Journal of Interpolation and Approximation in Scientific Computing

Volume 2016, No. 1 (2016), Pages 25-37

Article ID jiasc-00090, 13 Pages

doi: 10.5899/2016/jiasc-00090


Research Article


An algorithm for positive solution of boundary value problems of nonlinear fractional differential equations by Adomian decomposition method


Hytham. A. Alkresheh1 *, A. I. Md. Ismail1


1School of Mathematical Sciences, Universiti Sains Malaysia (USM), 11800, Penang, Malaysia


* Corresponding author. Email address: hytham.k@hotmail.com Tel:00962777442093


Received: 22 July 2015; Revised: 19 October 2015; Accepted: 12 November 2015


Copyright © 2016 Hytham. A. Alkresheh and A. I. Md. Ismail. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, an algorithm based on a new modification, developed by Duan and Rach, for the Adomian decomposition method (ADM) is generalized to find positive solutions for boundary value problems involving nonlinear fractional ordinary differential equations. In the proposed algorithm the boundary conditions are used to convert the nonlinear fractional differential equations to an equivalent integral equation and then a recursion scheme is used to obtain the analytical solution components without the use of undetermined coefficients. Hence, there is no requirement to solve a nonlinear equation or a system of nonlinear equations of undetermined coefficients at each stage of approximation solution as per in the standard ADM. The fractional derivative is described in the Caputo sense. Numerical examples are provided to demonstrate the feasibility of the proposed algorithm.


Keywords: Adomian decomposition method; Fractional boundary value problems; Duan-Rach approach; Caputo derivative.

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