Numerical Solution of the Fredholme-Volterra Integral Equation by the Sinc Function ()
1. Introduction
In recent years, many different methods have been used to approximate the solution of the Fredholme-Volterra Integral Equations, such as [1,2]. In this paper, we first present the Sinc Function and their properties. Then we consider the Fredholme-Volterra Integral Equation types in the forms
(1.1)
where
, 
and f(x) are known functions, but u(x) is an unknown function. Then we use the Sinc Function and convert the problem to a system of linear equations.
2. Sinc Function Properties
The sinc function properties are discussed thoroughly in [3-10]. The sinc function is defined on the real line by
(2.1)
For
, and 
The translated sinc functions with evenly spaced nodes are given by
(2.2)
The sinc function form for the interpolating point 
is given by
(2.3)
Let
(2.4)
If a function 
is defined on the real axis, then for h > 0 the series
(2.5)
called whittaker cardinal expansion of
, whenever this series converges. The properties of the whittaker cardinal expansion have been extensively studied in [8].
These properties are derived in the infinite stripe D of the complex wplane, where for
,
Approximations can be constructed for infinite, semiinfinite and finite intervals. To construct approximations on the interval [a,b], which is used in this paper, the eyeshaped domain in the z-plane.
Is mapped conformably onto the infinite strip D via
The basis functions on [a,b] are taken to be composite translated sinc functions,
(2.6)
Thus we may define the inverse images of the real line and of evenly spaced nodes 
as
and
(2.7)
We consider the following definitions and theorems in [8-10].
Definition 2.1:
Let 
be the set of all analytic functions, for which there exists a constant, C, such that
(2.8)
Theorem 2.1:
Let
, let N be appositive integer, and let h be
(2.9)
Then there exists positive constant C1, independent of N, such that
(2.10)
Proof: See [8,9].
Theorem 2.2:
Let 
Let N be a positive integer and let h be selected by the relation (2.9) then there exist positive constant C2, independent of N, such that
(2.11)
and also for 
be defined as in (2.4) then there exists a constant, 
which is independent of N, such that
(2.12)
Proof: See [8].
3. The Sinc Collocation Method
The solution of linear Fredholm-Volterra integral equation (1.1) is approximated by the following linear combination of the sinc functions and auxiliary functions:
(3.1)
where
(3.2)
where the basis functions 
defined by
(3.3)
(3.4)
(3.5)
We denote
then basis function must satisfy the following conditions:
(3.6)
(3.7)
Obviously by using Equations (2.3) and (3.1) we have
Lemma 3.1:
, let N be a positive integer and
, Then (see Equation (3.8)), where 
is defined in (2.14) and C4 is a positive constant, independent of N.
Proof: See [9].
Lemma 3.2:
For 
defined in (3.1), let
and h be selected from (2.9) then (see Equation(3.9))
(3.8)
(3.9)
Now let 
be the exact solution (1.1) that is approximated by following expansion.
(3.10)
Upon replacing 
in the Fredholm-Volterra integral equation (1.1)
, applying lemma 3.1 and Lemma 3.2, setting sinc collocation points 
and Then, considering 
we obtain the following system
(3.11)
We write the above system of equations in the matrix forms:
(3.12)
where
(3.13)
(3.14)
(3.15)
(3.16)
(3.17)
By solving the above system we obtain, 
, then, by using such solution we can obtain the approximate solution un as
(3.18)
4. Convergence Analysis
Now we discuss the convergence each of sinc collocation method. Suppose that 
is the exact solution of the Fredholme-Volterra integral equation (1.1). For each N, we can find 
which is our solution of the liner system (3.12), also by using 
we obtain the approximate solution
, In order to derive a bound for |u(x) - un(x)| we need to estimate the norm of the vector
, where 
is a vector defined by
where 
is the value of the exact solution of integral equation at the sinc points
. There for we need the following lemma.
Lemma 4.1:
Let u(x) be the exact solution of the integral (1.1) and 
let
for
, then there exists a constant C5 independent of N, such that
(4.1)
Proof: See [10].
Theorem 4.1:
Let us consider all assumptions of lemma 3.1 and let 
be the approximate solution of Fredholme-Volterra integral equation given by (3.3) then we have
(4.2)
where C6 a constant independent of N, and 
Proof:
Suppose 
defined this following form:
(4.3)
So we have
(4.4)
By using lemma 3.1, we obtain
Obviously by using equations (4.3) and (3.3) we have.
(4.5)
And we have from definition of the 
We obtain
(4.6)
That C7 a constant independent of N.
Now, by using equations (4.5) and (4.6) we get
(4.7)
In this case by using the system (3.12) and lemma 4.1 we obtain
(4.8)
Now by using Equations (4.7) and (4.8) we get
(4.9)
Obviously by using Equations (4.9) and (4.2) we obtain
5. Numerical Examples
In this section, we apply the sinc collocation method for solving Fredholm-Volterra integral equation example.
Example 5.1: consider the following Fredholm-Volterra integral equation of the second kind with exact solution u(x) = x.
We solved example 5.1 for different of
And we consider the sinc grid points as:
where
The errors on the given points are denoted by
(5.1)
Computational results are given in Tables 1-5.
Table 1. Results for Example 1 (N = 5).
Table 2. Results for Example 1 (N = 10).
Table 3. Results for Example 1 (N = 15).
Example 5.2: we consider the following FredholmVolterra integral equation of the second kind with exact solution
.
Computational results are given in Tables 6-9.
Table 4. Results for Example 1 (N = 20).
Table 6. Results for Example 2 (N = 5).
Table 7. Results for Example 2 (N = 7).
Table 8. Results for Example 2 (N = 9).