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 Of Differential Transform Method To Integral Equations Abdallah Habila Ali, 2 Shama Alamin Mustafa Sudan University of Science & Technology, College of Science, Department of Mathematics (Sudan), 2 University of Gezira, Faculty of Mathematical and Computer Science, Department of Mathematics (Sudan), Corresponding Author: Abdallah Habila Abstract: In this article, the Differential Transform Method is applied for solving integral equations. The approimate solution of the integral equation is calculated in the form of series with easily computable components. This powerful method catches the eact solution. Some integral equations are solved as eamples. The results of the differential transform method is in good agreement with those obtained by using the already eisting ones. The proposed method is promising to etensive class of linear and nonlinear problems. Keywords: Differential Transform, Nonlinear phenomena, Integral equations. ----------------------------------------------------------------------------------------------------------------------------- ---------- Date of Submission: --28 Date of acceptance: 27--28 ----------------------------------------------------------------------------------------------------------------------------- ---------- 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 [3,4,6]. The concept of the differential transform was first proposed by Zhou [5]. Integral equations arise in many scientific and engineering problems[,2].the potential theory contributed more than any field to give rise to integral equations. Mathematical physics models, such as diffraction problems, scattering in quantum mechanics, conformal mapping, and water waves also contributed to the creation of integral equations []. Many other applications in science and engineering are described by integral equations. The Volterra's population growth model [], biological species living together, propagation of stoced fish in a new lae, the heat transfer and the heat radiation are among many areas that are described by integral equations []. In this paper we presented a variety of integral equations that were handled using differential transform method. The problems we handled led in most cases to the determination of eact solutions. Because it is not always possible to find eact solutions to problems of physical sciences, much wor is devoted to obtaining qualitative approimations that high light the structure of the solution. It is the aim of this paper to handle some integral equations taen from a variety fields [7,8].. The Differential Transform Method Suppose that the solution u(; t) is analytic at (X ; Y), then the solution u(; t) can be represented by the Taylor series[]. u, t n! n!! + + n + u, t n u, t n i ( i i ) i (t t ) () Definition. Let us define the (n + ) dimensional differential transform U(, ) by w w w. a j e r. o r g Page 27
American Journal of Engineering Research (AJER) 28 Definition.2 U, t! n!! + + n + u, t n u, t (2) The differential inverse transform of U, t is define by u, t of the form in (). Thus u, t can be written by: u, t U(, ) ( i i ) i (t t ) (3) n i An arbitrary function f() can be epanded in Taylor series about a point as: n f d f d (4) The differential inverse transform of u, t is define by: F d f() d (5) Then the inverse differential transform is: F F() (6) 3 The fundamental operation of Differential Transformation Method [,3]: (3.) If y g ± then Y G() ± H() F d f d d g d ± d d d y d d g d d g ± d ± d d G() ± H() (3.2) If y αg() then Y αg(). (3.3)If y dg () d Y then Y()(+)G(), d y d d d g d d +! G + + G() (3.4) If y() d2 g() d 2 then Y + + 2 G + d + g d + (3.5) If y() dm g() dm then Y + + 2 + m G + m (3.6) If y then Y δ() (3.7) If y then Y δ( ) (3.8) If y m if m then Y δ m if m (3.9) If y g () then Y H G( m) (3.) If y e α then Y α (3.) If y + m m m m 2 (m +) then Y w w w. a j e r. o r g Page 272
American Journal of Engineering Research (AJER) 28 (3.2) If y sin(w + α), then Y constants. (3.3) If y cos(w + α), then Y constants. w w sin(π + α) where w and α are cos(π + α) where w and α are 4 Applications In this section, we apply the (DTM) to some to integral equations. For the following problems, we need the following theorem. 4. Theorem Suppose that U() and G() are the differential transformations of u() and g() respectively, then we have the following properties[2]: If f g t u t dt (7) then If F i G(i) U( i ), F (8) f g() u t dt (9) then F i G(i) U( i ), F () 4.2 Problem We consider the following linear integral equation : u + tu t dt () According to Theorem(4.3.) equation (7) and (8) and to the operation of differential transformation given in section(3) formula (3.8), we have the following recurrence relation: U δ + t δ t t U( t ) Consequently: when then U, when 2 then, U 2 2,,, U (2), when 3 then U 3 2,, U 4 2, U 5 2, U 6 2, U 7 2, U 8 2 (3) So the solution is: or u U() + + 2 2 + 3 2 + 4 2 + 5 2 + 6 2 + 7 2 + (4) u 2 2 (5) 4.3 Problem 2 w w w. a j e r. o r g Page 273
American Journal of Engineering Research (AJER) 28 Net we consider the following linear integral equation: u cos + 2 π 2 sin u t dt (6) According to Theorem(4.3.) equation (9) and ()and to the operation of differential transformation given in section(3) formula (3.2) and (3.3), we have the following recurrence relation: with U cos π + 2 n sin π n! U n n (7) U, (8) Thus: If then U, if 2 U 2, if 3 U 3, if 4 U 4. Thus: 2! 3! 4! u Therefore the required solution is: U + 2 2! 3 3! + 4 4! + + + 2 2! 3 3! + 4 4! + (9) u + e (2) 4.4 Problem 3 Net we consider the following linear integral equation : u e u 2 t dt, u, u (2) According to Theorem(4.3.) equation (9) and () and to the operation of differential transformation given in section(3) formula (3.7), (3.9) and (3.), we have : + ( + 2)U + 2 u m U m (22) Hence, we have the following recurrence relation: U + 2 with the following conditions: + ( + 2) u m U m (23) U, U (24) Thus: If then U 2 2 (25) Therefore, the solution is, if U 3 3!, if 2 U 4, if 3 U 5 4! 5!... w w w. a j e r. o r g Page 274
American Journal of Engineering Research (AJER) 28 u U + + 2 2! 3 3! + 4 4! 5 5! + + 2 + + 2 2! 3 3! + 4 4! 5 5! + (26) Therefore the required solution is: u + 2 + e 4.4 Problem 3 Net we consider the following linear integral equation : u + e e t u t dt, u, u (27) According to Theorem(4.3.) equation (9) and ()and to the operation of differential transformation given in section(3) formula (3.7), (3.9) and (3.), we have : + + 2 U + 2 Thus, the recurrence relation is: U + 2 m with the following conditions: δ + m m U m δ m + ( + 2) δ + m U m δ m (29) U, U (3) (28) Thus: If then U 2 2, if U 3 3!,... u U + + 2 2! + 3 3! + + + + 2 2! + 3 3! + (3) Therefore the required solution is: u + e V. Conclusion In this wor, we successfully apply the differential transform method to find eact solutions for linear and non-linear integral equations. The present methodreduces the computational difficulties of the other traditional methods and all thecalculations can be made simple manipulations. Several eamples were tested by applying the DTM and the results have shown remarable performance. Therefore, this method can be applied to many Non-linear integral equations without w w w. a j e r. o r g Page 275
American Journal of Engineering Research (AJER) 28 linearization, discretization or perturbation. References []. Abdallah Habila, Applications of Differential Transform Method To Initial Value Problems, American Journal of Engineering Research (AJER), 27 e-issn: 232-847 p-issn : 232-936 Volume-6, Issue-2, pp-365-37 [2]. A.M. Wazwaz, Linear and Non-linear Integral equations (methods and applications), Saint Xavier University Chicago (2). [3]. G.Adoman.R.Rac, On linear and nonlinear integro-differential equations. J. Math. Anal. Appl.3()(986), pp. 7-2. [4]. Kaya.N.T. Comparing numerical methods for solutions of ordinary differential equations, Appl.Math. Lett, 7(24)323-328. [5]. Liu.H and Y.Song, Differential Transform Method applied to higher inde differential- algebraic equations. Appl.Math. Comput,27,84-748-753. [6]. Zhou.J.K. Differential transformation Method and applications for electrical circuits, Huazhong University press, Wuhan, China, (986) [7]. Onur Kiymaz, An Algorithm for Solving Initial Value Problems, using Laplace Adomian Decomposition Method. Applied Mathematical Sciences, Vol. 3, 29, no. 3, 453-459 [8]. S.T. Mohyud-Din, M.A. Noor, K.I. Noor. Some relatively new techniques for nonlinear problems. Math. Porb.Eng. Article ID 234849, 25 pages, doi:.55/29/234849, 29. [9]. S.T. Mohyud-Din, A. Yildirim. Variational iteration method for solving Klein- Gordon equations. Journal of Applied Mathematics, Statistics and Informatics. 2. s Abdallah Habila Ali."Applications Of Differential Transform Method To Integral Equations. American Journal of Engineering Research (AJER), vol. 7, no., 28, pp. 27-276. w w w. a j e r. o r g Page 276