Solving $$(1+n)$$ ( 1 + n ) -Dimensional Fractional Bur
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ing (1 + n)-Dimensional Fractional Burgers Equation by Natural Decomposition Method M. Cherif1, 2* , D. Ziane1** , A. K. Alomari3*** , and K. Belghaba1**** 1
Laboratory of Mathematics and Its Applications (LAMAP), University of Oran1 Ahmed Ben Bella, Oran, Algeria 2 Oran’s Hight School of Electrical and Energetics Engineering (ESGEE), Oran, Algeria 3 Department of Mathematics, Faculty of Science, Yarmouk University, Irbid, Jordan Received January 21, 2019; in final form, June 7, 2019; accepted July 16, 2019
Abstract—In this paper, we combine the natural transform with the Adomian decomposition for solving nonlinear partial differential equations with time-fractional derivatives. We apply this method to obtain approximate analytical solutions of (1 + n)-dimensional fractional Burgers equations. Some illustrative examples are given, which show this to be a very efficient and accurate analytical method for solving nonlinear fractional partial differential equations. DOI: 10.1134/S1995423920040072 Keywords: Adomian decomposition method, natural transform, (1 + n)-dimensional Burgers equation, Caputo fractional derivative.
1. INTRODUCTION In recent years, many researchers became interested in solving linear and nonlinear differential equations. Investigation of exact solutions of nonlinear equations plays an important role in studying nonlinear physical phenomena. We find that many researchers have been interested in solving this kind of differential equations, including ODEs and PDEs. As some transformations, e.g., Laplace, Sumudu, Natural, and Alzaki ones, are unable to solve equations of this type, some researchers are working on combining these transformations with the Adomian decomposition method (ADM) in order to obtain new efficient methods to solve differential equations of this kind. Those include the Adomian decomposition method coupled with the Laplace transform method [15], Sumudu decomposition method for nonlinear equations [4], Elzaki transform decomposition algorithm [6], and natural decomposition method [5, 8, 9, 11, 14]. The objective of this paper is extending the scope of application of the natural decomposition method (NDM) to nonlinear fractional partial differential equations. We apply this method to solve the following (1 + n)-dimensional nonlinear fractional Burgers equation c
Dtα U (x, t) = βU Ux1 + α1 Ux1 x1 + α2 Ux2 x2 + α3 Ux3 x3 + .. + αn Uxn xn
(1)
subject to the initial condition U (x, 0) = U0 (x), ∂2U ∂x2n
(2)
, c Dtα is the Caputo fractional derivative of the where x = (x1 , x2 , x3 , . . . , xn ), Ux1 = ∂U ∂x1 , Uxn xn = order α, and αi , i = 1, 2, 3, . . . , n, and β are constants. This equation is used when studying cellular automata and systems of interacting particles. It can be also used as a model to describe water flows in soils. *
E-mail: [email protected] E-mail: [email protected] *** E-mail: [email protected] **** E-mail: [email protected] **
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THE (1 + n)-DIMENSIONAL FRACTIONAL BURGERS EQUATION
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2. PRELIMINARIES 2.1. Fractional Calculus There are several definiti
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