This paper presents a numerical method for solving a class of the delay Volterra integral equation of nonvanishing and vanishing types by applying the local radial basis function method. This method converts these types of integral equations into an easily solvable system of algebraic equations. To prove the method, we use the discrete collocation method and the local radial basis function method to approximate the delay Volterra integral equation. Also, we use the nonuniform Gauss–Legendre integration method to calculate the integral part appearing in the method. In addition, the existence, uniqueness, and convergence of the solution are investigated in this paper. Finally, some numerical examples are shown to observe the accuracy and effectiveness of the numerical method. Some problems have been plotted and compared with other methods. Obtained numerical results and their comparison with other methods show the reliability and accuracy of this method.

1. Introduction

The solution of the linear system of Volterra–Fredholm integral equations (VFIEs) has been a topic of a significant interest. Studies in the field of mathematical modeling, boundary value problems, and diverse biological and physical models led to superior explanations in the VFIEs. A survey of the formulation of such problems is shown in [1, 2] and the references therein. One of the main classes of mentioned areas is called as the delay Volterra integral equation (DVIE). Such a class has been found to be the best tool to model physical and chemical processes. DVIEs have been widely used for analysis and predictions in different areas of science such as biology, control, and electrodynamics [35]. These problems are applied in the mathematical modeling of different physical phenomena as dynamical systems in control theory, dynamics of multispan uniform continuous beams, heat conduction, signal processing, and the generalized voltage divider. In this paper, we are going to study the application of a locally radial basis function-discrete collocation (RBF-DC), a method with the Gauss–Legendre integration rule for obtaining a numerical solution of the following DVIE:where , , and on their domain are continuous functions. The DVIE explains a model of the particle motion in liquid and the crystallization of polymers in [6]. This model is a particular kind of the pantograph model and often emerges in several scientific models such as electrodynamics, number theory, and nonlinear dynamical systems. In recent years, due to such vital and important applications, there have been numerous studies in formulating some impressive numerical methods and their stability and convergence analysis. The convergence of the numerical solution of the delay Volterra integral equations has been provided by Zarebnia and Shiri [7]. In [8], the authors proposed a Bernoulli wavelet method for solving Volterra integral equations of the nonlinear fuzzy Hammerstein type including constant delay. Some papers studied the numerical solution of the DVIEs. In some cases, if DVFIEs do not have exact solutions, then we are interested in obtaining their approximate solution. Therefore, creating efficient numerical methods to obtain approximate solutions for the DVIEs is important. For example, in [9], the authors studied a geometric mesh and considered the corresponding collocation technique to solve the DVIE. Dastjerdi et al. [10] applied a moving least squares collocation method for DVIE. Ming et al. [3] studied a collocation method to solve the DVIE. In [11], a differential transform method was studied to solve the numerical solution of the DVIEs. In [12], proposed a sinc-collocation method for the solution of the DVIE. Some numerical methods have been used to approximate the solution of the DVIEs with different types of conditions, such as meshless methods [13], a local meshless method based on a radial basis function-finite difference [4, 5], a numerical method based on a radial basis function finite-difference [14], an efficient local meshless collocation algorithm [15], the Legendre spectral collocation method [16], an extrapolation method [17], a well-conditioned Jacobi spectral Galerkin method [18], and the Chebyshev spectral method [19]. The subject of radial basis functions was first studied by Hardy [20] and applied for topographical mapping. However, radial basis functions started to attract the attention of scientists, mathematicians, and researchers after their usage of them by Kansa [21] for finding approximate solutions to PDEs, and they are applied extensively to solve many problems coming from various fields of engineering and science.

A function whose independent variable is a distance, or with a more precise definition, a function that takes the distance from the center point to computing node , as an independent variable is called the radial basis function. In the last few years, radial basis functions as an impressive meshless scheme have been used for interpolating a function over a set of scattered points [2224]. Since the classical radial basis functions are global functions, the resultant coefficient matrix with respect to them will be full and large when many points in the domain are investigated for obtaining high-order accurate results. However, this coefficient matrix is nonsingular, but usually, it is ill-conditioned. To overcome this problem, the advantages of radial basis functions with local supports can be used which are called local radial basis functions studied in the papers [2528]. To make the approximation function by local radial basis functions, the only geometrical data demanded are applied fallen within the local influence domain. Therefore, local radial basis functions require much less computational work than the globally supported local radial basis functions, and eventually, the interpolation matrix of local radial basis functions will be well conditioned. The main purpose of this paper is to demonstrate the efficiency and implementation of a local radial basis function method based on the discrete collocation method with a nonuniform Gauss–Legendre integration method on numerical solutions of the DVIEs. This paper shows a computational method using the discrete collocation method together with the local radial basis function approach as a locally meshless method. To calculate integrals shown in the proposed method, we demonstrate a particular integration rule based on the application of the composite nonuniform Gauss–Legendre quadrature formula. The error analysis and convergence analysis for our scheme are studied. To the best of our knowledge, there are no reported studies based on the local radial basis function method for the DVIEs. Unlike other numerical methods such as the finite element method and the finite difference method, in the local radial basis function method there is no requirement to make a mesh or grid, only a set of scattered nodes is needed instead of the mesh or grid. This characteristic of the local radial basis function method makes them desirable, and particularly in complex geometries, getting rid of creating grids is a very good achievement that shows great flexibility to local radial basis function methods. This method provides sparse differentiation matrices and better-conditioned interpolation matrices. Owing to better-conditioned interpolation matrices, it is possible to obtain more stable results and sparse differentiation matrices that reduce both computational cost and storage requirements. According to the description given, the implementation of our numerical method on computers is simple and computationally attractive. Also, our numerical method is only independent of the pairwise distances between points and could be easily extended to higher dimension problems.

The remainder of the paper is organized as follows. In Section 2, we prove the existence and uniqueness of the solution for equation (1). In Section 3, we obtain a numerical method for solving DVIEs based on the local radial basis function approximation in the collocation method. The convergence analysis of the solution is provided in Section 4. In Section 5, numerical results using the proposed method and discussion for some numerical examples are provided with tables and figures. The conclusion of the paper is reported in Section 6.

2. Existence and Uniqueness

In this section, we show and state the existence and uniqueness of the solution for equation (1). Let be the Banach space. Then, we consider an operator by

If is a fixed point of the operator , then is a solution of equation (1). Assume ; then, we havewhere and . If , therefore the operator is a contraction and equation (1) has a unique fixed point which shows that there exists a unique solution of equation (1).

3. Explanation of the Proposed Method

In this section, we show the numerical method based on the local radial basis function collocation method for solving equation (1). Also, in this section, the locally supported RBFs so-called LRBFs are used to approximate a function in any dimension. A function is called to be radial if there exists a univariate function such thatin which . Assume , where are an open bounded cover for and centered at with the radius . Then, a function can be developed as the following series:where is the set of indexes corresponding to points fallen within the influence domain with the cardinal number which is shown in Figure 1. The most favorite radial functions [29] are reported in Table 1. The Gaussian and inverse multiquadrics are strictly positive definite radial basis functions on any [30]. It guarantees that the associated interpolation matrix in the expansion (5) is nonsingular.

In relation (5), the unknown coefficients for are calculated by enforcing the interpolation conditions.

So, we can rewrite equation (6) as the following matrix form:in whichand is the real symmetric coefficient matrix of size and given by

It should be noted that if a radial basis function is strictly positive definite, then the associated interpolation matrix is positive definite and so nonsingular. Let us consider

Then, equation (5) and the matrix can be displayed by

Let be an inverse matrix, then from equation (7), we have

By replacing equation (12) into equation (11), we obtain

By substituting equation (13) into equation (5), we obtainwhere

To discretize the integral in equation (1) by using the proposed method and applying a numerical integration method, we are required to estimate the solution of DVIEs for any point in the solution domain. Thus, we illustrate the local radial basis function collocation method to approximate a function at an arbitrary point by the following linear combination:where is called shape functions for the local radial basis function interpolation. In general, the shape functions can be represented as follows:

Then, we can assume that in which and the influence domain of the point are considered by . The choice of the parameter , named as the shape parameter, influences heavily the accuracy of the method [29]. For example in Gaussians, if we select a bigger shape parameter with a fixed number , the approximation will be more accurate, while the condition number of associated coefficient matrices will be large. On the other hand for fixed values of the shape parameter c, the condition number grows with . However, the selection of the optimal value for is still under check, and some values are suggested by many authors. Hardy has suggested in his original work on and interpolations using , where and is the distance from to its nearest neighbor [31]. We consider the following DVI operator :

Then, we can rewrite equation (1) as the following operator identity:

To solve equation (1) by using the numerical method, we need nodal points considered in the interval . Thus, for any arbitrary function , the function can be estimated by applying the local radial basis functions as

To calculate the coefficients by using the collocation method, we substitute equation (21) and put the nodal points into equation (1) and we obtain

Because the support of is for and the function satisfy the Kronecker delta condition, then we have

Assume be a local collocation operator which is defined bywhere the coefficients are calculated by

Taking the operator of equation (1) yields

To calculate the integral in equation (23), we can use the -point quadrature rule which is given bywhere , and . Therefore, we approximate the integrals given in equation (23) by using the -point quadrature rule aswhere and . Also, we obtainwhere and . Therefore, by substituting equations (28) and (29) into equation (23), we obtain a system of algebraic equations as follows:

By solving these algebraic equations, the unknown constants can be determined and the numerical solution given in equation (21) can be computed.

Definition 1. Let be a set of points, such that . Then, the fill distance of is given by

Theorem 1 (see [32]). Let be a positive definite radial basis function with infinite smoothness and that, is defined bywhere and are the Fourier transform of and , respectively. Then, for any and , there exist constants and such thatwhere .

With a similar process in [32], we can study the complexity of the proposed algorithm.

4. Convergence Analysis

In this section, we will check and study the convergence analysis of the DVIE based on the proposed method.

Theorem 2. Suppose that on be a differentiable function, such that . Then, over every subinterval we havewhere and are the exact solution of equation (1) and its local radial basis series expansion, respectively.

Proof. By using the mean value theorem for integral, we obtainAlso, for the function , by applying the mean value theorem, we haveBy substituting equation (36) into equation (35) and applying the mean value theorem, we getIf we consider , we obtainSince for we have , then,By substituting equation (37) into equation (39), we obtainTo prove the convergence analysis of equation (1), we consider the error function , that and are the exact solution and approximate solution of equation (1). Then, we havewhere is the remainder term. By subtracting equation (42) from equation (41), we obtain the following relation:When , from equation (40), we have , then . Therefore, we show the convergence analysis of equation (1).

5. Illustrative Examples

This part shows some numerical examples for demonstrating the efficiency and accuracy of the proposed method. Also, a comparison of the proposed method with previously known methods is shown. For numerical examples in this work, we use the Gaussian (GA), inverse multiquadric (IMQ), and multiquadric (MQ) in which the applicability and efficiency of them heavily depend on the shape parameters . Since local radial basis functions have much more freedom in selecting the shape parameter, we choose middle values for GAs and for IMQs and MQs. Also, in numerical examples, we use the linear and quadratic basis functions and the Gaussian weight functions for approximating integrals in the method. For assessing the performance and efficiency of the proposed method, we define the maximum of the absolute error and the mean error on a domain as follows:where is the exact solution and is the numerical solution of obtained by using the proposed method. Also, the convergence rate of the presented method has been defined by

Example 1. Consider the following DVIE:The exact solution of equation (45) is . Numerical solutions by using the previous numerical methods have difficulties in solving these kinds of DVIEs, but we can easily calculate the numerical solution for this example by applying GAs and MQs displayed in this paper based on some random nodes over the interval . The distribution of nodes selected randomly on the interval is depicted for some in Figure 2. We first use the proposed method for solving this example by choosing , and . Numerical solution and exact solution graphs by using the technique described in Section 3 for various values of are shown in Figure 3. Figure 4 shows the graphs of the absolute error function with different values of . The absolute error function with different values of for this problem using GA (left) and MQ (right) is shown in Figure 4. Figure 5 demonstrates the obtained errors by using the proposed method for various values of . Table 2 compares the absolute error function of the numerical solutions calculated by applying the proposed method when , and .

Example 2. Consider the DVFIE:For this example, the exact solution is . We solved this example by applying our suggested technique demonstrated in Section 3 by selecting different values of and . For this example, applying GAs and IMQs displayed in this paper based on some random nodes over the interval . Figure 6 represents the numerical solutions from the proposed method and the exact solution for different values of . Figure 7 shows the absolute error in by selecting different values of . The absolute error function with different values of for this problem using GA (left) and IMQ (right) is shown in Figure 7. Table 3 represents the maximum absolute error results of the studied method with various values of . Figure 8 demonstrates the obtained errors by using the proposed method for various values of . A comparison of our suggested technique and the numerical results shown in [3, 12] for different values of is given in Table 4. The obtained numerical results by using the method presented in the current paper are better than the numerical results shown in [3, 12]. We have also compared the CPU times by applying our suggested technique for different numbers of in Figure 9. The convergence rates of the proposed method are much higher than the convergence rates shown in [3, 12]. Also, the CPU times of the proposed method are lower than the numerical results shown in [3, 12].

6. Conclusion

This paper used the local radial basis function method to solve a class of the delay Volterra integral equation of nonvanishing and vanishing types with given initial and boundary conditions. The method used locally supported radial basis functions made on scattered points as a basis in the collocation method. By using the nonuniform Gauss–Legendre quadrature, we gave a numerical formula for obtaining integrals in the method. This method has reduced the solution of the delay Volterra integral equation to the solution of a linear system of algebraic equations. This method does not need any background approximation cells. Some numerical examples are shown to demonstrate that the proposed method is very accurate and effective. The results obtained are compared with the errors presented by other methods published in the scientific literature in this regard. The numerical results of our proposed method are more accurate than those presented in the articles indicated in Table 4 and the References.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.