Journal of Applied Mathematics

Journal of Applied Mathematics / 2020 / Article

Research Article | Open Access

Volume 2020 |Article ID 2360487 | https://doi.org/10.1155/2020/2360487

Ahmad Issa, Naji Qatanani, Adnan Daraghmeh, "Approximation Techniques for Solving Linear Systems of Volterra Integro-Differential Equations", Journal of Applied Mathematics, vol. 2020, Article ID 2360487, 13 pages, 2020. https://doi.org/10.1155/2020/2360487

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

Academic Editor: Yansheng Liu
Received22 Jul 2019
Revised08 Oct 2019
Accepted14 Oct 2019
Published04 Feb 2020

Abstract

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.

1. Introduction

Systems of integro-differential equations have motivated huge amounts of research in recent years. They arise in many physical phenomena like wind ripple in the desert, nano-hydrodynamics, population growth model, glass-forming process, and oceanography [13]. 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. They 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

The kernels and the function are given real valued functions and the unknown functions are to be determined. A comparison between these methods is carried out by solving some numerical examples.

The paper is organized as follows: In Section 2 the sinc-collocation method based on sinc functions is presented. The 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 software. Conclusions are followed in Section 6.

2. Sinc Collocation Method Based on Sinc Functions

The 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]). The sinc function is defined on the whole real line byas shown in Figure 1.

Definition 2 (see [23]). Let then the translated sinc basis functions are defined aswhich are called the th sinc functions.

Corollary 1 (see [5]). The sinc function for the interpolating points , is given by

Corollary 2 (see [5]). If is defined on the real axis and is a positive integer, then the seriesis called the Whittaker Cardinal expansion of .

The properties of the Whittaker Cardinal expansion have been extensively studied in [22]. These properties are derived in the infinite strip of the complex –planes where for any ,

To construct an approximation on the interval , we use the conformal map:

This map carries the eye-shaped region

For the sinc method, the basis functions on for are derived from the composite translated sinc functions,

where .

The inverse map of is

Also we define the range of on the real line as

and the interpolation points are then given by:

Definition 3 (see [23]). Let be the set of all analytic functions. Then, there exists a constant , such that:where .

Theorem 1 (see [23]). Let and is a natural number, and be selected by the formulawhereThen, there exists a positive constant , independent of , such that

Theorem 2 (see [5]). Let and is a natural number, and be given aswhereMoreover, let be defined asThen, there exists a positive constant , independent of , such that

Theorem 3 (see [5]). Let be a conformal injective map of the simply connected domain onto. Then

We consider the system of linear Volterra integro-differential equations of the form:

Subject to the initial conditions:

in the domain , and let . By using Theorem 1, is approximated as follows:

where

where are unknown coefficients and .

Integrating both sides of Equation (25) from to we get

and by differentiating both sides of Equation (25) with respect to we get

where

Substituting Equations (27) and (29) into Equation (23), and by evaluating the result at the sinc points

where , and using Theorems 2 and 3, we obtain a system of algebraic equations. Solving this system we obtain the unknown coefficients

and

3. Chebyshev Wavelets Method (CWM)

The main idea of using Chebyshev basis is that the problem under study reduces to a system of linear or nonlinear algebraic equations. This 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 [2426]. When dilation and translation vary continuously, we have the following family of continuous wavelets as

If we choose the dilation and translation , and , respectively, where , , then we have the following family of continuous wavelets as

where forms a wavelet basis for

where : set of all square integrable functions equipped with norm

For the particular case, when and , then forms an orthogonal basis.

Chebyshev wavelets have four parameters, , and c is the degree of Chebyshev polynomials of the first kind. They are defined on the interval by:

where

and , and .

are the famous Chebyshev polynomials of the first kind of degree which are orthogonal with respect to the weight function

and satisfy the following recurrence relation

Corollary 3 (see [6]). The set of Chebyshev wavelets is an orthogonal set with respect to the weight function

Definition 4 (see [6]). A function defined on the interval is called the wavelet series if this function is written in the following formwhereis the inner product in .

Corollary 4 (see [6]). The wavelet series in is convergent ifIf the wavelet series is truncated, then it can be written aswhere and are matrices given by

Corollary 5 (see [27]). The integral of the multiple of two Chebyshev wavelets vector functions with respect to from to is an identity matrix. Moreover, a function defined on can be approximated as:where is a matrix of the entries , that can be determined by: and .

The integral of the vector defined in Equation (46), is given as:

where is the operational matrix of integration [2].

This matrix has the form:

where , , and are matrices given by

The product characteristic of two Chebyshev wavelets vector functions are given as

where is a given vector and is an operational matrix. We consider the system of linear Volterra integro-differential equations of the form:

with the following conditions

Now we approximate by using Chebyshev wavelet space as follows

Therefore we have

where and are matrices given by

In virtue of Equations (55) and (56) we have the following approximations:

where and are known matrices for .

By substituting the approximations Equations (56) and (59) into the system equation (54), we obtain:

Therefore,

, where is the operational matrix of integration and are matrices.

We multiply both sides of Equation (61) by and integrating with respect to from to , we obtain a linear system in terms of input . Consequently, the vector functions elements are calculated by solving this system.

4. 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]).

Definition 5 (see [22]). If is a continuous initial function, then will denote the solution of (1) on . Frequently, it is sufficient to write . If , then is a solution of (1) called the zero solution. The norm on the initial function is given byThe definition of stability of the zero solution is given in Burton [18] and is restated below.

Definition 6 (see [22]). The zero solution of (1) is stable if for each and each , there exists such that on and imply .
We next define the statement “a Lyapunov functional for system (1)”. Let be defined for and and let be locally Lipschitz in . For each and every , we define the derivative along a solution of (1) bywhere is the unique solution of (1) with initial conditions and . Then the following result by Driver ([28]) gives a definition of the Lyapunov functional.

Theorem 4 (see [28]). If is defined for and every with(1).(2) continuous in and Lipschitz in .(3), where is a continuous function withand strictly increasing (positive definiteness).(4).then the zero solution of (1) is stable, andis called a Lyapunov functional for system (1).

Finally, we assumed that the functions in (1) are well behaved, that continuous initial functions generate solutions, and that solutions which remain bounded can be continued.

5. 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.

Example 1. Consider the system of Volterra integro-differential equations:together with the initial conditions

The exact solution of system (66) is

We start by implementing Algorithm 1 to solve system (66) using the Sinc collocation method based on sinc functions.

1:Input:(i)(ii)(iii)(iv)Initial Condition (v)Initial Condition
2:Define:(i)Sinc Function (ii)(iii)
3:Replace , in each equation
4:Calculate Sinc point
5:Evaluate
, for
 From this step, we get equation
6:Evaluate . From this step, we get equation
7:Evaluate . From this step, we get equation
8:Solving the algebraic system to get and
9:Set and in
10:Input
11:Plot
12:Define the error
13:Plot the error

Tables 1 and 2 contain the exact and numerical solutions together with the resulting error with .


Exact solution Numerical solution Exact solution Numerical solution Exact solution Numerical solution

0001122
0.1−0.1999−0.19989999864100370.9990.99899999961644052.31034183615129552.310341836911406
0.2−0.3984−0.398400004693820250.9920.9920000012685952.642805516320342.642805513660997
0.3−0.5919−0.59190007953928590.9730.97300002160990852.9997176151520062.9997175706314225
0.4−0.7744−0.774400228030920.9360.93600006159458373.38364939528254063.3836492678242998
0.5−0.9375−0.93750031568465420.8750.87500008588650243.79744254140025643.7974423632300143
0.6−1.0704−1.07040019185221010.7840.7840000567340324.2442376007810184.244237484652817
0.7−1.1599−1.15989988940311960.6570.65699998427898274.7275054149409534.72750545433764
0.8−1.1904−1.1903996437574780.4880.48799992822443435.2510818569849365.251082019406921
0.9−1.1439−1.14389961421316920.2710.27099992702593295.81920622231395.819206393371196


Absolute error Absolute error Absolute error

00−10
0.11.3589962488680383.8355951748059177.601106410959346
0.24.6938202213908651.2685950068203052.659342968058808
0.37.9539285824381522.1609908529285354.452058366410938
0.42.2803092902812246.1594583766755311.274582408505864
0.53.1568465419518078.5886502354526331.781702421155273
0.61.918522101185265.6734032027883071.161282012773767
0.71.105968803205571.5721017265768463.939668680175146
0.83.5624252125998627.1775565602116841.624219851947828
0.93.8578683070333857.2974067011344351.710572963276035

Figure 2 shows a comparison between the exact and numerical solutions for system (66). The maximum error corresponding to , and is , , and respectively.

1:Input:(i)(ii)(iii)(iv)Initial Condition (v)Initial Condition
2:Define:(i)Chebyshev Function (ii)weight function (iii)(iv)(v) as definition (4)(vi)operator as remark (5)
3:Calculate