Symbolic computation of analytic approximate solutions for nonlinear fractional differential equations

Abstract

The Adomian decomposition method (ADM) is one of the most effective methods to construct analytic approximate solutions for nonlinear differential equations. In this paper, based on the new definition of the Adomian polynomials, Rach (2008) [22], the Adomian decomposition method and the Padé approximants technique, a new algorithm is proposed to construct analytic approximate solutions for nonlinear fractional differential equations with initial or boundary conditions. Furthermore, a MAPLE software package is developed to implement this new algorithm, which is user-friendly and efficient. One only needs to input the system equation, initial or boundary conditions and several necessary parameters, then our package will automatically deliver the analytic approximate solutions within a few seconds. Several different types of examples are given to illustrate the scope and demonstrate the validity of our package, especially for non-smooth initial value problems. Our package provides a helpful and easy-to-use tool in science and engineering simulations. Program summaryProgram title: ADMPCatalogue identifier: AENE_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AENE_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 12011No. of bytes in distributed program, including test data, etc.: 575551Distribution format: tar.gzProgramming language: MAPLE R15.Computer: PCs.Operating system: Windows XP/7.RAM: 2 GbytesClassification: 4.3.Nature of problem:Constructing analytic approximate solutions of nonlinear fractional differential equations with initial or boundary conditions. Non-smooth initial value problems can be solved by this program.Solution method:Based on the new definition of the Adomian polynomials [1], the Adomian decomposition method and the Padé approximants technique, a new algorithm is proposed to construct analytic approximate solutions for nonlinear fractional differential equations.Restrictions:The problem [2] with some Robin boundary conditions can not be solved by this program.Running time:Seconds at most in routine uses of the program. Special tasks may take up to some minutes.