Abstract
A novel and efficient numerical method is developed based on interpolating scaling functions to solve 2D Fredholm integral equations (FIE). Using the operational matrix of integral for interpolating scaling functions, FIE reduces to a set of algebraic equations that one can obtain an approximate solution by solving this system. The convergence analysis is investigated, and some numerical experiments confirm the accuracy and validity of the method. To show the ability of the proposed method, we compare it with others.
1. Introduction
In this work, we focus on developing the wavelet Galerkin scheme for 2D Fredholm integral equations (2D-FIE), given bywhere the functions and are continuous functions on and with , respectively, as prescribed before. Function is given and can be linear or nonlinear. The unknown is sought.
Most of the mathematical models for physical phenomena use integral equations. Consequently, the application of integral equations is a momentous subject within applied mathematics. Also, this equation plays a significant role in reformulations of other mathematical problems. Principally, the Fredholm integral equation can be obtained from converting boundary value problems [1]. Equation (1) is an effective tool to model the problems that come from electromagnetic scattering, computer graphics, and aerodynamics [2, 3].
Many papers studied the Fredholm integral equations numerically. Among them, we can mention an effective numerical method based on block-pulse functions [3]. Effati et al. [4] utilized a neural network scheme for solving this type of integral equation. A sparse representation based on multiwavelets has been developed and utilized an efficient algorithm for solving the system of FIE [5]. In [2], numerous numerical methods are presented and investigated for solving the numerical solution of FIE. For some related works, we refer the readers to [6–8].
There are so many papers on developing and analyzing the numerical methods for solving two-dimensional FIE. In [9], a novel algorithm is developed to solve the problem using the integral mean value theorem. Kazemi et al. [10] proposed the iterative method based on quadrature formula to solve 2D-FEI. A numerical method based on developing the two-dimensional triangular orthogonal functions is applied in [11]. Aziz et al. [12] used Haar wavelet method to solve problem (1). The principal advantage of the method is that, unlike others, it does not apply numerical integration. In [13], an Euler-type method is proposed to solve considering problems. Singh et al. [14] proposed a numerical method based on Legendre scaling functions to solve the multidimensional Fredholm integral equations. In [15], an accurate scheme is developed for solving 2D Fredholm integral equation by using the cosine-trigonometric functions. In [16], an iterative algorithm based on sparse least-squares is used to solve the problem. For some related works, we refer the readers to [17–20].
Multiwavelets are found an interesting basis for solving a variety of equations [21–23]. Multiwavelets have some properties of wavelets, such as orthogonality, vanishing moments, and compact support. In contrast to the wavelets and biorthogonal wavelets, they can have high smoothness and high approximate order coupled with short support [24]. Contrary to biorthogonal wavelets, multiwavelets can have the high vanishing moments without enlarging their support [25]. In general, the multiwavelets are a very powerful tool for expressing a variety of operators. At the present work, we apply the interpolating scaling functions constructed in [6, 26].
An outline of the remaining part of the paper is as follows. In Section 2, we devote the properties of interpolating scaling functions and related projections. In Section 3, we define the wavelet Galerkin method and investigate the convergence analysis. In Section 4, we solve some numerical examples to confirm the accuracy and efficiency of the scheme and show its ability comparing with other methods. Section 5 contains a few concluding remarks.
2. Interpolating Scaling Functions
Let . We consider the uniform finite discretizations , where the subintervals are determined by the point , with . For , we introduce the subspace as a space of piecewise polynomial bases of degree less than multiplicity parameter that is spanned bywhere and are the translation and dilation operators, respectively, and are the primal interpolating scaling bases introduced by Alpert et al. [26]. Given nodes , which are the roots of Legendre polynomial of degree , the interpolating scaling bases are defined aswhere are the Lagrange interpolating polynomials at the point and are the Gauss-Legendre quadrature weights [22, 26]. These bases form orthonormal bases on with respect to the -inner product. According to the definition of the space , the spaces have dimension and obviously are nested
Every function can be represented in the formwhere is the orthogonal projection that maps onto the subspace . To find the coefficients that are determined by , we shall compute the integrals. We apply the -point Gauss-Legendre quadrature by a suitable choice of the weights and nodes for to avoid these integrals [5, 26] via
Convergence analysis of the projection is investigated for the -times continuously differentiable function [26].
For the full proof of this approximation and further details, we refer the readers to [6]. Thus, we can conclude that converges to with rate of convergence .
Assume that the vector function with includes the scaling functions and called multiscaling function. Approximation (5) may be rewritten using the vector that includes entries as follows:where is an -dimensional vector. The building blocks of these bases construction can be applied to approximate a higher-dimensional function. To this end, one can introduce the two-dimensional subspace that is spanned by
Thus, by this assumption, to derive an approximation of the function by the projection operator , we getwhere components of the square matrix of order are obtained bywhere . Consider the -th partial derivatives of are continuous. Utilizing this assumption, the error of this approximation can be bounded as follows:where is a constant [27, 28].
3. Wavelet Galerkin Scheme (WGS)
For the completeness of this paper, we describe the wavelet Galerkin scheme for two-dimensional Fredholm integral equation (1). In the operator form, equation (1) can be written aswhere is the identity operator and the Fredholm operatoris assumed to be compact on .
Suppose that the unique solution of (1) can be approximated as an expansion of the interpolating scaling functions viawhere is an matrix whose elements must be found and the projection operator maps on the subspace . Also, we can write the same expansion for function . These are substituted into (13), and the coefficients are determined by using the Galerkin method. For later use, introduce the residual aswhere , , and . Due to the Galerkin method, it is necessitous that , for . To derive the approximate solution, we solve the system of algebraic equations
This system can be linear or nonlinear and it depends on the function . The critical step of this method is how to approximate . For this purpose, assume thatwhere is an matrix whose entries are obtained by (11). Substituting (18) into (14), one can writewhere is an matrix and is the operational matrix of integral for interpolating scaling functions introduced in [22]. Due to this approximation, system (17) may be written aswhere .
3.1. Convergence Analysis
Theorem 1. Assume that is a compact operator, is injective, and the sequence is pointwise convergent to and collectively compact. Then, exists and is uniformly bounded and the solution of (13) satisfies the error estimate
Proof. Motivated by the hypothesis of the theorem, one can conclude thatUsing (22), for all sufficiently large , we can writeand as a consequence of this, is reversible.
Pick such thatDue to geometric series theorem, exists and is uniformly bounded, i.e.,The investigation of convergence analysis is founded by the approximation of by ,Taking norm of (26) and using (25), one can obtainApplying the operator to both sides of (13) and then rearrange to getSubtracting (28) of the original equation (13), we haveTaking norm and using (27),It is straightforward to show that as . Assume that is a sequence of continuous functions so that as . Since the orthonormal projection satisfies , we can obtainThus, , there exists a number so that, for any , one can write . This then implies thatThis implies that , for sufficiently large value of and because is arbitrary, consequently, as .
4. Numerical Examples
To illustrate the accuracy and efficiency of the scheme, some equations of form (1) solve by the proposed method. In this section, the maximum absolute value error,and -error are used to show the accuracy of the proposed method. All numerical computations are carried out simultaneously using Maple and MATLAB software.
Example 1. Let us consider the following 2D linear Fredholm integral equation:The exact solution is [9, 11].
The proposed method is compared with integral mean value theorem [9] and 2D-TFs method [11] in Table 1. The results in Table 1 show that the proposed method has better accuracy than other methods. Figure 1 shows the surfaces of the approximate solution and the absolute error when and . In Figure 2, we show the effect of the parameters and on -error. It is obvious that increasing the parameters and reduces the error. The order of convergence is presented numerically for Example 1 in Table 2. As we expected, the order of convergence must approach multiplicity .
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Example 2. Let us now consider the following 2D linear FIE:The exact solution is given in [9].
Table 3 illustrates a comparison among the numerical results of the proposed method, integral mean value theorem [9], and 2D-TFs method [11]. The results show that the proposed method is better than others. Figure 3 shows the surfaces of the approximate solution and the absolute error taking and . Table 4 shows the effects of the multiplicity parameter and refinement level for Example 2. The order of convergence is presented numerically for Example 2 in Table 5.
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Example 3. Let us consider the nonlinear 2D-FIE given in [10]:The exact solution is given in [10].
In Table 6, we compare the proposed method and quadrature formula proposed in [10]. It shows that the proposed method is much more flexible than the method presented in [10] and offers better accuracy using fewer computations. Figure 4 shows the plots of the approximate solution and the absolute error when and . Figure 5 shows the effects of the multiplicity parameter and refinement level for Example 3. The order of convergence is presented numerically for Example 3 in Table 7.
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Example 4. Consider the following nonlinear 2D-FIE given in [12, 18]:The exact solution is reported in [12] viaTable 8 illustrates a comparison among the numerical results of the proposed method, Nystrom method [18], and Haar wavelet method [12]. The approximate solution and the absolute error are plotted in Figure 6 when and .
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5. Conclusion
Interpolating scaling functions are applied to obtain the approximate solution of 2D Fredholm integral equations. The convergence analysis is investigated and the numerical results are compared with existing methods. It shows that the proposed method is much more flexible than others and the results show that the proposed method has better accuracy than other methods. To increase the accuracy, it is not necessary to increase the degree of polynomials (unlike bases such as Chebyshev and Legendre). By increasing the level of refinement , we can increase the accuracy. Also, these bases have the interpolation property and this helps us to avoid the direct integral for finding the coefficients and this reduces the computational cost.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares no known personal relationships or conflicts of interest that could have appeared to affect the work reported in this article.
Acknowledgments
This project was supported by Researchers Supporting Project no. RSP-2020/210 and King Saud University, Riyadh, Saudi Arabia.