amongst the ones generated from the previous examples. Table 3: Absolute error E 6 for variable x and t t/x Figure 5 and 6 shows error between solution obtained by ADM with four and eight terms, respectively, and the exact solution at x = 5. It can clearly be seen that the ADM is very accurate for this example e t Figure 5: Error between the solution obtained using ADM with four terms and the exact solution at x = 5 4 Conclusion In this paper, we presented the application of the ADM in solving nonlinear wave-like equations with variable coefficients. An advantage of the ADM is that

13 ADM for nonlinear wave-like equation 443 e t Figure 6: Error between the solution obtained using ADM with eight terms and the exact solution at x = 5 it solves the equation without any need to undergo any process of discretization, perturbation, transformation or linearization. The method was tested on three different examples where the numerical solutions clearly demonstrated that the ADM produces very accurate results which are very close to the exact solutions. Thus, the method is capable of providing fast solutions that are highly accurate and reliable in solving this type of equations. References [1] T. Achouri, K. Omrani, Numerical solutions for the damped generalized regularized long-wave equation with a variable coefficient by Adomian Decomposition method, Commun. Nonlinear. Sci. Numer. Simulat, 14 (9) 5-33 [] G. Adomian, Solving frontier problems of physics: The Decomposition method, Kluwer,Boston, [3] G. Adomian, A review of the decomposition method in applied mathematics, J. Math. Anal. Appl. 135 (1988) [4] G. Adomian, Nonlinear stochastic systems and applications to physics, Kluwer Academic Press, 1989 [5] J. Biazar, H. Ebrahimi, An approximation to the solution of the hyperbolic equation by the Adomian decomposition and comparison with characteristic method, Appl. Math. Comput. 163 (5) [6] Y. Cherruault, Convergence of Adomian s method, Math. Comput. Model. 14 (199)

14 444 M. Ghoreishi, A.I.B.Md. Ismail and N.H.M. Ali [7] Y. Cherruault, G. Adomianb, K. Abbaoui, R. Rach, Further remarks on convergence of decomposition method,international Journal of Bio-Medical Computing 38 (1995) [8] M. Chrysos, F. Sanchez, Y. Cherruault, Improvement of convergence of Adomian s method using Pade approximants, Kybernetes 31 () [9] A. Friedman, Partial differential equation of parabolic type. Prenter-Hall, New Jersy, [1] S. Momani, Analytical approximate solution for fractional heat-like and wave-like equations with variable coefficients using the decomposition method, 165 (5) [11] N. Ngarhasta, B. Some, K. Abbaoui, Y. Cherruault, New numerical study of Adomian method applied to a diffusion model, Kybernetes 31() [1] A. Soufyane, M. Boulmalf, Solution of linear and nonlinear parabolic equations by the decomposition method, Appl. Math. Comput. 16 (5) [13] A. Wazwaz, A. Gorguis, Exact solutions for heat-like and wave-like equations with variable coefficients, Appl. Math. Comput. 149 (4) 15-9 [14] A. M. Wazwaz, A new algoritm for calculating Adomian polynomials for nonlinear operator, Appl. Math. Comput. 111 () [15] D. Zwillinger, Handbook of differential equations, Wiley, New York, 199. Received: January, 1