Approximation Techniques for Solving Linear Systems of Volterra Integro-Differential Equations

In this paper, a collocation method using sinc functions and Chebyshev wavelet method is implemented to solve linear systems of Volterra integro-differential equations. To test the validity of these methods, two numerical examples with known exact solution are presented. Numerical results indicate that the convergence and accuracy of these methods are in good a agreement with the analytical solution. However, according to comparison of these methods, we conclude that the Chebyshev wavelet method provides more accurate results.


Introduction
Systems of integro-differential equations have motivated huge amounts of research in recent years. ey arise in many physical phenomena like wind ripple in the desert, nano-hydrodynamics, population growth model, glass-forming process, and oceanography [1][2][3]. Various numerical methods for solving systems of linear integro-differential equations have been developed by many researchers. Hesameddini and Rahimi [4] used the reconstruction of variational iteration method (RVIM) for solving systems of Volterra integro-differential equations. In [5], Hesameddini and Asadolhifard, implemented the sinc-collocation method to approximate the solution of systems of linear Volterra integro-differential equations with initial conditions. Aminikhah and Hosseini [6] applied the wavelet method for the numerical solution of systems of integro-differential equations. ey used the operational matrix of integration to solve these systems. Draidi and Qatanani [7] emplemented product Nystrom and sinc-collocation methods to solve Volterra integral equation with Carleman kernel. Hamaydi and Qatanani [8] used the Taylor expansion and the variational iteration methods to give approximate solution of Volterra integral equation of the second kind. In addition, Issa [9] has employed several numerical techniques for solving systems of Volterra integro-differential equations. Other numerical methods for systems of integro-differential equations are (power) functions and Chebyshev polynomials [10], single term Walsh series [11], Chebyshev collocation [12], rationalized Haar functions [13], differential transform [14], homotopy perturbation [15], power series [16], and finite difference approximation [17]. Regarding the stability of a system of Volterra integro-differential equations, some stability results are proposed for the linear system VIDEs in the 1980s, those of Burton are worthy to mention. His work [18,19] laid the foundation for a systematic treatment of the basic structure and stability properties of VIDEs via the direct method of Lyapunov. A more recent result is by Elaydi [20], who proposed a type of Lyapunov functional that is also applicable to delay equations. Moreover, Zhang [21] proposed recently a stability result from which certain well-known result could be derived. Also, Vanualailai and Nakagiri [22] have proposed a new stability criteria based on new and known forms of Lyapunov functionals for a system of Volterra integro-differential equations. In this article, we propose two numerical methods, namely, a collocation method using sinc functions and Chebyshev wavelet method to approximate the solution of a system of linear Volterra integro-differential equations given by subject to the initial conditions (1) e kernels 푘 (푥, 푡) and the function 푓 (푥) are given real valued functions and the unknown functions 푢 (푥) are to be determined. A comparison between these methods is carried out by solving some numerical examples. e paper is organized as follows: In Section 2 the sinc-collocation method based on sinc functions is presented. e Chebyshev wavelet method is addressed in Section 3. In Section 5, the proposed methods are implemented using two numerical examples with known analytical solution by applying MAPLE so ware. Conclusions are followed in Section 6.

Sinc Collocation Method Based on Sinc Functions
e sinc collocation method based on sinc functions is widely used for obtaining the approximate solution of ordinary and partial differential equations and integral equations [5]. It is well-known that the sinc approximate solution converges exponentially to the exact solution.
Definition 1 (see [23]). e sinc function is defined on the whole real line −∞ < 푥 < ∞ by as shown in Figure 1.
Corollary 1 (see [5]). e sinc function for the interpolating points 푥 = 푑푟, is given by Corollary 2 (see [5]). If 푝(푥) is defined on the real axis and is a positive integer, then the series is called the Whittaker Cardinal expansion of 푝(푥).
To construct an approximation on the interval (푎, 푏), we use the conformal map: is map carries the eye-shaped region Definition 3 (see [23]). Let be the set of all analytic functions. en, there exists a constant , such that: Theorem 1 (see [23]). Let 푢 ∈ 퐿 퐺 and is a natural number, and be selected by the formula en, there exists a positive constant 1 , independent of , such that Theorem 2 (see [5]). Let 푢/휑 ∈ 퐿 퐺 and is a natural number, and be given as where Moreover, let (−1) 푘푑 be defined as en, there exists a positive constant 2 , independent of , such that Theorem 3 (see [5]). Let be a conformal injective map of the simply connected domain onto . en We consider the system of linear Volterra integro-differential equations of the form: Subject to the initial conditions: in the domain [0, 1], and let 푢 (푥) ∈ 퐿 퐺 . By using eorem 1, 푢 (푥) is approximated as follows: where where 푗 = −푁 − 1, . . . , 푁, and using eorems 2 and 3, we obtain a system of algebraic equations. Solving this system we obtain the unknown coefficients and 3. Chebyshev Wavelets Method (CWM) e main idea of using Chebyshev basis is that the problem under study reduces to a system of linear or nonlinear algebraic equations. is may be done by truncated series of orthogonal basis functions for the solution of the problem and using the operational matrices [6]. Wavelets constitute a family of functions constructed from dilation and translation of a single function called the mother wavelet [24][25][26]. When dilation and translation vary continuously, we have the following family of continuous wavelets as If we choose the dilation and translation −푎 , and −푎 , respectively, where 푝 > 1, 푞 > 0, then we have the following family of continuous wavelets as where 푎,푏 forms a wavelet basis for

Numerical Examples and Results
In this section, some numerical examples are presented to show the validity of the proposed methods. In addition, the numerical results are compared with exact solution. We multiply both sides of Equation (61) by 푤 (푥)휓 (푥) and integrating with respect to from 0 to 1, we obtain a linear system in terms of input 퐷 , 푖 = 1, 2, 3, . . . , 푛. Consequently, the vector functions elements are calculated by solving this system.

Stability of Systems of Volterra Integro-Differential Equations (VIDEs)
In this section, we present some important results on the stability of VIDEs (1) (for more details see [22]).  Absolute error Journal of Applied Mathematics 8 Absolute error Exact solution We start by implementing Algorithm 1 to solve system (66) using the Sinc collocation method based on sinc functions. Tables 1 and 2 contain the exact and numerical solutions together with the resulting error with 푀 = 8. Figure 2 shows a comparison between the exact and numerical solutions for system (66). e maximum error corresponding to 푢 1 , 푢 2 , and 3 is 퐸 1 ≈ 3.9푒 −7 , 퐸 2 ≈ 8.5푒 −8 , and 퐸 3 ≈ 1.79푒 −7 , respectively.
Next, we implement Algorithm 2 to solve system (66) using the Chebyshev wavelets method. Tables 3 and 4 contain the exact and numerical solutions for system (66) together with the resulting error with 푀 = 8.

Conclusions
In this article, a collocation method using sinc functions and Chebyshev wavelets method is proposed to solve linear system of integro-differential equations. e numerical results show that the convergence and accuracy of both methods were in good agreement with the analytical solution. Comparison of numerical results mentioned in tables and figures shows clearly that the Chebyshev wavelets method provides more accurate results and is therefore more effective than any other methods for solving systems of integro-differential equations.

Data Availability
We do not have any objection of sharing the data, the findings and the results of our article with other authors in different means of communication.