Research Article
A Novel Method for Solving Nonlinear Volterra
Integro-Differential Equation Systems
Mohammad Hossein Daliri Birjandi, Jafar Saberi-Nadjafi , and Asghar Ghorbani
Department of Applied Mathematics, School of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran
Correspondence should be addressed to Jafar Saberi-Nadjaf; najaf141@gmail.com
Received 12 December 2017; Accepted 2 April 2018; Published 3 June 2018
Academic Editor: Patricia J. Y. Wong
Copyright © 2018 Mohammad Hossein Daliri Birjandi et al. Tis 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.
An efcient iteration method is introduced and used for solving a type of system of nonlinear Volterra integro-diferential equations.
Te scheme is based on a combination of the spectral collocation technique and the parametric iteration method. Tis method is
easy to implement and requires no tedious computational work. Some numerical examples are presented to show the validity and
efciency of the proposed method in comparison with the corresponding exact solutions.
1. Introduction
Systems of integro-diferential equations and their solu-
tions play a pivotal role in the felds of science, industrial
mathematics, control theory of fnancial mathematics, and
engineering [1–3]. Physical systems, such as biological appli-
cations in population dynamics, and genetics where impulses
arise naturally or are caused by control are modeled by a
system of integro-diferential equations [4, 5]. Te initial
value problem for a nonlinear system of integro-diferential
equations were used to model the competition between
tumor cells and the immune system [6]. In [7], two systems
of specifc inhomogeneous integro-diferential equations are
studied in order to examine the noise term phenomenon.
Tus applications of numerical methods for solving these
equations are attractive. Tis has led to a great deal of research
in recent years with the use of numerical methods such as
the variational iteration method [8], diferential transform
method [9], Bezier curves method [10], radial basis function
networks [11], biorthogonal systems [12], the block pulse
functions method [13], and a collocation method in combina-
tion with operational matrices of Bernstein polynomials [14].
Te parametric iteration method (PIM) is an analytic
approximate method that provides the solution of linear and
nonlinear problems as a sequence of iterations. In fact, the
PIM as a fxed- point iteration method is a reconstruction of
the variational iteration method. Since the implementation of
the PIM generally leads to the calculation of unneeded terms,
where more time is consumed in repeated calculations for
series solutions, so to overcome these shortcomings, a useful
improvement of the PIM was proposed in [15].
Te aim of this work is to present an efective algorithm,
requiring no tedious computational work, based on the
improved PIM and the spectral collocation technique to
obtaining an accurate solution for the following system of
Volterra integro-diferential equations as follows:
̇
()=
(,
1
(),
2
(),...,
())
+∫
0
(,)
(
1
(),
2
(),...,
()),
(0) = (
0
)
,
(1)
for = 1,..., and ∈ [0,]. Te functions
(,
1
(),
2
(),...,
()) are given real valued functions,
(,), =1,
2,...,, are kernels of the integral equations, and
(
1
(),
2
(),...,
()) are linear or nonlinear functions of
1
(),
2
(),...,
().
To demonstrate the utility of the proposed method, some
examples of system of Volterra integro-diferential equations
are given, which are solved using the established method. Te
Hindawi
Abstract and Applied Analysis
Volume 2018, Article ID 3569139, 6 pages
https://doi.org/10.1155/2018/3569139