Chebyshev Wavelet Method for Numerical Solution of Fredholm Integral Equations of the First Kind

A computational method for solving Fredholm integral equations of the first kind is presented. The method utilizes Chebyshev wavelets constructed on the unit interval as basis in Galerkin method and reduces solving the integral equation to solving a system of algebraic equations. The properties of Chebyshev wavelets are used to make the wavelet coefficient matrices sparse which eventually leads to the sparsity of the coefficients matrix of obtained system. Finally, numerical examples are presented to show the validity and efficiency of the technique.


Introduction
Many problems of mathematical physics can be stated in the form of integral equations.These equations also occur as reformulations of other mathematical problems such as partial differential equations and ordinary differential equations.Therefore, the study of integral equations and methods for solving them are very useful in application.In recent years, several simple and accurate methods based on orthogonal basic functions, including wavelets, have been used to approximate the solution of integral equation 1-5 .The main advantage of using orthogonal basis is that it reduces the problem into solving a system of algebraic equations.Overall, there are so many different families of orthogonal functions which can be used in this method that it is sometimes difficult to select the most suitable one.Beginning from 1991, wavelet technique has been applied to solve integral equations 6-10 .Wavelets, as very well-localized functions, are considerably useful for solving integral equations and provide accurate solutions.Also, the wavelet technique allows the creation of very fast algorithms when compared with the algorithms ordinarily used.
In various fields of science and engineering, we encounter a large class of integral equations which are called linear Fredholm integral equations of the first kind.Several Mathematical Problems in Engineering methods have been proposed for numerical solution of these types of integral equation.Babolian and Delves 11 describe an augmented Galerkin technique for the numerical solution of first kind Fredholm integral equations.In 12 a numerical solution of Fredholm integral equations of the first kind via piecewise interpolation is proposed.Lewis 13 studied a computational method to solve first kind integral equations.Haar wavelets have been applied to solve Fredholm integral equations of first kind in 14 .Also, Shang and Han 15 used Legendre multiwavelets for solving first kind integral equations.
Consider the linear Fredholm integral equations of the first kind: where f ∈ L 2 w 0, 1 and K ∈ L 2 w 0, 1 × 0, 1 , in which w x 1/2 x 1 − x , are known functions and u is the unknown function to be determined.In general, these types of integral equation are ill-posed for given K and f.Therefore 1.1 may have no solution, while if a solution exists, the response ration ∂u / ∂f to small perturbations in f may be arbitrary large 16 .
The main purpose of this article is to present a numerical method for solving 1.1 via Chebyshev wavelets.The properties of Chebyshev wavelets are used to convert 1.1 into a linear system of algebraic equations.We will notice that these wavelets make the wavelet coefficient matrices sparse which concludes the sparsity of the coefficients matrix of obtained system.This system may be solved by using an appropriate numerical method.
The outline of the paper is as follows: in Section 2, we review some properties of Chebyshev wavelets and approximate the function f and also the kernel function K x, y by these wavelets.Convergence theorem of the Chebyshev wavelet bases is presented in Section 3. Section 4 is devoted to present a computational method for solving 1.1 utilizing Chebyshev wavelets and approximate the unknown function u x .In Section 5, the sparsity of the wavelet coefficient matrix is studied.Numerical examples are given in Section 6.Finally, we conclude the article in Section 7.

Wavelets and Chebyshev Wavelets
Wavelets consist of a family of functions constructed from dilation and translation of a single function called the mother wavelet.When the dilation parameter a and the translation parameter b vary continuously, we have the following family of continuous wavelets 17 : If we restrict the parameters a and b to discrete values a a −k 0 , b nb 0 a −k 0 , a 0 > 1, b 0 > 0 where n and k are positive integers, then we have the following family of discrete wavelets: Mathematical Problems in Engineering 3 where ψ k,n t form a wavelet basis for L 2 R .In particular, when a 0 2 and b 0 1, then ψ k,n t forms an orthonormal basis 17, 18 .
Chebyshev wavelets ψ nm t ψ k, n, m, t have four arguments: n 1, 2, . . ., 2 k−1 , k is any nonnegative integer, m is the degree of Chebyshev polynomial of first kind, and t is the normalized time.The Chebyshev wavelets are defined on the interval 0, 1 by 19 where Here T m t , m 0, 1, . .., are Chebyshev polynomials of first kind of degree m, given by 20 in which θ arccos t.Chebyshev polynomials are orthogonal with respect to the weight function w t 1 /2 1/ √ 1 − t 2 , on −1, 1 .We should note that Chebyshev wavelets are orthonormal set with the weight function: where

Function Approximation
A function f x ∈ L 2 w 0, 1 may be expanded as where in which ., .w k denotes the inner product in L 2 w k 0, 1 .The series 2.7 is truncated as where C and Ψ x are two vectors given by

2.10
Similarly, by considering i M n − 1 m 1 and j M n − 1 m 1, we approximate K x, y ∈ L 2 w 0, 1 × 0, 1 as or in the matrix form where

Convergence of the Chebyshev Wavelet Bases
In this section, we indicate that the Chebyshev wavelet expansion of a function f x , with bounded second derivative, converges uniformly to f x .

Lemma 3.1. If the Chebyshev wavelet expansion of a continuous function f x converges uniformly, then the Chebyshev wavelet expansion converges to the function
where c nm f x , ψ nm x w k .Multiplying both sides of 3.1 by ψ pq x w k x , where p and q are fixed and then integrating termwise, justified by uniformly convergence, on 0, 1 , we have

Theorem 3.2. A function f x ∈ L 2
w 0, 1 , with bounded second derivative, say |f x | ≤ N, can be expanded as an infinite sum of Chebyshev wavelets, and the series converges uniformly to f x , that is, 3.3 Proof.From 2.8 it follows that

3.4
If m > 1, by substituting 2 k x − 2n 1 cos θ in 3.4 , it yields where Thus, we get

3.10
Since n ≤ 2 k−1 , we obtain Now, if m 1, by using 3.6 , we have Hence, the series ∞ n 1 ∞ m 1 c nm is absolutely convergent.It is understandable that for m 0, {ψ n0 } ∞ n 1 form an orthogonal system constructed by Haar scaling function with respect to the weight function w t , and thus ∞ n 1 c n0 ψ n0 x is convergence 22 .On the other hand, we have

3.13
Therefore, utilizing Lemma 3.1, the series ∞ n 1 ∞ m 0 c nm ψ nm x converges to f x uniformly.

Solution of First Kind Integral Equations
In this section, the Chebyshev wavelet method is used for solving 1.1 by approximating functions f x , u y , and K x, y in the matrix forms:

4.1
By substituting 4.1 into 1.1 , we obtain where L is a 2 k−1 M × 2 k−1 M matrix which is computed next, we have Our aim is to compute u 1 , u 2 , . . ., u x is made as small as possible such that where n 1, 2, . . ., 2 k−1 and m 0, 1, . . ., M − 1.Now, by using orthonormality of Chebyshev wavelets, we obtain the following linear system of algebraic equations: for unknowns U u 1 , u 2 , . . ., u 2 k−1 M .Here, we define two operator equations K and H as follows: for all u ∈ L 2 w 0, 1 and x ∈ 0, 1 .We assume that integral operator K as defined in 4.7 is compact, one-to-one, onto, and K −1 < ∞.We rewrite 1.1 and 4.2 in the operator form to obtain Ku f,

4.9
Combining the latter equations yields where e k,M u − T k,M u .Provided that K −1 exists, we obtain the error bound: The error depends, therefore, on the conditioning of the original integral equation, as is apparent from the term K −1 , on the fidelity of the finite-dimensional operator H to the integral operator K, and on the approximation of T kM f to f.
Suppose that the function f x , defined on 0, 1 , is M times continuously differentiable, f ∈ C M 0, 1 ; by using properties of Chebyshev wavelets and similar to 17 , we have where Q 2/2 M M! sup 0≤x≤1 |f M x | and C n k,M f x denotes the polynomial of degree M which agrees with f at the Chebyshev nodes of the order

Evaluating L
For numerical implementation of the method explained in previous part, we need to calculate matrix L L ij 1≤i,j≤2 k−1 M .For this purpose, by considering i M n − 1 m 1 and j M n − 1 m 1, we have 4.14 If n / n , then ψ i y ψ j y 0, because their supports are disjoint, yielding L ij 0. Hence, let n n ; by substituting 2 k x − 2n 1 cos θ in 4.14 , we obtain where , otherwise.

4.18
Consequently, L has the following form: where A A mm is an M × M matrix with the elements

Sparse Representation of the Matrix K
We proceed by discussing the sparsity of the matrix K, as an important issue for increasing the computation speed.

5.2
Now, let 2 k x − 2n 1 cos θ and 2 k y − 2n 1 cos α; then Similar to the proof of Theorem 3.2, since m, m > 1, we obtain

5.4
Remark 5.2.As an immediate conclusion from Theorem 5.1, when i or j → ∞, it follows that |K ij | → 0 and accordingly by increasing k or M, we can make K sparse which concludes the sparsity of the coefficient matrix of system 4.6 .For this purpose, we choose a threshold ε 0 and get the following system of linear equations whose matrix is sparse:

5.6
Now, we can solve 5.5 instead of 4.6 .

Numerical Examples
In order to test the validity of the present method, three examples are solved and the numerical results are compared with their exact solution 11, 14, 15 .In addition, in Examples  1 shows the numerical results for this example with k 2, M 2, ε 0 10 −5 and k 2, M 4, ε 0 10 −4 .Also, the approximate solution for k 2, M 4, ε 0 10 −4 is graphically shown in Figure 1, which agrees with exact solution and results are compared with those of 15 .
Example 6.2.In this example we solve integral equation by the present method, where the exact solution is u ex x e x 11 .Table 2 gives the absolute error for this example with k 2, M 3, ε 0 10 −5 and k 3, M 4, ε 0 10 −4 where u denote the approximation of u ex .The approximate solution for k 3, M 4, ε 0 10 −4 in collocation points x j j −1/2 /50, j 1, 2, . . ., 50, is graphically shown in Figure 2 .It is seen that the numerical results are improved, as parameter k increases.Also, results are compared with those of 14 .Table 2: Absolute error of exact and approximated solution of Example 6.2.The proposed method was applied to approximate the solution of Fredholm integral equation 6.3 with some values of k and M. Table 3 represents the error estimate for the result obtained of .∞ and .

Conclusion
Integral equations are usually difficult to solve analytically, and therefore, it is required to obtain the approximate solutions.In this study we develop an efficient and accurate method for solving Fredholm integral equation of the first kind.The properties of Chebyshev wavelets are used to reduce the problem into solution of a system of algebraic equations whose matrix is sparse.However, to obtain better results, using the larger parameter k is recommended.The convergence accuracy of this method was examined for several numerical examples.

Theorem 5 . 1 .
Suppose that K ij is the Chebyshev wavelet coefficient of the continuous kernel K x, y , where i M n−1 m 1 and j M n −1 m 1.If mixed partial derivative is ∂ 4 K x, y /∂x 2 ∂y 2 bounded by N and m, m > 1, then one has

2 .
The following norms are used for the errors of the approximation u x of u x :u − u ∞ max{|u x − u x |, 0 ≤ x ≤ 1}, error e x u x − u x for k 2, M 3, and k 3, M 4 is graphically shown in Figures3 and 4for 0, 1/2 and 1/2, 1 , respectively.

x
Error with k 2, M 3 Error with k 3, M 4

x
Error with k 2, M 3 Error with k 3, M 4

Table 1 :
Some numerical results for Example 6.1., our results are compared with numerical results in 14, 15 .It is seen that good agreements are achieved, as dilation parameter a 2 −k decreases.

Table 3 :
Some error estimates for Example 6.3.