Applied Mathematics, 2013, 4, 204-209
http://dx.doi.org/10.4236/am.2013.41A031 Published Online January 2013 (http://www.scirp.org/journal/am)
Numerical Solution of Freholm-Volterra Integral
Equations by Using Scaling Function Interpolation Method
Yousef Al-Jarrah, En-Bing Lin
Department of Mathematics, Central Michigan University, Mt. Pleasant, USA
Email: enbing.lin@cmich.edu
Received October 15, 2012; revised November 15, 2012; accepted November 23, 2012
ABSTRACT
Wavelet methods are a very useful tool in solving integral equations. Both scaling functions and wavelet functions are
the key elements of wavelet methods. In this article, we use scaling function interpolation method to solve Volterra in-
tegral equations of the first kind, and Fredholm-Volterra integral equations. Moreover, we prove convergence theorem
for the numerical solution of Volterra integral equations and Freholm-Volterra integral equations. We also present three
examples of solving Volterra integral equation and one example of solving Fredholm-Volterra integral equation. Com-
parisons of the results with other methods are included in the examples.
Keywords: Wavelets; Coiflets; Scaling Function Interpolation; Volterra Integral Equation; Fredholm-Volterra Integral
Equation
1. Introduction
The study of finite-dimensional linear systems is well
developed. As an infinite-dimensional counter part of
finite-dimensional linear systems, one can view integral
equations as extensions of linear systems of algebraic
equations. An integral equation maybe interpreted as an
analogue of a matrix equation which is easier to solve.
There are many different ways to transform integral
equations to linear systems. Many different methods have
been used for solving Volterra integral equations and
Freholm-Velterra integral equations numerically.
In this paper, we first recall the method of scaling
function interpolation. Then we solve linear Volterra inte-
gral equation of the form:
,
x
a
d f x kxt yt
t (1)
and Fredholm-Volterra integral equations of the form:
1
2
, d
, d
x
a
b
a
yx gx k xt yt t
k xt yt t
(2)
where the functions and
1
, , , kxt k xt
2
, k xt are
known functions and called kernels. The function f x
is known, and the function y t is to be determined.
One of the motivations in this study arose from equations
in theoretical physics. In fact, there are many applica-
tions in several disciplines as well. We will use scaling
function interpolation method to solve integral equations.
As a natural question, one would wonder any possible
convergence properties and how this method would com-
pare with other methods. We will prove two convergence
theorems and present several examples.
2. Approximation
Wavelets and scaling functions are a useful tool in ap-
proximation methods of solutions of differential and in-
tegral equations [1]. We first recall Multiresolution analy-
sis (MRA) [2]. We assume the scaling function and
wavelet function , Ψ are sufficiently smooth and satisfy
MRA with compact support and Ψ has N vanishing mo-
ments (defined below). The scaling function x is
defined as
,
2
j
p p
p p
jp
x x p x
(3)
for some coefficients
,
p
p Z . By using this dilation
and translation we defined a nested of sequence spaces
,
j
V j Z which is called MRA of with the
following properties:
2
L R
1
,
j j
V V j Z
(4)
0
j
j Z
V V
(5)
j
j Z
V
is dense in (6) L R
2
1
2
j j
x V x
V . (7)
For the subspace is built by
1
V 2 x p , p Z
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