Numerical Investigation of the Fredholm Integral Equations with Oscillatory Kernels Based on Compactly Supported Radial Basis Functions

Abstract

The integral equations with oscillatory kernels are of great concern in applied sciences and computational engineering, particularly for large-scale data points and high frequencies. Therefore, the interest of this work is to develop an accurate, efficient, and stable algorithm for the computation of the Fredholm integral equations (FIEs) with the oscillatory kernel. The oscillatory part of the FIEs is evaluated by the Levin quadrature coupled with a compactly supported radial basis function (CS-RBF). The algorithm exhibits sparse and well-conditioned matrix even for large-scale data points, as compared to its counterpart, multi-quadric radial basis function (MQ-RBF) coupled with the Levin quadrature. Usually, the RBFs behave with spherical symmetry about the centers, known as radial. The comparison of convergence and stability analysis of both types of RBFs are performed and numerically verified. The proposed algorithm is tested with benchmark problems and compared with the counterpart methods in the literature. It is concluded that the algorithm in this work is accurate, robust, and stable than the existing methods in the literature based on MQ-RBF and the Chebyshev interpolation matrix.

1. Introduction

In natural sciences, the theory of integrals is much demanding for the practical computation of the volumes of rotating objects, areas of shapes, distance among objects, analysis of telecommunication signals, study of earthquakes, etc. An interesting application of integrals involves boundaries with symmetric functions. The concept of symmetry to the Fredholm integral equations and Volterra integral equations along with applications can be found in detail [1,2,3]. Many problems in computational biology, communications, mechanics, and the ecosystem are modeled by the symmetric integrals [3].

The integral equations with oscillatory kernels have been a concerning research problem in the literature as per their applications in engineering and applied sciences, particularly in acoustic scattering, heat conduction problem, electromagnetics, diffusion problems, mechanical problems, pipe flow, laser engineering, and quantum mechanics [4,5,6,7,8]. Therefore, this paper consider the Fredholm integral equations (FIEs) of the type

$$v\left(t\right)=u\left(t\right)+{\int }_c^{d}K(t,\tau )v\left(\tau \right)\mathrm{d}\tau ,$$

(1)

where $v\left(t\right)$ is the unknown function, and $K(t,\tau )$ represents kernel function. In the oscillation form, the above equation can be represented as

$$v\left(t\right)=u\left(t\right)+{\int }_c^{d}f(t,\tau )e^{i\omega g(t,\tau )}v\left(\tau \right)\mathrm{d}\tau ,$$

(2)

for $K(t,\tau )=f(t,\tau )e^{i\omega g(t,\tau )}$, which is the oscillatory part of the kernel function, and $\omega \gg 1$. The complete analysis of the integral Equation (2) is based on the proper computation of the oscillatory integral. Therefore, the existing literature on the highly oscillatory integrals would be of great concern. The accurate, efficient, and stable computation of the highly oscillatory integrals (HOIs) is much demanding in engineering and applied sciences. It is difficult to compute the highly oscillatory integrals as demanded in the applied sciences with traditional quadratures, such as the Simpson rule and Gaussian quadrature, particularly for large-frequency $\omega$. Therefore, substantial efforts have been carried out towards the better computation of highly oscillatory integrals. There are several methods in the literature, for this purpose, including the Filon-type method [9,10,11,12], Levin-type method [4,5,13,14,15,16], asymptotic method [17], and steepest descent method [18].

In the literature, the computation of FIEs with oscillatory kernels is performed in [6,7,8]. In [8], the author considered the asymptotic properties of the FIEs with the oscillatory kernel $f(t,\tau )e^{i\omega |t-\tau |}$. In [6], the authors considered the analysis of the spectral problem with oscillatory Fredholm integral operators. Moreover, in [19], the authors implemented a collocation method while using the Clenshaw–Curtis points for the solution of FIEs having oscillatory kernels; the oscillatory part of the integral equation was evaluated by the efficient Filon method. The authors in [20] constructed structure oscillatory spaces which capture the oscillatory components of the FIEs having oscillatory kernels. The authors implemented the Galerkin method for solving the FIEs and obtained an optimal convergence rate with the claim of stability. In [21], the authors developed a spectral Levin method for computing the generalized Fourier transforms. The barycentric weights of the Hermite interplant are evaluated by the method which produces differential matrices. In [22], the authors applied the Clenshaw–Curtis algorithm to the Jacobi weights for functions having endpoint singularities. The theoretical results were verified with several numerical experiments. Furthermore, in [23], the authors applied different splitting algorithms for computing Cauchy-type singular integrals. Error analysis of the algorithms were demonstrated and verified with several numerical results. Further, in [24], the authors presented a sparse and accurate Levin method based on Chebyshev operators for HOIs and extended the applicability of the method to Volterra integral equations (VIEs) with oscillatory kernels. Later on, the work [25] observed several facts about the method [24], which usually produced ill-conditioned and dense coefficient matrices. To cover this gap, the author applied the Jacobi spectral Levin method for highly oscillatory integrals to highly oscillatory integrals containing end-point singularities and integral equations with oscillatory kernels and weakly singular kernels [25].

In various scientific disciplines, HOIs exhibit many practical phenomena and are the core subject of interest for different computational algorithms. The modified Fourier series utilizes HOIs for a function approximation scheme, resulting a faster convergence than the Fourier series [26]. In acoustics, the problem is modeled by the oscillatory integrals for computing the oscillatory integral equations with boundary element method (BEM) [27]. A modified version of the BEM is available in the literature, known as the dual interpolation boundary face method (DiBFM), exhibits superior behavior in accuracy and computational time. The DiBFM has been applied to potential [28], elasticity problems [29], and the Poisson equation [30,31,32].

In [33], the authors applied a modified Levin method based on the Chebyshev differentiation matrix for the numerical analysis of highly oscillatory integrals. The authors used TSVD in the case of an ill-conditioned system. The method in [33] was extended to FIEs with oscillatory kernels by the authors in [7]. Further, the authors in [34] applied the Filon–Clenshaw–Curtis method for the analysis of oscillatory integrals along with Sommariva’s results, and a modified Chebyshev collocation method was presented for computing FIEs with oscillatory kernels. The authors claimed a lower computational complexity and accurate behavior of the method. The numerical results in [34] have advantages over the results in [7], but the method in [34] is limited to a certain class of functions, and it is difficult to compute the modified moments for a complex phase function. Keeping a few observations on the methods in [7], the authors in [35] applied the Levin method based on MQ-RBFs and obtained accurate and efficient results for computing FIEs with oscillatory kernels.

The RBFs used in [35] are global basis functions, which exhibit ill-conditioned matrix. As a result, the analysis of the FIEs with oscillatory kernels performed by the method in [35] causes computational and stability issues. The methods in [7,35] are suitable for computing the FIEs with oscillatory kernels for small data points and low frequencies. As the data points or frequency increases, the methods in [7,35] confront the computational and stability issues to compute FIEs with oscillatory kernels. For this purpose, in this paper, we apply the local basis functions known as compactly supported radial basis functions (CS-RBFs), which exhibit well-conditioned behavior than the global RBFs. The Levin method is coupled with the CS-RBFs to compute the FIEs with oscillatory kernels. The method behaves well in accuracy, efficiency, and stability, even for large-scale data points and high frequency. The numerical results are included to verify the performance of the proposed method.

The rest of the paper is categorized as follows. A detailed description of the Levin method with the CS-RBFs is outlined in Section 2. The stability and error estimates of the global RBFs and local RBFs are described in Section 3. The validation of the proposed method is performed by solving benchmark problems in Section 4, and compared with the competitive methods [7,35]. Finally, a few remarks about the conclusions are given in Section 5.

2. Evaluation of the Integral Equation with Levin Method

To evaluate the integral Equation (2), first we have to discretize it and write as follows:

$$v\left(t_i\right)=u\left(t_i\right)+{\int }_c^{d}f(t_i,\tau )e^{i\omega g(t_i,\tau )}v\left(\tau \right)\mathrm{d}\tau ,i=0,1,\dots ,n.$$

(3)

The main hurdle in the integral Equation (3) is the computation of the oscillatory integral

$$I_i={\int }_c^{d}f(t_i,\tau )e^{i\omega g(t_i,\tau )}v\left(\tau \right)\mathrm{d}\tau ,i=0,1,\dots ,n.$$

(4)

To compute the integral $I_i$, we applied the modified Levin method [4,25]. Based upon the idea of the Levin method, we have to find an approximate polynomial $\psi \left(\tau \right)$ that satisfies the following ODE:

$${\psi }^{{}^{\prime }}\left(\tau \right)+i\omega g^{{}^{\prime }}(t_i,\tau )\psi \left(\tau \right)=f(t_i,\tau )v\left(\tau \right).$$

(5)

First, to find the unknown function $\psi \left(\tau \right)$ from (5), and applying the well-known Levin procedure, we get

$$I_i=\left[\psi \left(d\right)e^{i\omega g(t_i,d)}-\psi \left(c\right)e^{i\omega g(t_i,c)}\right]\mathbf{V}.$$

(6)

The ODE (5) in the form of differentiation matrix $\mathbf{D}$ (see Section 2.1) can be represented as

$$\mathbf{D}\psi \left(\tau \right)+i\omega g^{{}^{\prime }}(t_i,\tau )\psi \left(\tau \right)=f(t_i,\tau )v\left(\tau \right),i=0,1,\dots ,n,$$

(7)

which produces the following system of linear equations:

$$(\mathbf{D}+i\omega {\mathbf{Y}}_i){\mathsf{\Psi }}_i=diag\left({\mathbf{F}}_i\right)\mathbf{V},i=0,1,\dots ,n,$$

(8)

• here ${\mathsf{\Psi }}_i={\left[\begin{array}{c}\psi \left({\tau }_o\right),\psi \left({\tau }_1\right),\dots ,\psi \left({\tau }_n\right)\end{array}\right]}^T$,
• $\mathbf{V}={\left[\begin{array}{c}v\left({\tau }_o\right),v\left({\tau }_1\right),\dots ,v\left({\tau }_n\right)\end{array}\right]}^{\mathbf{T}}$,
• ${\mathbf{F}}_i={\left[\begin{array}{c}f(t_i,{\tau }_o),f(t_i,{\tau }_1),\dots ,f(t_i,{\tau }_n)\end{array}\right]}^T$,
• ${\mathbf{Y}}_i$ represents a diagonal matrix with the following entries: $g^{{}^{\prime }}(t_i,{\tau }_o),\dots ,g^{{}^{\prime }}(t_i,{\tau }_n)$, and $diag\left({\mathbf{F}}_i\right)\mathbf{V}={\mathbf{F}}_i\ast \mathbf{V}$ (where ’*’ represents the Hadamard product) is a vector of entries $f(t_i,{\tau }_j)v\left(j\right),j=0,1,\dots ,n.$

Therefore, Equation (8) takes the form

$${\mathsf{\Psi }}_i={(\mathbf{D}+i\omega {\mathbf{Y}}_i)}^{-1}diag\left({\mathbf{F}}_i\right)\mathbf{V}.$$

(9)

The solution of Equation (6) is based on the first and last value of the solution vector, for which we have a vector ${\mathbf{P}}_i=\left[\begin{array}{c}{e}^{i\omega g(t_i,d)},0,\dots ,0,-e^{i\omega g(t_i,c)}\end{array}\right]$.

Consequently, the integral (4) along with (9) can be written as

$$I_i={\mathbf{P}}_i{\mathsf{\Psi }}_i={\mathbf{P}}_i{(\mathbf{D}+i\omega {\mathbf{Y}}_i)}^{-1}diag\left({\mathbf{F}}_i\right)\mathbf{V},$$

(10)

or

$$I_i={\mathbf{W}}_i\mathbf{V},$$

(11)

where ${\mathbf{W}}_i={\mathbf{P}}_i{(\mathbf{D}+i\omega {\mathbf{Y}}_i)}^{-1}diag\left({\mathbf{F}}_i\right)$ and vector $\mathbf{V}$ is of the order $(n+1)\times 1$. Finally, the FIE (2) can be written as

$$v\left(t_i\right)=u\left(t_i\right)+I_i,$$

(12)

or

$$\mathbf{V}=\mathbf{U}+\mathbf{WV},$$

(13)

where the vector $\mathbf{U}$ is of the order $(n+1)\times 1$, and the matrix $\mathbf{W}$ is of order $(n+1)\times (n+1)$. Consequently, the solution of the FIE can be written as

$$\mathbf{V}={(\mathbf{I}-\mathbf{W})}^{-\mathbf{1}}\mathbf{U}.$$

(14)

As in the papers [7,35], the solution vector $\mathbf{V}$ was obtained with LU-factorization for the well-conditioned system of equations, and truncated singular value decomposition (TSVD) was applied for the ill-conditioned system of equations. In our work, as the system of equations is always stable and there is no risk of being ill-conditioned, we have applied LU-factorization for the solution vector $\mathbf{V}$.

2.1. Differential Matrix D

This subsection provides details about the differential matrix $\mathbf{D}$ used in Equation (7). An RBF approximation usually take the following form:

$$\mathsf{\Psi }\left(x\right)=\sum _{k=0}^n{\gamma }_k\varphi \left(\right||x-y_k^c\left|\right|),x\in R^d,$$

(15)

where ${\gamma }_k,k=0,\dots ,n$ are the coefficients to be determined. For the case of CS-RBFs $\varphi \left(r\right)={(1-r)}_{+}^3(3r+1)$, where $r_j=\sqrt{{(x_j-y_k^c)}^2},j,k=0,\dots ,n$. The substitution of $\varphi \left(r\right)$ in Equation (15) yields:

$$\mathsf{\Psi }\left(x\right)=\sum _{k=0}^n{\gamma }_k\varphi \left(r\right),$$

(16)

which produces the following matrix form

$$\mathbf{A}\gamma =\mathbf{g},$$

(17)

here the matrix $\mathbf{A}$ is of order $(n+1)\times (n+1)$, and the vector $\mathbf{g}$ is of order $(n+1)\times 1$. The elements of the matrix $\mathbf{A}$ are to be written as:

$$a_{jk}={(1-r_j)}_{+}^3(3r_j+1).$$

The function $\mathsf{\Psi }\left(x\right)$ with mth derivatives can be of the form:

$$\frac{d^m}{d{x}^m}\mathsf{\Psi }\left(x\right)=\sum _{k=0}^n{\gamma }_k\frac{d^m}{d{x}^m}\varphi \left(r\right),$$

(18)

which, on collocation points $x_j,j=0,\dots ,n$, can take the form:

$$\frac{d^m}{d{x}_j^m}\mathsf{\Psi }\left(x_j\right)=\sum _{k=0}^n{\gamma }_k\frac{d^m}{d{x}_j^m}{(1-r_j)}_{+}^3(3r_j+1).$$

(19)

Consequently, the matrix form of the above equation is

$$\frac{d^m}{d{x}_j^m}\mathbf{g}=\frac{d^m}{d{x}_j^m}\mathbf{A}\gamma .$$

(20)

Substituting the value of $\gamma$ from Equation (17), the differential weights can be obtained by:

$${\mathbf{D}}_j^m=\frac{d^m}{d{x}_j^m}\mathbf{A}{\mathbf{A}}^{-\mathbf{1}}.$$

(21)

Using the above procedure, we can get a differentiation matrix $\mathbf{D}$ with the order $(n+1)\times (n+1)$.

In the literature, the methods based on MQ-RBF usually produce the dense and ill-conditioned matrix $\mathbf{A}$. This behavior of matrix $\mathbf{A}$ causes computational and stability issues in practical computation. For this purpose, we have applied CS-RBFs $\varphi \left(r\right)={(1-r)}_{+}^3(3r+1)$ instead of MQ-RBF to obtain a stable and well-conditioned system.

3. Error and Stability Behavior of the Interpolation Functions

This section deals with different types of RBFs, their interpolation behavior, and the analysis of error estimates and stability performance.

3.1. Radial Basis Function Interpolation

A univariate function $\varphi :[0,\infty ]\to R$ with radial distances is known as the radial basis function. There are two types of RBFs: one is a globally defined basis function (such as the MQ-RBF $\varphi (r,ϵ)=\sqrt{{ϵ}^2+r^2}$), and the other is a local basis function with compact support (such as Wendland’s CS-RBFs $\varphi \left(r\right)={(1-r)}_{+}^3(3r+1)$). For n centers $y_1^c,\dots ,y_n^c$ in $R^d$, the RBF interpolation can take the form

$$Q\left(x\right)=\sum _{k=1}^n{\gamma }_k\varphi \left(\right||x-y_k^c\left|\right|),x\in R^d.$$

(22)

The coefficients ${\gamma }_k$, can be obtained with the following condition

$$Q\left(x_j\right)=s_j,j=1,\dots ,m,$$

(23)

to be satisfied for the points to be coincided, i.e., $n=m$. From Equation (23), the following system of linear equations is obtained:

$$A\gamma =s,$$

which can be solved by LU-factorization for the unknowns ${\gamma }_k,k=1,\dots ,n$. The coefficient matrix A has the following type of entries:

$$a_{jk}={\varphi }_{jk}=\varphi \left|\right|x_j-y_k^c{\left|\right|}_2,j,k=1,\dots ,n.$$

3.2. Error and Stability Analysis

This subsection describes the stability analysis and error bounds of the two types of RBFs. One of the famous bounds are derived in the work of Schaback [36] in the native space ℵ; these bounds are represented as

$$|Q_s{\left(x\right)-s\left(x\right)|\le |s|}_{\aleph }.P\left(x\right),$$

(24)

here the power function $P\left(x\right)$ represents a norm of the error functional on ℵ and can be evaluated at x with

$${\left|s\right|}_{\aleph }^2={\int }_{R^d}\frac{|\widehat{s}{\left(\omega \right)|}^2}{\widehat{\psi }\left(\omega \right)}d\omega <\infty ,$$

and

$$\aleph =\{s\in L^2\left(R^n\right){\int }_{R^d}\frac{|\widehat{s}{\left(\omega \right)|}^2}{\widehat{\psi }\left(\omega \right)}d\omega <\infty \}.$$

In the above equation, $\widehat{\psi }$ is the generalized Fourier transform of $\psi$. For different RBFs, the upper bounds are different and represented by $F\left(h\right(x\left)\right)$ for the power function $P\left(x\right)$, with different values in each case. In the work of Schaback [36], the author derived lower bounds in terms of eigenvalues and represented them by $G\left(h\right(x\left)\right)$. For the specific values of each upper and lower bounds, readers are referred to (Tables 1 and 2 in [36]) and (Tables 11.1 and 12.1 in [37]). A few lower bounds of the multi-quadric (MQ), Gaussian (GA), and CS-RBFs are as follows:

$$\mathrm{MQ}:G\left(h\right)=h{e}^{\left(\frac{-12.76ϵd}{h}\right)}$$

$$\mathrm{GA}:G\left(h\right)=h^{-d}{e}^{\left(\frac{-40.71{ϵ}^2{d}^2}{h^2}\right)}$$

$$\mathrm{CS-RBFs}:G\left(h\right)=h^{2k+1},$$

where $h=su{p}_{y\in \Omega }mi{n}_{x\in X}\left|\right|x-y\left|\right|$, d represents the dimension, X is the interpolation set, and $ϵ$ is the shape parameter.

Furthermore, in the work of Wendland [38], the error estimate of CS-RBF is derived and represented as

$$\left|\right|s-Q_{s,X}{\left|\right|}_{L^{\infty }(\Omega )}\le C_1{\left(\frac{h}{\alpha }\right)}^{k+\frac{1}{2}}{\left|\right|s\left|\right|}_{H^z\left(R^d\right)},$$

(25)

where $z=\frac{d}{2}+k+\frac{1}{2}$, s is a function from $H^z\left(R^d\right)$, and $C_1$ is a constant. Moreover, in the book of Wendland (see Table 11.1, [37]), the upper bounds $F\left(h\right(x\left)\right)$ are computed, which are given as

$$\mathrm{MQ}:F\left(h\right)=e^{\left(\frac{-ϵ}{h}\right)}$$

$$\mathrm{GA}:F\left(h\right)=e^{\left(\frac{-\left|\log h\right|}{hϵ}\right)}$$

$$\mathrm{CS-RBFs}:F\left(h\right)=h^{2k+1}.$$

This goes along-with the upper bound based on the condition number of the interpolation matrix performed by (see [37], Corollary 12.8)

$$C_2{\left(\frac{p_X}{\alpha }\right)}^{-d-2k-1},$$

(26)

where $C_2$ is a constant independent of $p_X=\frac{1}{2}mi{n}_{i\ne j}\left|\right|x_i-x_j\left|\right|$.

4. Numerical Experiments

This section presents the practical implementation of the proposed method on benchmark problems. In each case, the relative error ${ϵ}_r$, conditioned number, and CPU time have been reported and compared with the existing methods in the literature [7,35]. The method in [35] is based on the MQ-RBF, which has a shape parameter, and the proper selection of a shape parameter is still an open problem. For simplicity, we have taken the shape parameter $ϵ=0.5$. On the other hand, the CS-RBFs also involved a scaling parameter, which affects the performance of the method. In this work, we have applied $\varphi \left(\frac{r}{\alpha }\right)={(1-\frac{r}{\alpha })}_{+}^3(3\frac{r}{\alpha }+1)$ CS-RBF in the numerical experiments with scaling parameter $\alpha =0.5$. The details of the scaling parameter and its effects on accuracy and stability can be found in [39,40,41].

Example 1. 

Take the integral equation

$$v\left(t\right)=u\left(t\right)+{\int }_{-1}^{1}f(t,\tau )v\left(\tau \right)e^{i\omega g(t,\tau )}\mathrm{d}\tau ,$$

(27)

where $u\left(t\right)=1+{(t-1/2)}^2\cos \left(10t\right)-e^{-t^2-7}[e^{i\omega (t^2/20+121/25)}-e^{i\omega (t^2/20+1/25)}],g(t,\tau )=t^2/20+{(\tau +6/5)}^2$, and $f(t,\tau )=\frac{-28{\tau }^3+i\omega (2\tau +12/5)}{1+{(\tau -1/2)}^2\cos \left(10\tau \right)}{e}^{-t^2-7{\tau }^4}$.

The analysis of the FIEs (27) are performed with the methods in [7,35] and by the proposed method. For low nodal points, the method in [35] demonstrated better in accuracy than the counterpart methods but worst in conditioned number and computational time (Figure 1, Figure 2 and Figure 3 reflect this behavior). As the nodal points increased, the proposed method attained its superiority in accuracy, efficiency, and stability over the counterpart methods [7,35] for the same problem (Figure 4, Figure 5 and Figure 6 confirm this claim). For the fixed nodes and varying frequency, the computations were performed, obtaining the supremacy of the proposed method in all respects (Figure 7, Figure 8 and Figure 9 reflects this behavior). The exact solution of Equation (27) is $v\left(t\right)=1+{(t-1/2)}^2\cos \left(10t\right)$.

Example 2. 

To evaluate the integral equation

$$v\left(t\right)=u\left(t\right)+{\int }_{-1}^{1}f(t,\tau )v\left(\tau \right)e^{i\omega g(t,\tau )}\mathrm{d}\tau ,$$

(28)

where $u\left(t\right)={\tan }^2\left(\cos \left(5\sqrt{t+1}\right)\right)+\frac{1}{\omega }{e}^{\frac{1}{15}i\omega t^4}\left(e^{121/25i\omega }-e^{1/25i\omega }\right),g(t,\tau )=t^4/15+{(\tau +6/5)}^2$, and $f(t,\tau )=\frac{i\omega (2\tau +12/5)}{{\tan }^2\left(\cos \left(5\sqrt{t+1}\right)\right)}$.

The exact solution of Equation (28) is $v\left(t\right)={\tan }^2\left(\cos \left(5\sqrt{t+1}\right)\right)$. The FIEs (28) are evaluated by the methods in [7,35] and by the proposed method. For low nodal points and fixed frequency, the performance of the method [35] is better in accuracy but worse in conditioned number and computational time (Figure 10, Figure 11 and Figure 12 reflects this behavior). As the nodal points increased, the proposed method gained its goal of better accuracy, well-conditioned behavior, and fast computation (Figure 13 and Figure 14 confirms this claim). The proposed method maintained its good performance for the large frequency and fixed nodal points as well (Figure 15 and Figure 16 verify this).

Example 3. 

Here, we consider computation of the integral equation

$$v\left(t\right)=u\left(t\right)+{\int }_{-1}^{1}f(t,\tau )v\left(\tau \right)e^{i\omega g(t,\tau )}\mathrm{d}\tau ,$$

(29)

where $u\left(t\right)=\cos \left(10t\right)-e^{-t^2}\left(e^{i\omega (t^2/20+121/25)}-e^{i\omega (t^2/20+1/25)}\right),g(t,\tau )=t^2/20+{(\tau +6/5)}^2$, and $f(t,\tau )=\frac{i\omega (2\tau +12/5)}{\cos \left(10\tau \right)}{e}^{-t^2}$.

The numerical experiments for the evaluation of the integral (29) are performed. For low nodal points and frequency, the performance of the methods [7,35] is better in terms of accuracy, but worse in conditioned number and computational time (Figure 17, Figure 18 and Figure 19 reflects this behavior). For a large number of nodes and frequency, the proposed method gained better accuracy, well-conditioned and demonstrated fast computational behavior (Figure 20, Figure 21 and Figure 22 confirms this claim). Finally, the analysis is carried out for a large frequency, in which the proposed method performed well in all aspects of the computation (Figure 23, Figure 24 and Figure 25 confirms this behavior). From Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23, Figure 24 and Figure 25, it is concluded that the proposed method performed better in all respects than the competitive methods [7,35]. The exact solution of the Equation (29) is $v\left(t\right)=\cos \left(10t\right)$.

5. Conclusions

In the sequel, the Levin method based on CS-RBF is presented for better computation of the FIEs with oscillatory kernels. In the literature, the Levin method based on MQ-RBF and the Chebyshev differentiation matrix have been reported in the work [7,35]. These methods are better in accuracy for a lower number of nodal points and lower frequencies. On increasing the nodal points or frequency, these methods produce dense and ill-conditioned matrices, which causes computational complexities and stability issues. For this purpose, we have adopted CS-RBF as a basis function, which has produced sparse and well-conditioned interpolation matrices, even for large-scale data points and large frequencies. The proposed method has demonstrated superior performance in accuracy and stability and has reduced computational time for the FIEs with oscillatory kernels. Numerical experiments were conducted and compared with the counterpart methods [7,35] to ensure the better performance of the proposed method.