A Wavelet Based Method for the Solution of Fredholm Integral Equations ()
1. Introduction
Integral equations play an important role in both mathematics and other applicable areas. Many physical phenomena can be modeled by differential equations. In fact, a differential equation can be replaced by an integral equation that incorporates its boundary conditions. Integral equations are also useful in many branches of pure mathematics as well. Here we study Fredholm integral equations [1-3].
Wavelets have been applied in a wide range of engineering and physicaldisciplines, and it is an exciting tool for mathematicians. In this paper we will find a numerical solution for the second kind Fredholm integral equation of the form
(1)
where the function
and are
given, and the unknown function
is to be determined.
1.1. Wavelets
In this subsection we will provide a brief account of wavelet transform and Multiresolution analysis (MRA). We first define the scaling function
and the sequence
such that
(2)
By using this dilation and translation [4], we defined a nested sequence spaces
which is called MRA of
with the following properties
(3)
(4)
is dense in
(5)
(6)
For the subspace
is built by
then
and since
we can write
![](https://www.scirp.org/html/7-1100078\2f9324a0-8681-4a0a-a3c7-8fa57e410bb7.jpg)
In general,
![](https://www.scirp.org/html/7-1100078\bd5fff42-9c3e-49f9-b1bc-097f735b4694.jpg)
(7)
Any function
can be approximated by scaling functions in one of the subspace in the given nested sequence. In fact, for each j we define the orthogonal complement subspace
of
in the subspace
. The orthogonal basis of
is denoted by
(8)
and the wavelet function can be obtained by
. (9)
Some interesting properties of scaling and wavelet functions make wavelet method more efficiently than quadrature formula methods and spline approximations in solving Integral equations. A lot of computational time and storage capacity can be saved since we do not require a huge number of arithmetic operations partly due to the following properties.
1) Vanishing Moments:
(10)
and in this case the wavelet is said to have a vanishing moments of order m2) Semiorthogonality:
(11)
The set of scaling functions
are orthogonal at the same level n, which means:
(12)
Coiflet (of order L) has more symmetries and is an orthogonal multiresolution wavelet system with,
(13)
(14)
where
are the moments of the scaling functions.
1.2. Scaling Function Interpolation
The function
can be interpolated by using the basis functions in the subspace
as follows.
(15)
where
are the coefficients evaluated by using equation (12) such that
(16)
Hence the equation (15) becomes:
.
On the other hand, one can use sampling values of
at certain points to approximate the function
It is proved in [5], namely, an interpolation theoremusing coiflet such that if
and
are sufficiently smooth and satisfy the equations (10)-(14) and the function
, where
is a bounded open set in
![](https://www.scirp.org/html/7-1100078\1a2478a0-0251-40f5-b228-30cb29c2ee58.jpg)
Then,
(17)
where the index set is
![](https://www.scirp.org/html/7-1100078\17d0a62a-6d72-4837-9d07-48835726ed79.jpg)
Sup denote the support of a function.
In addition, the moment
satisfies
![](https://www.scirp.org/html/7-1100078\c5ac4846-92db-47b5-9813-90a1470ab22c.jpg)
Then,
![](https://www.scirp.org/html/7-1100078\4381e667-ba28-4662-b8e1-f629df1386a5.jpg)
where
is a constant depending only on
, diameter of
and
![](https://www.scirp.org/html/7-1100078\256dd134-e030-48f2-9f4b-78aadd009ee3.jpg)
For the function with one variable, we have
(18)
and
(19)
where
(20)
2. Solve Fredholm Integral Equations Using Coiflet
In this section we will apply coiflet and the interpolation formula (18) to solve the Fredholm integral equation (1). The unknown function
in equation (1) can be expanded in term of the scaling functions
in the subspace
such that
(21)
Consider the equation (1) and the function
which is defined on the interval [a, b] and the scaling function
defined on the interval
then we have the index:
![](https://www.scirp.org/html/7-1100078\1b356252-252f-46fd-8253-c1abbaea0f6f.jpg)
By applying equation (21) into equation (1), we get the system,
(22)
which is equivalent to the following system,
(23)
where thecoefficients
can be evaluated by substituting
into the system (23). Moreover, the system (23) can be expressed in compact form,
(24)
where
![](https://www.scirp.org/html/7-1100078\46564f65-4199-4901-ae82-b15ba3bbbef1.jpg)
![](https://www.scirp.org/html/7-1100078\6ef6a8e2-3705-4056-a03f-63e3d3730d76.jpg)
Then ![](https://www.scirp.org/html/7-1100078\8ef6cf36-4fb9-4877-92ed-62352493193e.jpg)
This gives rise to coefficients in (21) and we obtain a numerical solution of (1). In what follows, we will derive a convergence theorem of this numerical solution.
3. Error Analysis
In this section will discuss the convergence rate of our method for solving linear Fredholm integral equation (1).
Theorem 1. In equation (1), supposethat the function
and the functions
and
, for ![](https://www.scirp.org/html/7-1100078\b9504c6c-5c56-485d-95be-c27be1ae0593.jpg)
(25)
is an approximate solution of the equation (1) with the coefficients obtained in (24). Then,
(26)
where,
![](https://www.scirp.org/html/7-1100078\55d537de-268d-4fbf-b178-0dde814ba2ea.jpg)
Proof. Subtracting equation (25) from equation (1) and taking the norm for both sides, we get the following
(27)
where ![](https://www.scirp.org/html/7-1100078\2e6b8663-4a07-438d-8b1e-d3b126856553.jpg)
By [5], the unknown function
can be interpolated by using coiflet such that:
(28)
Let
in equation (28) then add and subtract it in equation (27), we get the following inequalities.
![](https://www.scirp.org/html/7-1100078\546688d9-ad63-46de-93b5-b0efbbe59cf8.jpg)
which equals to the equation
(29)
By [5], we have
(30)
Since
is finite we define it as
. (31)
Using the above results and the orthonomality of the scaling functions
, we conclude that
(32)
4. Numerical Examples
In the following examples, we will solve linear Fredholm integral equation (1) using coiflet of order 5 and provide errors between exact solutions and our numerical solutions at different resolution levels. Both examples are also presented in [6] by using different method.
Example 1.
Consider
where
![](https://www.scirp.org/html/7-1100078\c00e37f1-0aed-4a07-b436-18790b8e2c3e.jpg)
The exact solution is
and ![](https://www.scirp.org/html/7-1100078\fb48a141-f4e8-4f86-9b9e-4764a32ae436.jpg)
Example 2.
Consider
![](https://www.scirp.org/html/7-1100078\8e860ffe-6c0a-4998-aaef-5ecf41a9add7.jpg)
where
and
are given on the interval [0,1] such that,
![](https://www.scirp.org/html/7-1100078\4aebd631-ec09-41d8-bd90-641f93221e21.jpg)
The exact solution is ![](https://www.scirp.org/html/7-1100078\9752ef79-7243-409f-8b21-41bf273aa120.jpg)
We use our interpolation method to solve the above integral equations, and find the errors in Table 1.
5. Conclusion
In this work, we use our interpolation method by using coiflets to solve Fredholm integral equations, and compare our results with those in [6]. It turns out our method is more efficient with better accuracy. Moreover, our method can be applied to different kind of integral equations as well as integral-algebraic equations. Although the results in the above examples don’t seem to have
![](https://www.scirp.org/html/7-1100078\01a85b9a-0e54-4f0b-9e62-4b504c0eb1cb.jpg)
Table 1. The error e(x) for examples 1 and 2 with different values of j.
correlation with the level of resolutions but they basically validate our theorem. In fact, we can also interpolate the given functions in the integral equation. This would simplify the calculations in finding numerical solutions of integral equations. It would be interesting to use our method to solve nonlinear integral equations as well.
6. Acknowledgements
The authors would like to thank the anonymous referees for their helpful comments.