Analytical and numerical solution of multi-term nonlinear differential equations of arbitrary orders
type: Item
instance of: scholarly article
title: Analytical and numerical solution of multi-term nonlinear differential equations of arbitrary orders
zbMATH Open document ID: 1204.34006
author: J. Martínez
published in: Journal of Applied Mathematics and Computing
publication date: 2010-06-29T00:00:00Z
review text: The authors study a nonlinear multi-term fractional differential equation (FDE) of the form \[ D^\alpha x = f(t, D^{\alpha_{0}} x, D^{\alpha_{1}} x, \cdots D^{\alpha_{n}}x), \quad x(0) = 0, \] where \(t \in J = [0,T]\), \(T \in \mathbb{R}^+\), \(\alpha_0 = 0,\) and the fractional derivatives are considered in the Riemann-Liouville sense. Here, \(\alpha_i \in [0,1)\), \(\alpha > \{\alpha_0, \alpha_1, \dots, \alpha_n\}\). It is proved that, if the function \(f\) satisfies a Lipschitz condition uniformly in \(t\), then the above IVP has a unique continuous solution on \(J\). Using the Adomian polynomials formula, the convergence of the Adomian decomposition method is established. Based on this convergence analysis, the maximum absolute truncated error of the Adomian series is estimated. Numerical solutions are obtained via a numerical method introduced by [\textit{I. Podlubny}, Fractional differential equations. Mathematics in Science and Engineering. 198. San Diego, CA: Academic Press (1999; Zbl 0924.34008)] and the results are compared with those of the ADM solutions.
reviewed by: Gnana Bhaskar Tenali
Mathematics Subject Classification ID: 34A08
Mathematics Subject Classification ID: 34A12
zbMATH DE Number: 5728304
zbMATH Keywords: nonlinear fractional differential equation
zbMATH Keywords: Riemann-Liouville derivative
zbMATH Keywords: convergence analysis
zbMATH Keywords: Adomian method
zbMATH Keywords: Podlubny numerical method
MaRDI profile type: Publication
full work available at URL: s12190-009-0291-0
OpenAlex ID: W2048921731
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DOI: 10.1007/S12190-009-0291-0
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description: scientific article; zbMATH DE number 5728304