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

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.

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:¹₀𝐾(𝑥,𝑦)𝑢(𝑦)d𝑦=𝑓(𝑥),0≤𝑥≤1, (1.1) where 𝑓∈𝐿²_𝑤[0,1] and 𝐾∈𝐿²_𝑤([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 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.

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 parameter 𝑎 and the translation parameter 𝑏 vary continuously, we have the following family of continuous wavelets [17]:𝜓_𝑎,𝑏(𝑡)=|𝑎|^−1/2𝜓(𝑡−𝑏𝑎),𝑎,𝑏∈ℝ,𝑎≠0. (2.1) 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:𝜓_𝑘,𝑛(𝑡)=|𝑎₀|^𝑘/2𝜓(𝑎^𝑘₀𝑡−𝑛𝑏₀), (2.2) where 𝜓_𝑘,𝑛(𝑡) form a wavelet basis for 𝐿²(ℝ). In particular, when 𝑎₀=2 and 𝑏₀=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]𝜓_𝑛𝑚2(𝑡)=^𝑘/2𝑇_𝑚(2^𝑘𝑡−2𝑛+1),𝑛−12^𝑘−1𝑛≤𝑡<2^𝑘−1,0,otherwise, (2.3) where𝑇_𝑚1(𝑡)=√𝜋,𝑚=0,2𝜋𝑇_𝑚(𝑡),𝑚>0, (2.4) and 𝑚=0,1,…,𝑀−1,𝑛=1,2,…,2^𝑘−1. Here 𝑇_𝑚(𝑡), 𝑚=0,1,…, are Chebyshev polynomials of first kind of degree 𝑚, given by [20]𝑇_𝑚(𝑡)=cos𝑚𝜃, (2.5) 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: 𝑤_𝑘𝑤(𝑡)=^1,𝑘1(𝑡),0≤𝑡<2^𝑘−1,𝑤_2,𝑘1(𝑡),2^𝑘−12≤𝑡<2^𝑘−1,𝑤⋮⋮_{2𝑘−1,𝑘}2(𝑡),^𝑘−1−12^𝑘−1≤𝑡<1, (2.6) where 𝑤_𝑛,𝑘(𝑡)=𝑤(2^𝑘−1𝑡−𝑛+1).

2.2. Function Approximation

A function 𝑓(𝑥)∈𝐿²_𝑤[0,1] may be expanded as𝑓(𝑥)=^∞∞_𝑛=1𝑚=0𝑐_𝑛𝑚𝜓_𝑛𝑚(𝑥), (2.7) where𝑐_𝑛𝑚=⟨𝑓(𝑥),𝜓_𝑛𝑚(𝑥)⟩_𝑤𝑘, (2.8) in which ⟨.,.⟩_𝑤𝑘 denotes the inner product in 𝐿²_𝑤𝑘[0,1]. The series (2.7) is truncated as𝑓(𝑥)≃𝑇_𝑘,𝑀(𝑓(𝑥))=^2𝑘−1_𝑛=1^𝑀−1_𝑚=0𝑐_𝑛𝑚𝜓_𝑛𝑚(𝑥)=𝐶^𝑡Ψ(𝑥), (2.9) where 𝐶 and Ψ(𝑥) are two vectors given by𝐶=[𝑐₁₀,𝑐₁₂,…,𝑐_1(𝑀−1),𝑐₂₀,…,𝑐_2(𝑀−1),…,𝑐₍₂^𝑘−1₎₀,…,𝑐₍₂^𝑘−1_)(𝑀−1)]^𝑡=[𝑐₁,𝑐₂,…,𝑐₂^𝑘−1_𝑀]^𝑡,Ψ(𝑥)=[𝜓₁₀(𝑥),𝜓₁₂(𝑥),…,𝜓_1(𝑀−1)(𝑥),𝜓₂₀(𝑥),…,𝜓_2(𝑀−1)(𝑥),…,𝜓_(2𝑘−1)0(𝑥),…,𝜓_{(2𝑘−1)(𝑀−1)}(𝑥)]^𝑡=[𝜓₁(𝑥),𝜓₂(𝑥),…,𝜓_{2𝑘−1𝑀}(𝑥)]^𝑡. (2.10) Similarly, by considering 𝑖=𝑀(𝑛−1)+𝑚+1 and 𝑗=𝑀(𝑛^′−1)+𝑚^′+1, we approximate 𝐾(𝑥,𝑦)∈𝐿²_𝑤([0,1]×[0,1]) as𝐾(𝑥,𝑦)≃^{2𝑘−1𝑀2}_𝑖=1^𝑘−1𝑀_𝑗=1𝐾_𝑖𝑗𝜓_𝑖(𝑥)𝜓_𝑗(𝑦), (2.11) or in the matrix form𝐾(𝑥,𝑦)≃Ψ^𝑡(𝑥)𝐊Ψ(𝑦), (2.12) where 𝐊=[𝐾_𝑖𝑗]_{1≤𝑖,𝑗≤2}^𝑘−1_𝑀 with the entries𝐾_𝑖𝑗=⟨𝜓_𝑖(𝑥),⟨𝐾(𝑥,𝑦),𝜓_𝑗(𝑦)⟩_𝑤𝑘⟩_𝑤𝑘. (2.13)

3. 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 𝑓(𝑥).

Proof. Let 𝑔(𝑥)=^∞∞_𝑛=1𝑚=0𝑐_𝑛𝑚𝜓_𝑛𝑚(𝑥), (3.1) where 𝑐_𝑛𝑚=⟨𝑓(𝑥),𝜓_𝑛𝑚(𝑥)⟩_𝑤𝑘. 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 ¹₀𝑔(𝑥)𝜓_𝑝𝑞(𝑥)𝑤_𝑘(𝑥)d𝑥=¹₀^∞∞_𝑛=1𝑚=0𝑐_𝑛𝑚𝜓_𝑛𝑚(𝑥)𝜓_𝑝𝑞(𝑥)𝑤_𝑘=(𝑥)d𝑥_{∞∞𝑛=1𝑚=0}𝑐_𝑛𝑚¹₀𝜓_𝑛𝑚(𝑥)𝜓_𝑝𝑞(𝑥)𝑤_𝑘(𝑥)d𝑥=𝑐_𝑝𝑞. (3.2) Thus ⟨𝑔(𝑥),𝜓_𝑛𝑚(𝑥)⟩_𝑤𝑘=𝑐_𝑛𝑚 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 function 𝑓(𝑥)∈𝐿²_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, 𝑓(𝑥)=^∞∞_𝑛=1𝑚=0𝑐_𝑛𝑚𝜓_𝑛𝑚(𝑥). (3.3)

Proof. From (2.8) it follows that 𝑐_𝑛𝑚=¹₀𝑓(𝑥)𝜓_𝑛𝑚(𝑥)𝑤_𝑘(𝑥)d𝑥=^{𝑛/2𝑘−1}_{(𝑛−1)/2𝑘−1}2^𝑘/2𝑓(𝑥)𝑇_𝑚(2^𝑘𝑥−2𝑛+1)𝑤(2^𝑘𝑥−2𝑛+1)d𝑥. (3.4) If 𝑚>1, by substituting 2^𝑘𝑥−2𝑛+1=cos𝜃 in (3.4), it yields 𝑐_𝑛𝑚=12^𝑘/2𝜋₀𝑓(cos𝜃+2𝑛−12^𝑘)2𝜋=√cos𝑚𝜃d𝜃22^𝑘/2√𝜋𝑓(cos𝜃+2𝑛−12^𝑘)(sin𝑚𝜃𝑚)]^𝜋₀+√22^3𝑘/2𝑚√𝜋^𝜋₀𝑓^′(cos𝜃+2𝑛−12^𝑘=1)sin𝑚𝜃sin𝜃d𝜃2^3𝑘/2𝑚√𝑓2𝜋^′(cos𝜃+2𝑛−12^𝑘)(sin(𝑚−1)𝜃−𝑚−1sin(𝑚+1)𝜃𝑚+1)]^𝜋₀+12^5𝑘/2𝑚√2𝜋^𝜋0𝑓^′′(cos𝜃+2𝑛−12^𝑘)ℎ_𝑚(𝜃)d𝜃, (3.5)(3.6)(3.7) where ℎ_𝑚(𝜃)=sin𝜃(sin(𝑚−1)𝜃−𝑚−1sin(𝑚+1)𝜃𝑚+1). (3.8) Thus, we get |𝑐_𝑛𝑚1|=|2^5𝑘/2𝑚√2𝜋^𝜋0𝑓^′′(cos𝜃+2𝑛−12^𝑘)ℎ_𝑚1(𝜃)d𝜃|≤(2^5𝑘/2𝑚√)2𝜋^𝜋0|𝑓^′′(cos𝜃+2𝑛−12^𝑘)ℎ_𝑚𝑁(𝜃)|d𝜃≤(2^5𝑘/2𝑚√)2𝜋^𝜋0|ℎ_𝑚(𝜃)|d𝜃. (3.9) However ^𝜋₀|ℎ_𝑚(𝜃)|d𝜃=^𝜋₀|sin𝜃(sin(𝑚−1)𝜃−𝑚−1sin(𝑚+1)𝜃𝑚+1)|d𝜃≤^𝜋₀|sin𝜃sin(𝑚−1)𝜃𝑚−1|+|sin𝜃sin(𝑚+1)𝜃≤𝑚+1|d𝜃2𝑚𝜋(𝑚².−1) (3.10) Since 𝑛≤2^𝑘−1, we obtain |𝑐_𝑛𝑚√|<2𝜋𝑁(2𝑛)^5/2(𝑚².−1) (3.11) Now, if 𝑚=1, by using (3.6), we have |𝑐_𝑛1√|<2𝜋(2𝑛)^3/2max_0≤𝑥≤1|𝑓^′(𝑥)|. (3.12) Hence, the series ∑^∞_𝑛=1∑^∞_𝑚=1𝑐_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=1𝑐_𝑛0𝜓_𝑛0(𝑥) is convergence [22]. On the other hand, we have |^∞∞_𝑛=1𝑚=0𝑐_𝑛𝑚𝜓_𝑛𝑚(𝑥)|≤|^∞_𝑛=1𝑐_𝑛0𝜓_𝑛0(𝑥)|+^∞∞_𝑛=1𝑚=1|𝑐_𝑛𝑚||𝜓_𝑛𝑚(𝑥)|≤|^∞_𝑛=1𝑐_𝑛0𝜓_𝑛0(𝑥)|+^∞∞_𝑛=1𝑚=1|𝑐_𝑛𝑚|<∞. (3.13) Therefore, utilizing Lemma 3.1, the series ∑^∞_𝑛=1∑^∞_𝑚=0𝑐_𝑛𝑚𝜓_𝑛𝑚(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:𝑓(𝑥)≃𝐹^𝑡Ψ(𝑥),𝑢(𝑦)≃𝑈^𝑡Ψ(𝑦),𝐾(𝑥,𝑦)≃Ψ^𝑡(𝑥)𝐊Ψ(𝑦). (4.1) By substituting (4.1) into (1.1), we obtainΨ^𝑡(𝑥)𝐊(¹₀Ψ(𝑦)Ψ^𝑡(𝑦)d𝑦)𝑈−Ψ^𝑡(𝑥)𝐹=𝑅₂^𝑘−1_𝑀(𝑥), (4.2) where 𝑅₂^𝑘−1_𝑀(𝑥) is the residual. By letting¹₀Ψ(𝑦)Ψ^𝑡(𝑦)d𝑦=𝐋, (4.3) where 𝐋 is a 2^𝑘−1𝑀×2^𝑘−1𝑀 matrix which is computed next, we haveΨ^𝑡(𝑥)𝐊𝐋𝑈−Ψ^𝑡(𝑥)𝐹=𝑅₂^𝑘−1_𝑀(𝑥). (4.4) Our aim is to compute 𝑢₁,𝑢₂,…,𝑢₂^𝑘−1_𝑀 such that 𝑅₂^𝑘−1_𝑀(𝑥)≡0, but in general, it is not possible to choose such 𝑢_𝑖,𝑖=1,2,…,2^𝑘−1𝑀. In this work, 𝑅₂^𝑘−1_𝑀(𝑥) is made as small as possible such that⟨𝜓_𝑛𝑚(𝑥),𝑅₂^𝑘−1_𝑀(𝑥)⟩_𝑤𝑘=0, (4.5) 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:𝐊𝐋𝑈=𝐹, (4.6) for unknowns 𝑈=[𝑢₁,𝑢₂,…,𝑢₂^𝑘−1_𝑀].

Here, we define two operator equations 𝒦 and ℋ as follows:𝒦(𝑢(𝑥))=¹₀𝐾(𝑥,𝑦)𝑢(𝑦)d𝑦,ℋ(𝑢(𝑥))=¹₀Ψ^𝑡(𝑥)𝐊Ψ(𝑦)𝑢(𝑦)d𝑦, (4.7)(4.8) for all 𝑢∈𝐿²_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 obtain𝒦𝑢=𝑓,ℋ𝑇_𝑘,𝑀(𝑢)=𝑇_𝑘,𝑀(𝑓). (4.9) Combining the latter equations yields𝒦𝑒_𝑘,𝑀=(𝒦−ℋ)𝑇_𝑘,𝑀(𝑢)+(𝑓−𝑇_𝑘,𝑀(𝑓)), (4.10) where 𝑒_𝑘,𝑀=𝑢−𝑇_𝑘,𝑀(𝑢). Provided that 𝒦⁻¹ exists, we obtain the error bound:‖𝑒_𝑘𝑀‖≤‖𝒦⁻¹‖‖(𝒦−ℋ)𝑇_𝑘𝑀(𝑢)+(𝑓−𝑇_𝑘𝑀(𝑓))‖. (4.11) 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 ‖𝑓(𝑥)−𝑇_𝑘,𝑀(𝑓(𝑥))‖₂=(¹₀[𝑓(𝑥)−𝑇_𝑘,𝑀(𝑓(𝑥))]²d𝑥)^1/2≤(^2𝑘−1_𝑛=1^{𝑛/2𝑘−1}_{(𝑛−1)/2𝑘−1}¹₀[𝑓(𝑥)−𝐶^𝑛_𝑘,𝑀(𝑓(𝑥))]²d𝑥)^1/2≤2^−𝑘𝑀𝑄, (4.12) where 𝑄=(2/2^𝑀𝑀!)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 𝑀 on [(𝑛−1)/2^𝑘−1,𝑛/2^𝑘−1]. Therefore, if we want to have ‖𝑓(𝑥)−𝑇_𝑘,𝑀(𝑓(𝑥))‖₂<𝜀, we can choose 𝑘 as 𝑄𝑘=[𝜀ln(2)𝑀]+1. (4.13)

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 𝐿_𝑖𝑗=¹₀𝜓_𝑖(𝑦)𝜓_𝑗(𝑦)d𝑦. (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^𝑘𝑥−2𝑛+1=cos𝜃 in (4.14), we obtain 𝐿_𝑖𝑗=𝐶_𝑚𝑚′^𝜋₀cos𝑚𝜃cos𝑚^′𝜃sin𝜃d𝜃, (4.15) where 𝐶_𝑚𝑚′=−1𝜋,𝑚=𝑚^′=0,−2𝜋,𝑚≠0≠𝑚^′,−√2𝜋,otherwise. (4.16) Now, if |𝑚+𝑚^′|=1, then ^𝜋₀cos𝑚𝜃cos𝑚^′𝜃sin𝜃d𝜃=0, (4.17) implies that 𝐿_𝑖𝑗=0, and if |𝑚+𝑚^′|≠1, then 𝐿_𝑖𝑗=𝐶_𝑚𝑚′4×(−cos(𝑚−𝑚^′+1)𝜃𝑚−𝑚^′++1cos(−𝑚+𝑚^′+1)𝜃−𝑚+𝑚^′−+1cos(𝑚+𝑚^′−1)𝜃𝑚+𝑚^′+−1cos(−𝑚−𝑚^′+1)𝜃−𝑚−𝑚^′+1)]^𝜋₀. (4.18) Consequently, L has the following form: 𝐋=diag(𝐴,𝐴,…,𝐴_{2𝑘−1times}), (4.19) where 𝐴=[𝐴_𝑚𝑚′] is an 𝑀×𝑀 matrix with the elements 𝐴_𝑚𝑚′=𝐶_𝑚𝑚′2(𝑚²+𝑚^′2−1)1+𝑚⁴−2𝑚^′2𝑚²−2𝑚^′2−2𝑚²+𝑚^′4,𝑚+𝑚^′iseven,0,𝑚+𝑚^′isodd. (4.20)

5. 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.

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 𝜕⁴𝐾(𝑥,𝑦)/𝜕𝑥²𝜕𝑦² bounded by 𝑁 and 𝑚,𝑚^′>1, then one has |𝐾_𝑖𝑗|<𝜋𝑁2⁴(𝑛𝑛^′)^5/2(𝑚²−1)(𝑚^′2.−1) (5.1)

Proof. From (2.13), we obtain |𝐾_𝑖𝑗|=2^𝑘|^{𝑛/2𝑘−1}_{(𝑛−1)/2𝑘−1}^{𝑛′/2𝑘−1}_{(𝑛′−1)/2𝑘−1}𝐾(𝑥,𝑦)𝑇_𝑚(2^𝑘𝑥−2𝑛+1)𝑤(2^𝑘×𝑇𝑥−2𝑛+1)_𝑚′(2^𝑘𝑦−2𝑛^′+1)𝑤(2^𝑘𝑦−2𝑛^′+1)d𝑦d𝑥|. (5.2) Now, let 2^𝑘𝑥−2𝑛+1=cos𝜃 and 2^𝑘𝑦−2𝑛^′+1=cos𝛼; then |𝐾_𝑖𝑗2|=2^𝑘𝜋|^𝜋₀𝐾(cos𝜃+2𝑛−12^𝑘,cos𝛼+2𝑛^′−12^𝑘)cos𝑚𝜃cos𝑚^′𝛼d𝛼d𝜃|. (5.3) Similar to the proof of Theorem 3.2, since 𝑚,𝑚^′>1, we obtain |𝐾_𝑖𝑗1|≤2^5𝑘+1𝜋𝑚𝑚^′|^𝜋₀𝜕⁴𝐾((cos𝜃+2𝑛−1)/2^𝑘,(cos𝛼+2𝑛^′−1)/2^𝑘)𝜕𝑡²𝜕𝑠²ℎ_𝑚(𝜃)ℎ_𝑚′≤𝑁(𝛼)d𝛼d𝜃|2^5𝑘+1𝜋𝑚𝑚^′𝜋₀|ℎ_𝑚(𝜃)|d𝜃^𝜋₀|ℎ_𝑚′<(𝛼)|d𝛼𝜋𝑁2⁴(𝑛𝑛^′)^5/2(𝑚²−1)(𝑚^′2.−1) (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 𝜀₀ and get the following system of linear equations whose matrix is sparse: 𝐊𝐋𝑈=𝐹, (5.5) where 𝐊=[𝐾_𝑖𝑗]₂^𝑘−1_𝑀×2^𝑘−1_𝑀 with the entries 𝐾_𝑖𝑗=𝐾_𝑖𝑗,|𝐾_𝑖𝑗|≥𝜀₀,0,otherwise. (5.6) 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 ¹₀sin(𝑥𝑦)𝑢(𝑦)d𝑦=sin(𝑥)−𝑥cos(𝑥)𝑥²,0≤𝑥≤1, (6.1) with the exact solution 𝑢_𝑒𝑥(𝑥)=𝑥 [15].
Table 1 shows the numerical results for this example with 𝑘=2,𝑀=2,𝜀₀=10⁻⁵ and 𝑘=2,𝑀=4,𝜀₀=10⁻⁴. Also, the approximate solution for 𝑘=2, 𝑀=4, 𝜀₀=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 Example 6.1.

Figure 1 

Approximate solution for Example 6.1 with $k=2,M=4,{\varepsilon }_0=10^{-4}.$

Example 6.2. In this example we solve integral equation 𝑢(𝑥)−¹₀e^𝑥𝑦e𝑢(𝑦)d𝑦=^𝑥+1−1𝑥+1,0≤𝑥≤1, (6.2) 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,𝜀₀=10⁻⁵ and 𝑘=3,𝑀=4,𝜀₀=10⁻⁴ where 𝑢 denote the approximation of 𝑢_𝑒𝑥. The approximate solution for 𝑘=3,𝑀=4,𝜀₀=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.

Figure 2 

Approximate solution for Example 6.2 with $k=3,M=4,{\varepsilon }_0=10^{-4}.$

Example 6.3. As our final example let ¹₀(𝑦−𝑥)²1+𝑦²𝑢(𝑦)d𝑦=0.179171−0.532108𝑥+0.487495𝑥²,0≤𝑥≤1, (6.3) 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 𝑢(𝑥) of 𝑢(𝑥): ‖𝑢−𝑢‖_∞=max{|𝑢(𝑥)−𝑢(𝑥)|,0≤𝑥≤1},‖𝑢−𝑢‖₂=(¹₀|𝑢(𝑥)−𝑢(𝑥)|²d𝑥)^1/2. (6.4) Also, the error 𝑒(𝑥)=𝑢(𝑥)−𝑢(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 Example 6.3.

Figure 3 

Error distributions for Example 6.3 with $k=2\text{,}$$M=3$ and $k=3\text{,}$$M=4$ on $[0,1/2].$

Figure 4 

Error distributions for Example 6.3 with $k=2\text{,}$$M=3$ and $k=3\text{,}$$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 parameter 𝑘 is recommended. The convergence accuracy of this method was examined for several numerical examples.