Research Article
Convergence Analysis of Legendre Pseudospectral Scheme
for Solving Nonlinear Systems of Volterra Integral Equations
Emran Tohidi,
1
O. R. Navid Samadi,
1
and S. Shateyi
2
1
Young Researchers and Elite Club, Mashhad Branch, Islamic Azad University, Mashhad, Iran
2
Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State,
Bloemfontein 9300, South Africa
Correspondence should be addressed to S. Shateyi; stanford.shateyi@univen.ac.za
Received 27 December 2013; Revised 1 July 2014; Accepted 16 July 2014; Published 12 August 2014
Academic Editor: Hagen Neidhardt
Copyright © 2014 Emran Tohidi 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.
We are concerned with the extension of a Legendre spectral method to the numerical solution of nonlinear systems of Volterra
integral equations of the second kind. It is proved theoretically that the proposed method converges exponentially provided that
the solution is sufciently smooth. Also, three biological systems which are known as the systems of Lotka-Volterra equations are
approximately solved by the presented method. Numerical results confrm the theoretical prediction of the exponential rate of
convergence.
1. Introduction
Volterra-type integral equations (VIEs) are the mathematical
model of many evolutionary problems with memory aris-
ing from biology, chemistry, physics, and engineering. For
instance, in several heat transfer problems in physics, the
equations are usually replaced by systems of Volterra integral
equations (SVIEs). Since just few of these equations (i.e., VIEs
and SVIEs) can be solved analytically, it is ofen necessary to
apply appropriate numerical techniques.
Among numerical approaches, spectral methods are very
powerful tools for approximating the solutions of many kinds
of diferential equations arising in various felds of science
and engineering [1–5]. Spectral (exponential) accuracy and
ease of applying these methods are two efective properties
which have encouraged many authors to use them for integral
equations (IEs) too. Spectral methods have been widely used
by many authors in numerical analysis [6–13] for diferent
kinds of IEs. In [11], Tang et al. proposed a Legendre spectral
method (LSM) and its error analysis for the linear VIEs of
the second kind. In this paper, we extend the LSM [11] to the
numerical solution of the SVIEs of the second kind, including
giving a convergence analysis for the nonlinear case. Tus, we
consider the following nonlinear SVIEs:
()=∫
−1
(,,())+(), −1≤≤1, (1)
where (,,()) = [
1
(,,()),
2
(,,())]
and
() = [
1
(),
2
()]
are given, whereas () = [(),
V()]
is the unknown function. We will consider the case
that the solution of (1) is sufciently smooth.
Te remainder of this paper is organized as follows. Te
LSM is introduced in Section 2. Convergence analysis of the
proposed method is discussed in Section 3. Section 4 states
three applications of the desired equation in the biological
systems. In Section 5, four types of biological models that are
known as Lotka-Volterra system of equations are solved by
the LSM to show the efciency of the presented method and
to verify the theoretical results obtained in Section 3. Also
some comparisons are made with the existing results. Finally,
Section 6 includes some concluding remarks.
Hindawi Publishing Corporation
Advances in Mathematical Physics
Volume 2014, Article ID 307907, 12 pages
http://dx.doi.org/10.1155/2014/307907