THE ADOMIAN DECOMPOSITION METHOD FOR SOLVING DELAY DIFFERENTIAL EQUATION

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1 International Journal of Computer Mathematics Vol. 00, No. 0, Month 004, pp. 1 6 THE ADOMIAN DECOMPOSITION METHOD FOR SOLVING DELAY DIFFERENTIAL EQUATION D. J. EVANS a and K. R. RASLAN b, a Faculty of Engineering and Computing, Nottingham Trent University, UK; b Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City, Cairo, Egypt (Revised 14 March 004; In final form 8 April 004 A numerical method based on the Adomian decomposition method which has been developed by Adomian [Adomian, G. (1994. Solving Frontier Problems of Physics: The Decomposition Method. Kluwer Academic Publishers, Boston, MA] is introduced in this paper for the approximate solution of delay differential equation (DDE. The algorithm is illustrated by studying an initial value problem. The results obtained are presented and show that only few terms are required to obtain an approximate solution which is found to be accurate and efficient. Keywords: Delay differential equation; Adomian decomposition method C.R. Categories: G.1.7; G.1. 1 INTRODUCTION The decomposition method has been shown [1 4] to solve effectively, easily, and accurately a large class of linear and nonlinear, ordinary, partial, deterministic or stochastic differential equations with approximate solutions which converge rapidly to accurate solutions. In recent years, many papers were devoted to the problem of approximate solution of delay differential equations (DDEs [5 9]. The basic motivation of this work is to apply the Adomian decomposition method to the DDE. It is well known now in the literature that this algorithm provides the solution in a rapidly convergent series [3]. The implementation of the Adomian method in Ref. [1] amongst others has shown reliable results in that few terms only are needed to obtain accurate solutions. ANALYSIS OF THE METHOD Consider the DDE written in the form Ly(x = f (x, y(x, y(g(x, 0 x 1 Corresponding author. Community College in Riyadh, King Saud University, 438 Salah El-Deen El-Auoby, Malaz, P.O. Box 8095, Riyadh, Saudi Arabia; kamel raslan@yahoo.com ISSN print; ISSN online c 004 Taylor & Francis Ltd DOI: / Techset Composition Ltd, Salisbury GCOM TeX Page#: 6 pages Printed: 17/7/004

2 D. J. EVANS AND K. R. RASLAN where the differential operator L is given by y (i (0 = y0 i, i = 0, 1,...,N 1, (1 y(x = (x, x 0, L( = dn ( N, ( the inverse operator L 1 is therefore considered a N-fold integral operator defined by Q1 L 1 (# = x operating with L 1 on Eq. (1, it then follows y(x = N 1 0 (# Ntimes, (3 α j j! xj + L 1 (f (x, y(x, y(g(x, (4 where the α j are constants that describe the boundary conditions. The Adomian decomposition method assumes that the unknown function y(x can be expressed by an infinite series of the form y(x = y n (x, (5 so that the components y n (x will be determined recursively. Moreover, the method defines the nonlinear term f (x, y(x, y(g(x by the Adomian polynomials f (x, y(x, g(y(x = A n, (6 where A n are Adomian polynomials that can be generated for all forms of nonlinearity [3] as A n = 1 d n f x, λ j y n! dλ n j (x, λ j y j (g(x. λ=0 Substituting Eqs. (5 and (6 into Eq. (4 gives y n (x = N 1 α j j! xj + L 1 ( A n, (7 to determine the components y n (x, n 0. First, we identify the zero component y 0 (x by all terms that arise from the boundary conditions at x = 0 and from integrating the source term if it exists. Second, the remaining components of y(x can be determined in a way such that each component is determined by using the preceding components. In other words, the method introduces the recursive relation: y 0 (x = N 1 α j j! xj, y n+1 (x = L 1 (A n, n 0, (8

3 DELAY DIFFERENTIAL EQUATION 3 for the determination of the components y n (x, n 0ofy(x, the series solution of y(x follows immediately with the constants α j,j = 0, 1,...,N 1 are as yet undetermined. The above analysis yields the following theorem. THEOREM 1 The solution of the DDE in the form (1 can be determined by the series (5 with the iterations (8. 3 APPLICATION In this section, we discuss the linear DDE (LDDE and the nonlinear DDE (NDDE by using the Theorem 1 as given in Section. Example 3.1 [5,6] dy(x Consider the LDDE of first-order = 1 ex/ y + 1 y(x, 0 x 1, y(0 = 1, (9 which has the exact solution y(x = e x. Theorem 1 gives x ( 1 y 0 (x = 1, y n+1 (x = ex/ y n + 1 y n(x, n 0. (10 0 When 13 terms of the solution of this DDE are compared with other methods [5, 6], we found that the present method is best as shown Table I. The differences between the exact and numerical solutions are given in Table I. Example 3. [7] Consider the LDDE of second-order d y(x = 3 4 y(x + y x +, 0 x 1, y(0 = 0, dy(0 = 0. (11 By applying the present method, we get x x ( y 0 (x = x x4 3 1, y n+1(x = y n(x + y n, n 0, (1 then we get y 0 (x = x x 4, y 1 (x = x x 6,...If we take eight terms of the series, we get the difference between the exact and numerical solution given in Table II. TABLE I X h = [6] h = [5] Present method E E E E 15.E E E 14.E E 11.13E E E E E 15

4 4 D. J. EVANS AND K. R. RASLAN TABLE II X Present solution Exact solution Example 3.3 Consider the LDDE of third-order d 3 y(x 3 = y(x y(x e x+0.3, 0 x 1, y(0 = 1, dy(0 = 1, d y(0 = 1, y(x = e x, x 0, (13 Theorem 1 gives the scheme y 0 (x = α 0 + α 1 x + α x e x+0.3, y n+1 (x = x x x ( y n (x y n (x 0.3, n 0, (14 where α 0 = , α 1 = , α = In Table III, we make a comparison between the present scheme and the exact. When we take only six terms from the Adomian series, we obtain the results given in Table III. In the following examples, we apply the method to NDDE. Example 3.4 [8, 9] Consider the NDDE of first-order dy(x as in Section, we get the recurrence relations ( = 1 y x, 0 x 1, y(0 = 0, (15 x y 0 (x = x, y n+1 (x = A n, n 0, (16 0 TABLE III X Present method Exact Difference E E E E E E 14

5 DELAY DIFFERENTIAL EQUATION 5 where A n,n 0 are Adomian polynomials that represent the nonlinear term. We list the set of Adomian polynomial as A 0 (x = y 0 A (x = y 1 A 3 (x = y 1,A 1 (x = y 0 y 1, + 4y 0 y, y + y 0 y 3,... (17 The solution in a series form is given by y(x = x x3 6 + x5 10 x x x x x x 17 ( and using Taylor series, the exact solution y(x = sin x is readily obtained. Example 3.5 Consider the NDDE of third-order d 3 y(x 3 ( = 1 + y x, 0 x 1, y(0 = 0, we get the recurrence relations dy(0 = 1, d y(0 = 0, (19 x x x y 0 (x = x x3 6, y n+1(x = A n, n 0, ( then the solution in series form is given by y(x = y 0 (x + y 1 (x + y (x + y 3 (x + (1 for few terms only the comparison between the exact solution and approximate solution is shown in Table IV and it shows that the errors are very small. From our examples, we can say that Adomian decomposition method gives approximate solutions in very good agreement with the exact solutions for only a few terms when solving DDEs. The programs for each case are written in the programming language Mathematica. TABLE IV X Our scheme Exact solution

6 6 D. J. EVANS AND K. R. RASLAN 4 CONCLUSION A new technique, using Adomian decomposition method, to numerically solve the DDEs is presented. All the numerical results obtained by using the Adomian decomposition method described earlier show very good agreement with the exact solutions for only a few terms. Comparing the decomposition method with several other methods that have been advanced for solving DDEs shows that the new technique is reliable, powerful and promising. We believe that the efficiency of the decomposition method gives it much wider applicability which been to be explored further. References Q3 [1] Adomian, G. (1994. Solving Frontier Problems of Physics: The Decomposition Method. Kluwer Academic Publishers, Boston, MA. [] Adomian, G. and Rach, R. (1993. Analytic solution of nonlinear boundary-value problems in several dimensions by decomposition. J. Math. Anal. Appl., 174, [3] Wazwaz, A.-M. (000. Approximate solutions to boundary value problems of higher order by the modified decomposition method. Comput. Math. Appl., 40, [4] Ismail, H. N., Raslan, K. R. and Salem, G. S. (003. Solitary wave solutions for the general KdV equation by Adomian decomposition method. Appl. Math. Comput., in press. [5] El-Safty, A., Salim, M. S. and El-Khatib, M. A. (003. Convergent of the spline functions for delay dynamic system. Int. J. Comput. Math., 80(4, [6] Shadia, M. (199. Numerical solution of delay differential and neutral differential equations using spline methods. Ph. D Thesis, Assuit University. [7] El-Safty, A. (1993. Approximate solution of the delay differential equation y = f (x, y(x, y(α(x with cubic spline functions. Bull. Fac. Sci., Assuit Univ., (-c, [8] Ibrahim, M. A.-K., EI-Safty, A. and Abo-Hasha, S. M. (1995. h-step spline method for the solution of delay differential equations. Comput. Math. Appl., 9(8, 1 6. [9] EI-Safty, A. and Abo-Hasha, S. M. (1990. On the application of spline functions to initial value problem with retarded argument. Int. J. Comput. Math., 3,

7 Journal International Journal of Computer Mathematics Article ID GCOM TO: CORRESPONDING AUTHOR AUTHOR QUERIES - TO BE ANSWERED BY THE AUTHOR The following queries have arisen during the typesetting of your manuscript. Please answer the queries. Q1 Please check Eq. (3. Please supply captions for Tables I IV. Q3 Please update Ref. [4]. Production Editorial Department, Taylor & Francis Ltd. 4 Park Square, Milton Park, Abingdon OX14 4RN Telephone: +44 ( Facsimile: +44 (