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International Journal of Differential Equations
Volume 2018, Article ID 7237680, 10 pages
https://doi.org/10.1155/2018/7237680
Research Article

An Analytical and Approximate Solution for Nonlinear Volterra Partial Integro-Differential Equations with a Weakly Singular Kernel Using the Fractional Differential Transform Method

1Department of Applied Mathematics, Ferdowsi University of Mashhad, International Campus, Mashhad, Iran
2Department of Applied Mathematics, School of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran

Correspondence should be addressed to Jafar Saberi-Nadjafi; moc.liamg@141ifajan

Received 17 August 2017; Revised 24 November 2017; Accepted 18 January 2018; Published 1 April 2018

Academic Editor: Jaume Giné

Copyright © 2018 Rezvan Ghoochani-Shirvan et al. 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

An analytical-approximate method is proposed for a type of nonlinear Volterra partial integro-differential equations with a weakly singular kernel. This method is based on the fractional differential transform method (FDTM). The approximate solutions of these equations are calculated in the form of a finite series with easily computable terms. The analytic solution is represented by an infinite series. We state and prove a theorem regarding an integral equation with a weak kernel by using the fractional differential transform method. The result of the theorem will be used to solve a weakly singular Volterra integral equation later on.