Numerical Approximation of Fractional-Order Volterra Integrodifferential Equation

Qiang, Xiaoli; Kamran, undefined; Mahboob, Abid; Chu, Yu-Ming

doi:https://doi.org/10.1155/2020/8875792

Journal of Function Spaces

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Abstract Introduction Results and Discussion Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

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Fractional Problems with Variable-Order or Variable Exponents

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Research Article | Open Access

Volume 2020 | Article ID 8875792 | https://doi.org/10.1155/2020/8875792

Numerical Approximation of Fractional-Order Volterra Integrodifferential Equation

Xiaoli Qiang,¹ Kamran,²Abid Mahboob,³and Yu-Ming Chu^4,5

Academic Editor: Jiabin Zuo

Received21 Sept 2020

Revised27 Oct 2020

Accepted26 Nov 2020

Published16 Dec 2020

Abstract

Laplace transform is a powerful tool for solving differential and integrodifferential equations in engineering sciences. The use of Laplace transform for the solution of differential or integrodifferential equations sometimes leads to the solutions in the Laplace domain that cannot be inverted to the real domain by analytic methods. Therefore, we need numerical methods to invert the solution to the real domain. In this work, we construct numerical schemes based on Laplace transform for the solution of fractional-order Volterra integrodifferential equations in the sense of the Atangana-Baleanu Caputo derivative. We propose two numerical methods for approximating the solution of fractional-order linear and nonlinear Volterra integrodifferential equations. In our scheme, the inverse Laplace transform is approximated using a contour integration method and Stehfest method. Some numerical experiments are performed to check the accuracy and efficiency of the methods. The results obtained using these methods are compared.

1. Introduction

Fractional calculus has a large number of applications in engineering and mathematical sciences [1–5]. Problems from engineering and other sciences which involve derivatives or integrals of noninteger orders are large in number and still growing. That is why the research community has showed great interest in the area of fractional calculus. One can find various applications of fractional calculus in numerous phenomena such as frequency-dependent damping behavior of viscoelastic materials [6], control theory [7], economics [8], mass and heat diffusion processes, electromagnetic theory, biological species [9], robotics, and many other engineering problems [10]. Accurate models of systems under consideration can be obtained using fractional differential and integrodifferential equations [11]. Many remarkable works on fractional calculus are available in literature for the approximation of the solution fractional differential or integrodifferential equations, for example, the sinc-collocation method [12], Legendre collocation method [13], Laguerre polynomials [14], Adomian decomposition method [15], Variational iteration method [16–18], and their references.

Numerous essential phenomena in nature are resolved using the solutions of fractional integrodifferential equations [11, 19]. For example, one can find applications of fractional integrodifferential equations (FIDEs) in electromagnetics [20], heat conduction [21], etc. Due to the wide range of applications, FIDEs have attracted researchers. Therefore, a large number of analytic and numerical methods have been developed for finding the solutions of FIDEs [22–26]. For example, the authors [27] utilized the fractional differential transform method for the solution of fractional integrodifferential equations. A Legendre wavelet method [28] is proposed for the solution of FIDEs. In [29], the analytic solution of FIDEs is obtained using the homotopy analysis method. The author in [30] obtained the analytic solution of FIDEs using the fractional residual power series method.

In many cases, the analytic solutions of fractional differential or integrodifferential equations are hard to obtain, so numerical methods must be used. In this connection, the researchers have developed numerous numerical methods. For instance, the authors in [19, 31] have solved FIDEs using the reproducing kernel Hilbert space method. A hybrid collocation method [32] is proposed to solve the FIDEs. The authors in [33] solved the first-order Volterra equation by the collocation method. The author in [34] utilized the shifted Chebyshev polynomial and least squares method for the approximation of the solution of FIDEs. Mahdy and Shwayyea [35] obtained the approximate solution of FIDEs using the shifted Laguerre polynomial pseudospectral method and least squares method. The authors in [36] approximated the solution of Volterra-type FIDEs via Bernoulli wavelet approximation. In [14], Laguerre polynomials are utilized to approximate FIDEs of Volterra type. More work on the solution of FIDEs can be found in [31, 33, 37, 38] and their references.

There are various definitions of fractional derivatives such as Caputo-Liouville’s and Riemann-Liouville’s [11, 39]. These derivatives have certain disadvantages due to nonlocal and nonsingular kernel functions involved in these derivatives. In order to avoid these difficulties, it is better to use the ABC derivative. The ABC derivative contains a nonsingular kernel and therefore can model a phenomenon which cannot be handled by fractional-order derivatives having singular kernels [40]. Some recent articles on the application of the ABC derivative can be found in References [41–45]. In the present work, we consider a FIDEs of Volterra type with the ABC derivative of the form where

Here, is the ABC derivative of order .

1.1. Preliminaries

Definition 1. The ABC fractional derivative of order , of with base point at , is defined as [46] where is a Sobolev-space of first-order fitted with -norm defined over the defined as follows: where is defined as follows: also, is a one-parameter Mittage-Leffler (ML) function defined as

Definition 2. The ABC fractional integral of order of with base point at is defined as [46]

Definition 3. The Laplace transform of a piecewise continues function is defined as

Definition 4. If then the Laplace transform of the ABC derivative is defined by

2. Proposed Method

We provide a description of our proposed schemes. Our numerical scheme has three main steps. In our first step, we apply the Laplace transform to the given model, with which the problem is reduced to an algebraic equation. In the second step, we solve the reduced equation in Laplace space for the unknown. Finally, the solution of the original problem is obtained using numerical inversion of Laplace transform. In this work, the numerical approximation of Laplace transform is obtained using two schemes. One is the contour integration method (CIM), and second is the Stehfest method (SM). We discuss these methods in the next sections. First, we apply Laplace transform to Eq. (1); we get which can be rearranged as where

In order to obtain the solution of problem (1), we apply Laplace inverse on (12), and the Laplace inverse is then approximated using the contour integration method and the Stehfest method, which are discussed as follows.

3. Contour Integration Method (CIM)

In CIM, we represent the solution of (1) as Bromwich integral: where and is an initially appropriately chosen line in a complex plane parallel to the axis, with . Then, in (14), is the inverse transform of , satisfying the condition that all the singularities of lie to the left of . However, for our purposes, we assume that may be continued analytically in an appropriate way; we shall want to take for a deformed contour in which has asymptotic behavior in the left complex plane, with when , which force to decay towards both ends of . In our work, we choose as where

Letting , we see that (15) represents the left branch of the hyperbola given by where are the asymptotes for (17) and is its -intercept. It is confirmed from Eq. (16) that lies in the sector and grows into the left half plane. From (15) and (14), we get

The approximation of Eq. (18) using the trapezoidal rule with uniform step is given as

3.1. Error Analysis

While solving the fractional-order problem defined in Eqs. (1)–(3), the first step is the application of Laplace transform to the given problem. The Laplace transform reduces the problem to a time-independent problem in Laplace space, and in this process, no error occurs. Next, the problem is solved in complex space for the unknown, and this process also incurs no error. In the final step, the solution of the given problem is retrieved using the Laplace inverse. We represent our solution as integral (18). We approximate this integral by the trapezoidal rule. During this process, integral (18) converges at different time rates which depends on . In the process of numerical approximation of the Bromwich integral (18), the orders of convergence greatly depend on the step of the trapezoidal rule and the temporal domain . For useful results and strong convergence, a suitable temporal domain is chosen. The next theorem gives the order of quadrature error.

Theorem 5 (see [47], Theorem 2.1). Let be the solution of (1) with analytic in . Let and define , for , where , , , and let . Then, for Eq. (19), with we have for , , , , and Hence, we have

4. Stehfest Method (SM)

In the Stehfest method, the approximate value of is given as where the weights are given by

Solving (12) for the corresponding Laplace parameters

The solution of the original problem can be obtained using (21).

4.1. Error Analysis

The authors [48, 49] performed a large number of numerical experiments to study the effect of the parameter on numerical accuracy, and they summarized there experimental work as the following: “If significant digits are desired, then let be the positive integer . Given , set the system precision at . Given and the system precision, calculate the weights , , using (22). Then, given the transform and the argument , calculate in (21).” According to these conclusions, the error is , where [50].

5. Results and Discussion

In order to check the accuracy and efficiency of the two numerical schemes. We consider linear and nonlinear and system of fractional-order VIDEs. In our numerical computations, the accuracy is achieved using the optimal parameters. The optimal values of the parameters utilized for CIM in our work are . The MATLAB command is used to generate the quadrature nodes for CIM. The parameter used in our numerical computation for the Stehfest method is .

5.1. Problem 1

Here, we consider the fractional linear Volterra integrodifferential equation [31]:

The exact solution of this problem is . In Table 1, the approximate solutions of for various values of and using the two proposed methods are presented, and in Table 2, the absolute errors are shown. The plot of error function for various quadrature nodes is shown in Figure 1(a), and the plot of the error function for is shown in Figure 1(b). The graphs of approximate solutions for are presented in Figure 2(b), and Figure 2(a) displays the plot absolute error vs. error estimate for . The numerical results using the Stehfest method are obtained by using . All the plots are obtained using the CIM. It is evident that these methods can solve such type of equations efficiently.

(a)

(b)

(a)

(b)

5.2. Problem 2

Here, we consider the fractional nonlinear Volterra integrodifferential equation [31]:

The problem has exact solution . The Stehfest parameter used is . The numerical results for various values of and are presented in Table 3. The graphs of numerical solutions for various values of are shown in Figure 3(a), whereas Figure 3(b) presents the graph of error estimate vs. absolute error for . For and , the error function is shown in Figure 4(a), and the error function for various , and is shown in Figure 4(b). A similar performance is observed as was observed for Problem 1.

(a)

(b)

(a)

(b)

5.3. Problem 3

Here, we consider the fractional nonlinear Volterra integrodifferential equation [51]:

The exact solution of this problem is . The approximate solutions for different values of and are presented in Table 4. The Stehfest parameter used in this experiment is . The graphs of approximate solutions for various values of are depicted in Figure 5(a), whereas Figure 3(b) presents the plot of error estimate vs. observed error for . In Figure 6(a), for and , the error function is shown, and in Figure 6(b) for different nodes and for is shown.

(a)

(b)

(a)

(b)

5.4. Problem 4

Here, we consider the fractional system of nonlinear Volterra integrodifferential equations [51]:

The exact solutions of this system are and . The approximate solution of for various values of and and is using the two numerical schemes , corresponding to Problem 3 presented in Table 5, and the approximate solution of for various values of and and is presented in Table 6. In Table 7, the errors obtained using the two proposed schemes are shown. For and various nodes for , the error function is shown in Figure 7(a), and for , the error function is shown in Figure 7(b). The plots of numerical solutions of for different fractional-order are displayed in Figure 8(a), and the plots of numerical solutions of for various values of are displayed in Figure 8(b).

(a)

(b)

(a)

(b)

5.5. Problem 5

Here, we consider the fractional system of nonlinear Volterra integrodifferential equations [31]:

The exact solutions are , and . The numerical solutions for various values of and are presented in Table 8.

The absolute errors for , and quadrature nodes using the CIM are presented in Table 9. The plots of numerical solutions of for different fractional-orders are displayed in Figure 9(a), and the plots of numerical solutions of for various values are displayed in Figure 9(b). In Figure 10(a), the plots of error function of for , and are displayed, and in Figure 10(b), the plots of error function of for , and are displayed. The plots of error function of for various nodes and and are displayed in Figure 11(a), and the plots of error function of for quadrature nodes and are displayed in Figure 11(b).

(a)

(b)

(a)

(b)

(a)

(b)

6. Conclusion

In this work, we have successfully applied the Laplace transform to fractional VIDEs. The numerical inversion of Laplace transform is performed using the CIM and SM methods. In our numerical experiments, we considered linear, nonlinear, and system of fractional VIDEs with the ABC derivative. The obtained results led us to the conclusion that the two proposed methods are capable of solving such type of equations. However, it was observed that the CIM is more accurate and converges faster than SM. Hence, the two proposed numerical schemes are the best alternative methods for solving such type of equations.

Data Availability

All data required for this paper is included within this paper.

Conflicts of Interest

The authors do not have any competing interests.

Authors’ Contributions

Xiaoli Qiang wrote the paper, Kamran proved the results, Abid Mahboob revised the paper and arranged the funding, and Yu-Ming Chu supervised this work.

Acknowledgments

The research was supported by the National Natural Science Foundation of China (Grant Nos. 11971142, 11871202, 61673169, 11701176, 11626101, and 11601485).

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