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

Authors: Hojatollah Adibi; Pouria Assari
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

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.


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## Body

## 1. 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 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:(1.1)∫01K(x,y)u(y)dy=f(x),0≤x≤1,
where f∈Lw2[0,1] and K∈Lw2([0,1]×[0,1]), in which w(x)=1/2x(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 Section2, 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.


## 2. Properties of Chebyshev Wavelets

### 2.1. 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 parametera and the translation parameter b vary continuously, we have the following family of continuous wavelets [17]:(2.1)ψa,b(t)=|a|-1/2ψ(t-ba),a,b∈ℝ,a≠0.
If we restrict the parameters a and b to discrete values a=a0-k,b=nb0a0-k,a0>1,b0>0 where n and k are positive integers, then we have the following family of discrete wavelets:(2.2)ψk,n(t)=|a0|k/2ψ(a0kt-nb0),
where ψk,n(t) form a wavelet basis for L2(ℝ). In particular, when a0=2 and b0=1, then ψk,n(t) forms an orthonormal basis [17, 18].Chebyshev waveletsψnm(t)=ψ(k,n,m,t) have four arguments: n=1,2,…,2k-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](2.3)ψnm(t)={2k/2T̃m(2kt-2n+1),n-12k-1≤t<n2k-1,0,otherwise,
where(2.4)T̃m(t)={1π,m=0,2πTm(t),m>0,
and m=0,1,…,M-1,n=1,2,…,2k-1. Here Tm(t), m=0,1,…, are Chebyshev polynomials of first kind of degree m, given by [20](2.5)Tm(t)=cosmθ,
in which θ=arccost. Chebyshev polynomials are orthogonal with respect to the weight function w((t+1)/2)=1/1-t2, on [-1,1]. We should note that Chebyshev wavelets are orthonormal set with the weight function: (2.6)wk(t)={w1,k(t),0≤t<12k-1,w2,k(t),12k-1≤t<22k-1,⋮⋮w2k-1,k(t),2k-1-12k-1≤t<1,
where wn,k(t)=w(2k-1t-n+1).


### 2.2. Function Approximation

A functionf(x)∈Lw2[0,1] may be expanded as(2.7)f(x)=∑n=1∞∑m=0∞cnmψnm(x),
where(2.8)cnm=〈f(x),ψnm(x)〉wk,
in which 〈.,.〉wk denotes the inner product in Lwk2[0,1]. The series (2.7) is truncated as(2.9)f(x)≃Tk,M(f(x))=∑n=12k-1∑m=0M-1cnmψnm(x)=CtΨ(x),
where C and Ψ(x) are two vectors given by(2.10)C=[c10,c12,…,c1(M-1),c20,…,c2(M-1),…,c(2k-1)0,…,c(2k-1)(M-1)]t=[c1,c2,…,c2k-1M]t,Ψ(x)=[ψ10(x),ψ12(x),…,ψ1(M-1)(x),ψ20(x),…,ψ2(M-1)(x),…,ψ(2k-1)0(x),…,ψ(2k-1)(M-1)(x)]t=[ψ1(x),ψ2(x),…,ψ2k-1M(x)]t.
Similarly, by considering i=M(n-1)+m+1 and j=M(n'-1)+m'+1, we approximate K(x,y)∈Lw2([0,1]×[0,1]) as(2.11)K(x,y)≃∑i=12k-1M∑j=12k-1MKijψi(x)ψj(y),
or in the matrix form(2.12)K(x,y)≃Ψt(x)KΨ(y),
where K=[Kij]1≤i,j≤2k-1M with the entries(2.13)Kij=〈ψi(x),〈K(x,y),ψj(y)〉wk〉wk.


## 2.1. 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 parametera and the translation parameter b vary continuously, we have the following family of continuous wavelets [17]:(2.1)ψa,b(t)=|a|-1/2ψ(t-ba),a,b∈ℝ,a≠0.
If we restrict the parameters a and b to discrete values a=a0-k,b=nb0a0-k,a0>1,b0>0 where n and k are positive integers, then we have the following family of discrete wavelets:(2.2)ψk,n(t)=|a0|k/2ψ(a0kt-nb0),
where ψk,n(t) form a wavelet basis for L2(ℝ). In particular, when a0=2 and b0=1, then ψk,n(t) forms an orthonormal basis [17, 18].Chebyshev waveletsψnm(t)=ψ(k,n,m,t) have four arguments: n=1,2,…,2k-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](2.3)ψnm(t)={2k/2T̃m(2kt-2n+1),n-12k-1≤t<n2k-1,0,otherwise,
where(2.4)T̃m(t)={1π,m=0,2πTm(t),m>0,
and m=0,1,…,M-1,n=1,2,…,2k-1. Here Tm(t), m=0,1,…, are Chebyshev polynomials of first kind of degree m, given by [20](2.5)Tm(t)=cosmθ,
in which θ=arccost. Chebyshev polynomials are orthogonal with respect to the weight function w((t+1)/2)=1/1-t2, on [-1,1]. We should note that Chebyshev wavelets are orthonormal set with the weight function: (2.6)wk(t)={w1,k(t),0≤t<12k-1,w2,k(t),12k-1≤t<22k-1,⋮⋮w2k-1,k(t),2k-1-12k-1≤t<1,
where wn,k(t)=w(2k-1t-n+1).


## 2.2. Function Approximation

A functionf(x)∈Lw2[0,1] may be expanded as(2.7)f(x)=∑n=1∞∑m=0∞cnmψnm(x),
where(2.8)cnm=〈f(x),ψnm(x)〉wk,
in which 〈.,.〉wk denotes the inner product in Lwk2[0,1]. The series (2.7) is truncated as(2.9)f(x)≃Tk,M(f(x))=∑n=12k-1∑m=0M-1cnmψnm(x)=CtΨ(x),
where C and Ψ(x) are two vectors given by(2.10)C=[c10,c12,…,c1(M-1),c20,…,c2(M-1),…,c(2k-1)0,…,c(2k-1)(M-1)]t=[c1,c2,…,c2k-1M]t,Ψ(x)=[ψ10(x),ψ12(x),…,ψ1(M-1)(x),ψ20(x),…,ψ2(M-1)(x),…,ψ(2k-1)0(x),…,ψ(2k-1)(M-1)(x)]t=[ψ1(x),ψ2(x),…,ψ2k-1M(x)]t.
Similarly, by considering i=M(n-1)+m+1 and j=M(n'-1)+m'+1, we approximate K(x,y)∈Lw2([0,1]×[0,1]) as(2.11)K(x,y)≃∑i=12k-1M∑j=12k-1MKijψi(x)ψj(y),
or in the matrix form(2.12)K(x,y)≃Ψt(x)KΨ(y),
where K=[Kij]1≤i,j≤2k-1M with the entries(2.13)Kij=〈ψi(x),〈K(x,y),ψj(y)〉wk〉wk.


## 3. Convergence of the Chebyshev Wavelet Bases

In this section, we indicate that the Chebyshev wavelet expansion of a functionf(x), with bounded second derivative, converges uniformly to f(x).Lemma 3.1.
If the Chebyshev wavelet expansion of a continuous functionf(x) converges uniformly, then the Chebyshev wavelet expansion converges to the function f(x).Proof.
Let(3.1)g(x)=∑n=1∞∑m=0∞cnmψnm(x),
where cnm=〈f(x),ψnm(x)〉wk. Multiplying both sides of (3.1) by ψpq(x)wk(x), where p and q are fixed and then integrating termwise, justified by uniformly convergence, on [0,1], we have
(3.2)∫01g(x)ψpq(x)wk(x)dx=∫01∑n=1∞∑m=0∞cnmψnm(x)ψpq(x)wk(x)dx=∑n=1∞∑m=0∞cnm∫01ψnm(x)ψpq(x)wk(x)dx=cpq.
Thus 〈g(x),ψnm(x)〉wk=cnm for n=1,2,… and m=0,1,…. Consequently f and g have same Fourier expansions with Chebyshev wavelet basis and therefore f(x)=g(x);(0≤x≤1) [21].Theorem 3.2.
A functionf(x)∈Lw2([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)f(x)=∑n=1∞∑m=0∞cnmψnm(x).Proof.
From (2.8) it follows that
(3.4)cnm=∫01f(x)ψnm(x)wk(x)dx=∫(n-1)/2k-1n/2k-12k/2f(x)T̃m(2kx-2n+1)w(2kx-2n+1)dx.
If m>1, by substituting 2kx-2n+1=cosθ in (3.4), it yields
(3.5)cnm=12k/2∫0πf(cosθ+2n-12k)2πcosmθdθ=22k/2πf(cosθ+2n-12k)(sinmθm)]0π(3.6)+223k/2mπ∫0πf′(cosθ+2n-12k)sinmθsinθdθ(3.7)=123k/2m2πf′(cosθ+2n-12k)(sin(m-1)θm-1-sin(m+1)θm+1)]0π+125k/2m2π∫0πf′′(cosθ+2n-12k)hm(θ)dθ,
where
(3.8)hm(θ)=sinθ(sin(m-1)θm-1-sin(m+1)θm+1).
Thus, we get
(3.9)|cnm|=|125k/2m2π∫0πf′′(cosθ+2n-12k)hm(θ)dθ|≤(125k/2m2π)∫0π|f′′(cosθ+2n-12k)hm(θ)|dθ≤(N25k/2m2π)∫0π|hm(θ)|dθ.
However
(3.10)∫0π|hm(θ)|dθ=∫0π|sinθ(sin(m-1)θm-1-sin(m+1)θm+1)|dθ≤∫0π|sinθsin(m-1)θm-1|+|sinθsin(m+1)θm+1|dθ≤2mπ(m2-1).
Since n≤2k-1, we obtain
(3.11)|cnm|<2πN(2n)5/2(m2-1).
Now, if m=1, by using (3.6), we have
(3.12)|cn1|<2π(2n)3/2max0≤x≤1|f′(x)|.
Hence, the series ∑n=1∞∑m=1∞cnm is absolutely convergent. It is understandable that for m=0, {ψn0}n=1∞ form an orthogonal system constructed by Haar scaling function with respect to the weight function w(t), and thus ∑n=1∞cn0ψn0(x) is convergence [22]. On the other hand, we have
(3.13)|∑n=1∞∑m=0∞cnmψnm(x)|≤|∑n=1∞cn0ψn0(x)|+∑n=1∞∑m=1∞|cnm||ψnm(x)|≤|∑n=1∞cn0ψn0(x)|+∑n=1∞∑m=1∞|cnm|<∞.
Therefore, utilizing Lemma 3.1, the series ∑n=1∞∑m=0∞cnmψnm(x) converges to f(x) uniformly.


## 4. 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)f(x)≃FtΨ(x),u(y)≃UtΨ(y),K(x,y)≃Ψt(x)KΨ(y).
By substituting (4.1) into (1.1), we obtain(4.2)Ψt(x)K(∫01Ψ(y)Ψt(y)dy)U-Ψt(x)F=R2k-1M(x),
where R2k-1M(x) is the residual. By letting(4.3)∫01Ψ(y)Ψt(y)dy=L,
where L is a 2k-1M×2k-1M matrix which is computed next, we have(4.4)Ψt(x)KLU-Ψt(x)F=R2k-1M(x).
Our aim is to compute u1,u2,…,u2k-1M such that R2k-1M(x)≡0, but in general, it is not possible to choose such ui,i=1,2,…,2k-1M. In this work,  R2k-1M(x) is made as small as possible such that(4.5)〈ψnm(x),R2k-1M(x)〉wk=0,
where n=1,2,…,2k-1 and m=0,1,…,M-1. Now, by using orthonormality of Chebyshev wavelets, we obtain the following linear system of algebraic equations:(4.6)KLU=F,
for unknowns U=[u1,u2,…,u2k-1M].Here, we define two operator equations𝒦 and ℋ as follows:(4.7)𝒦(u(x))=∫01K(x,y)u(y)dy,(4.8)ℋ(u(x))=∫01Ψt(x)KΨ(y)u(y)dy,
for all u∈Lw2[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(4.9)𝒦u=f,ℋTk,M(u)=Tk,M(f).
Combining the latter equations yields(4.10)𝒦ek,M=(𝒦-ℋ)Tk,M(u)+(f-Tk,M(f)),
where ek,M=u-Tk,M(u). Provided that 𝒦-1 exists, we obtain the error bound:(4.11)∥ekM∥≤∥𝒦-1∥∥(𝒦-ℋ)TkM(u)+(f-TkM(f))∥.
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 TkM(f) to f.Suppose that the functionf(x), defined on [0,1], is M times continuously differentiable, f∈CM([0,1]); by using properties of Chebyshev wavelets and similar to [17], we have (4.12)∥f(x)-Tk,M(f(x))∥2=(∫01[f(x)-Tk,M(f(x))]2dx)1/2≤(∑n=12k-1∫(n-1)/2k-1n/2k-1∫01[f(x)-Ck,Mn(f(x))]2dx)1/2≤2-kMQ,
where Q=(2/2MM!)sup0≤x≤1|f(M)(x)| and Ck,Mn(f(x)) denotes the polynomial of degree M which agrees with f at the Chebyshev nodes of the order M on [(n-1)/2k-1,n/2k-1]. Therefore, if we want to have ∥f(x)-Tk,M(f(x))∥2<ɛ, we can choose k as (4.13)k=[Qɛln(2)M]+1.EvaluatingL
For numerical implementation of the method explained in previous part, we need to calculate matrixL=[Lij]1≤i,j≤2k-1M. For this purpose, by considering i=M(n-1)+m+1 and j=M(n'-1)+m'+1, we have
(4.14)Lij=∫01ψi(y)ψj(y)dy.
If n≠n′, then ψi(y)ψj(y)=0, because their supports are disjoint, yielding Lij=0. Hence, let n=n'; by substituting 2kx-2n+1=cosθ in (4.14), we obtain
(4.15)Lij=Cmm′∫0πcosmθcosm′θsinθdθ,
where
(4.16)Cmm'={-1π,m=m'=0,-2π,m≠0≠m′,-2π,otherwise.
Now, if |m+m'|=1, then
(4.17)∫0πcosmθcosm′θsinθdθ=0,
implies that Lij=0,  and if |m+m'|≠1, then
(4.18)Lij=Cmm′4×(-cos(m-m′+1)θm-m′+1+cos(-m+m′+1)θ-m+m′+1-cos(m+m′-1)θm+m′-1+cos(-m-m′+1)θ-m-m′+1)]0π.
Consequently, L has the following form:
(4.19)L=diag(A,A,…,A︸2k-1times),
where A=[Amm'] is an M×M matrix with the elements
(4.20)Amm′={Cmm'2(m2+m'2-1)1+m4-2m'2m2-2m'2-2m2+m'4,m+m′iseven,0,m+m′isodd.


## 5. Sparse Representation of the MatrixK

We proceed by discussing the sparsity of the matrixK, as an important issue for increasing the computation speed.Theorem 5.1.
Suppose thatKij 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 ∂4K(x,y)/∂x2∂y2 bounded by N and m,m'>1, then one has
(5.1)|Kij|<πN24(nn′)5/2(m2-1)(m'2-1).Proof.
From (2.13), we obtain
(5.2)|Kij|=2k|∫(n-1)/2k-1n/2k-1∫(n′-1)/2k-1n′/2k-1K(x,y)T̃m(2kx-2n+1)w(2kx-2n+1)×T̃m′(2ky-2n′+1)w(2ky-2n′+1)dydx∫(n′-1)/2k-1n′/2k-1|.
Now, let 2kx-2n+1=cosθ and 2ky-2n'+1=cosα; then
(5.3)|Kij|=22kπ|∬0πK(cosθ+2n-12k,cosα+2n′-12k)cosmθcosm′αdαdθ|.
Similar to the proof of Theorem 3.2, since m,m'>1, we obtain
(5.4)|Kij|≤125k+1πmm'|∬0π∂4K((cosθ+2n-1)/2k,(cosα+2n′-1)/2k)∂t2∂s2hm(θ)hm′(α)dαdθ|≤N25k+1πmm'∫0π|hm(θ)|dθ∫0π|hm′(α)|dα<πN24(nn′)5/2(m2-1)(m'2-1).Remark 5.2.
As an immediate conclusion from Theorem5.1, when i or j→∞, it follows that |Kij|→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 ɛ0 and get the following system of linear equations whose matrix is sparse:
(5.5)K¯LU=F,
where K¯=[K¯ij]2k-1M×2k-1M with the entries
(5.6)K¯ij={Kij,|Kij|≥ɛ0,0,otherwise.
Now, we can solve (5.5) instead of (4.6).


## 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 6.1 and 6.2, 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.Example 6.1.
As the first example, let(6.1)∫01sin(xy)u(y)dy=sin(x)-xcos(x)x2,0≤x≤1,
with the exact solution uex(x)=x [15].
Table1 shows the numerical results for this example with k=2,M=2,ɛ0=10-5 and k=2,M=4,ɛ0=10-4. Also, the approximate solution for k=2, M=4, ɛ0=10-4 is graphically shown in Figure 1, which agrees with exact solution and results are compared with those of [15].Table 1
Some numerical results for Example6.1.
xExact solutionApproximate solutionk=2,M=2,ɛ0=10-5Approximate solutionk=2,M=4,ɛ0=10-4Legendre  wavelets [15]0.00.00000000000.0000000002-0.0000080915-0.00006232030.10.10000000000.09994671450.09999190840.09993998030.20.20000000000.19986283690.19999190830.19994228100.30.30000000000.29977895930.29999190830.29994458160.40.40000000000.39969508170.39999190820.39994688230.50.50000000000.49940171040.49998211260.50008367480.60.60000000000.59961972200.59999084260.60005833210.70.70000000000.69944893770.69999148080.70003298940.80.80000000000.79927815330.79999211900.80000764660.90.90000000000.89910736900.89999275720.89998230391.01.00000000000.99893658470.99999339550.9999569612Figure 1
Approximate solution for Example6.1 with k=2,M=4,ɛ0=10-4.Example 6.2.
In this example we solve integral equation(6.2)u(x)-∫01exyu(y)dy=ex+1-1x+1,0≤x≤1,
by the present method, where the exact solution is uex(x)=ex [11].
Table2 gives the absolute error for this example with k=2,M=3,ɛ0=10-5 and k=3,M=4,ɛ0=10-4 where ũ denote the approximation of uex. The approximate solution for k=3,M=4,ɛ0=10-4 in collocation points xj=(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 Example6.2.
x|ũ(x)-uex(x)||ũ(x)-uex(x)||ũ(x)-uex(x)|k=2,M=3,ɛ0=10-5k=3,M=4,ɛ0=10-4Haar wavelets [14]0.00.488296e-30.146932e-40.785334e-20.10.937569e-30.150830e-40.173943e-20.20.265918e-40.173487e-40.569956e-20.30.108134e-20.186327e-40.635611e-20.40.110062e-20.157720e-40.231400e-20.50.125395e-30.658547e-50.129479e-10.60.211862e-20.130256e-50.286785e-20.70.226888e-20.232601e-50.939698e-20.80.728295e-30.152448e-40.104794e-10.90.383486e-30.983848e-50.381514e-21.00.127610e-20.104492e-40.498723e-2Figure 2
Approximate solution for Example6.2 with k=3,M=4,ɛ0=10-4.Example 6.3.
As our final example let(6.3)∫01(y-x)21+y2u(y)dy=0.179171-0.532108x+0.487495x2,0≤x≤1,
with the exact solution u(x)ex=x [14].
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 ∥.∥2. The following norms are used for the errors of the approximation û(x) of u(x):
(6.4)∥u-û∥∞=max{|u(x)-û(x)|,0≤x≤1},∥u-û∥2=(∫01|u(x)-û(x)|2dx)1/2.
Also, the error e(x)=u(x)-û(x)for k=2, M=3, and k=3, M=4 is graphically shown in Figures 3 and 4 for [0,1/2] and [1/2,1], respectively.Table 3
Some error estimates for Example6.3.
kM∥u-û∥∞∥u-û∥222<0.15005271920.2898279425e-123<0.9003163152e-10.1298595687e-124<0.6430830822e-10.7461435276e-233<0.6366197694e-10.6493091346e-234<0.4547284061e-10.3730718373e-235<0.3536776479e-10.2438633959e-2Figure 3
Error distributions for Example6.3 with k=2, M=3 and k=3, M=4 on [0,1/2].Figure 4
Error distributions for Example6.3 with k=2, M=3 and k=3, M=4 on [1/2,1].


## 7. 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 parameterk is recommended. The convergence accuracy of this method was examined for several numerical examples.



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*Source: 138408-2010-06-28.xml*