Applications of Differential Transform Method To Initial Value Problems

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1 American Journal of Engineering Research (AJER) 207 American Journal of Engineering Research (AJER) e-issn: p-issn : Volume-6, Issue-2, pp Research Paper Open Access Applications of Differential Transform Method To Initial Value Problems Abdallah Habila Ali Sudan University of Science & Technology, College of Science, Department of Mathematics (Sudan), Corresponding Author: Abdallah Habila Ali ABSTRACT: In this article the Differential Transform method is employed for obtaining solutions for initial value problems. This method gives the series of solutions which can be easily converted to exact ones. The differential transform method was successfully applied to initial value problems. The findings of the study has demonstrated that the method is easy, effective and flexible. The results of the differential transform method is in good agreement with those obtained by using the already existing ones. The proposed method is promising to a broad class of linear and nonlinear problems. Keywords: Differential Transform, Nonlinear, Initial value problems Date of Submission: Date of acceptance: I. INTRODUCTION Nonlinear phenomena have important effects in applied mathematics, physics and related to engineering; many such physical phenomena are modeled in terms of nonlinear differential equations [3,4,0]. A variety of numerical and analytical methods have been developed to obtain accurate approximate and analytic solutions for the problems in the literature [3,7,8,0,,2]. The classical Taylor's series method is one of the earliest analytic techniques to many problems, especially ordinary differential equations. However, since it requires a lot of symbolic calculation for the derivatives of functions, it taes a lot of computational time for higher derivatives. Here, we introduce the update version of the Taylor series method which is called the differential transform method (DTM)[4,5]. The (DTM) is the method to determine the coefficients of the Taylor series of the function by solving the induced recursive equation from the given differential equation. The basic idea of the (DTM) was introduced by Zhou [5]. In what follows we introduce a few notations for the (DTM). II. THE DIFFERENTIAL TRANSFORM METHOD Suppose that the solution u(x; t) is analytic at (X ; Y), then the solution u(x; t) can be represented by the Taylor series[]. u x, t n 0 n 0 0! n!! + + n + u x, t x x n x u x, t ( (x i x i ) i )(t t ) () i Definition. Let us define the (n + ) dimensional differential transform U(, ) by U x, t! n!! + + n + u x, t x x n x u x, t (2) w w w. a j e r. o r g Page 365

2 American Journal of Engineering Research (AJER) 207 Definition.2 The differential inverse transform of U x, t is define by u x, t of the form in (). Thus u x, t can be written by: u x, t 0 n 0 0 U(, ) n i (x i x i ) i An arbitrary function f(x) can be expanded in Taylor series about a point x 0 as: (t t ) (3) f x 0 x d f (4) The differential inverse transform of u x, t is define by: F x d f(x) (5) Then the inverse differential transform is: F x x 0 F() (6) 3 The fundamental operation of Differential Transformation Method [2]: (3.) If y x g x ± x then Y G() ± H() F d f d g x ± d x d y x d g x d g x ± x ± d x G(x) ± H(x) (3.2) If y x αg(x) then Y αg(x). (3.3)If y x dg (x) dx Y then Y()(+)G(), d y x d d g x +! G + + G() (3.4) If y(x) d2 g(x) dx 2 then Y G + d + g x + (3.5) If y(x) dm g(x) dxm then Y m G + m (3.6) If y x then Y δ() (3.7) If y x x then Y δ( ) (3.8) If y x x m if m then Y δ m 0 if m (3.9) If y x g x (x) then Y H G( m) (3.0) If y x e αx then Y α (3.) If y x + x m m m m 2 (m +) then Y (3.2) If y x sin(wx + α), then Y constants. w sin(π + α) where w and α are w w w. a j e r. o r g Page 366

3 American Journal of Engineering Research (AJER) 207 (3.3) If y x cos(wx + α), then Y constants. w cos(π + α) where w and α are IV. APPLICATIONS In this section, we apply the (DTM) to some ordinary differential equations and then to Voltera equation. 4. Problem Consider the following initial value problem [6] + x dx2 dx y 4 3 y x2 (7) y 5 3 By using the transformation x e t the problem is converted to: d 2 y y + e2t dt2 y (8) y Y + 2 Y + 2 (9) Y Y() + 2 (0) with the following conditions: Y 0 4 3, Y 5 3 () when 0 then Y 2 7, when Y 3 6 8,when 2 Y ,... The solution is: This can be written as: 0 t Y() t t2 + 8 t t4 + (2) t t + 2 t t2 + 6 t t t t4 + + t + t2 2! + t3 3! t + 2t + + 2t 3 + 2t ! 3! 4! w w w. a j e r. o r g Page 367

4 American Journal of Engineering Research (AJER) 207 e t + 3 e2t (3) The required answer is: y x x + x Problem 2 We, next consider the following nonlinear problem [6]: x dx2 dx + y + y2 xln(x) 2 y 0 y (4) using the transformation x e t the problem converted to: d 2 y dt 2 2 dy dt + y + y2 t 2 e 2t y 0 0 y 0 (5) Y Y + + Y + Y Y m δ(m 2) 2 m m! (6) or Y Y + Y Y Y m + δ(m 2) 2 m m! (7) with the following conditions Y 0 0 and Y (8) If 0 then Y 2, when Y 3 2 when 2 Y 4 6, 0 t Y 0 + t + t t3 + 6 t4 + t + t + t 2 + t tet (9) Therefore the required solution is: y x xln(x) (20) For the next problem, we need the following theorem: w w w. a j e r. o r g Page 368

5 American Journal of Engineering Research (AJER) Problem 3 Consider the following initial value problem [6] 4x dx2 dx y 2 + 6y x5 (2) y 3 2 By using the transformation x e t the problem is converted to: d 2 y dt 2 5 d2 y 6y + e5t dx2 y 0 2 (22) y Y Y + 6Y + 5 Y + 2 with the following conditions: Y + 6Y + (23) Y 0 2, Y 3 2 (24) Consequently: If 0 then Y , when Y 3 when 2 Y 4, or 0 t U() t t t t4 + (25) 3 + 2t + 2t ! + 2t 3 3! + 5t + 5t 2 2! + 2t 4 4! + 5t 3 3! + + 5t 3 4! 3 e2t + 6 e5t 26 Therefore the required solution is: w w w. a j e r. o r g Page 369

6 American Journal of Engineering Research (AJER) 207 y x 3 x2 + 6 x5 (27) 4.4 Problem 4 Consider the following initial value problem [6] + 3x dx2 dx + y2 ln x 3 y y 2 (28) By using the transformation x e t the problem is converted to: d 2 y dy 2 dt2 dt y2 + t 3 y 0 y 0 2 (29) Y Y + Y Y m +6δ 3 (30) Y + 2 with the following conditions: Y + Y Y m + δ 3 (3) Y 0, Y 2 (32) Consequently: If 0 then Y 2 5, when Y 3 2 when 2 Y 4, if when 2 Y , 0 t U() + 2t 5 2 t t3 48 t t5 + (33) The required solution is: w w w. a j e r. o r g Page 370

7 American Journal of Engineering Research (AJER) 207 y x + 2 ln x 5 2 ln x ln x 3 48 ln x ln x 5 + (35) V. CONCLUSION The observations of the present study have shown that the (DTM) is easy to implement and effective. As a result, the conclusion comes through this wor, is that the Differential Transform Method can be applied to a wide class of differential equations, due to the efficiency in the application to get the possible results. REFERENCES []. Ayaz.F, Solutions of system of differential equation by Differential Transform Method.Appl.Math.comput 2004.pp [2]. G.Adoman.R.Rac, On linear and nonlinear integro-differential equations. J. Math. Anal. Appl.3()(986), pp [3]. Kaya.N.T. Comparing numerical methods for solutions of ordinary differential equations, Appl.Math. Lett, 7(2004) [4]. Liu.H and Y.Song, Differential Transform Method applied to higher index differential- algebraic equations. Appl.Math. Comput,2007, [5]. Zhou.J.K. Differential transformation Method and applications for electrical circuits, Huazhong University press, Wuhan, China, (986) [6]. Onur Kiymaz, An Algorithm for Solving Initial Value Problems, using Laplace Adomian Decomposition Method. Applied Mathematical Sciences, Vol. 3, 2009, no. 30, [7]. E. Hesameddini, H. Latifizadeh. A new vision of the He s homotopy perturbation method. International Journal of Nonlinear Sciences and Numerical Simulation [8]. E. Hesameddini, H. Latifizadeh. Reconstruction of variational iteration algorithms using the Laplace transform. International Journal of Nonlinear Sciences and Numerical Simulation [9]. S.T. Mohyud-Din, M.A. Noor, K.I. Noor. Some relatively new techniques for nonlinear problems. Math. Porb.Eng. Article ID , 25 pages, doi: 0.55/2009/234849, [0]. S.T. Mohyud-Din, A. Yildirim. Variational iteration method for solving Klein- Gordon equations. Journal of Applied Mathematics, Statistics and Informatics.200. []. Jagdev Singh, Devendra Kumar and Sushila Rathore, Application of Homotopy Perturbation Transform Method for Solving Linear and Nonlinear [2]. Klein- Gordon Equations, Journal of Information and Computing Science,ISSN England, UK,Vol. 7, No. 2, 202, pp [3]. A. Yildirim. An Algorithm for Solving the Fractional Nonlinear Schröndinger Equation by Means of the Homotopy Perturbation Method. International Journal of Nonlinear Science and Numerical Simulation Abdallah Habila Ali. Applications of Differential Transform Method To Initial Value Problems. American Journal of Engineering Research (AJER), vol. 06, no. 2, 207, pp w w w. a j e r. o r g Page 37

Applications Of Differential Transform Method To Integral Equations

Applications Of Differential Transform Method To Integral Equations American Journal of Engineering Research (AJER) 28 American Journal of Engineering Research (AJER) e-issn: 232-847 p-issn : 232-936 Volume-7, Issue-, pp-27-276 www.ajer.org Research Paper Open Access Applications