# Numerical Approximation of Fractional-Order Volterra Integrodifferential Equation

Authors: Xiaoli Qiang; Kamran; Abid Mahboob; Yu-Ming Chu
Journal: Journal of Function Spaces
 (2020)

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## 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.


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## Body

## 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
(1)0ABCDtβζt=Ht,ζt,Tζt,(2)Tζt=∫0tBt,τQζtdτ,(3)ζ0=α,where β∈0,1,0≤t,τ≤1,α∈R,H,B,Q∈C0,1.Here,0ABCDtβ is the ABC derivative of order β.


### 1.1. Preliminaries

Definition 1.
The ABC fractional derivative of order0<β<1, of ζ∈S1a,b with base point a at t∈a,b, is defined as [46]
(4)0ABCDtβζt=Mβ1−β∫atζ′sEβ−β1−βt−sβds,where S1 is a Sobolev-space of first-order fitted with L2-norm defined over the Ω⊂R defined as follows:
(5)S1Ω=ζ∈L2Ω:ζ′∈L2Ω,where Mβ is defined as follows:
(6)Mβ=βΓβ+1−β;also, Eβt is a one-parameter Mittage-Leffler (ML) function defined as
(7)Eβt=∑m=0∞tmΓmβ+1,β>0,−∞<t<∞.Definition 2.
The ABC fractional integral of order0<β<1 of ζ∈S1a,b with base point a at t∈a,b is defined as [46]
(8)Itβζt=1−βMβζt+βMβΓβ∫atζst−sβ−1ds.Definition 3.
The Laplace transform of a piecewise continues functionζt is defined as
(9)Lζt=ζ^s=∫0∞e−stζtdt.Definition 4.
If0<β≤1, then the Laplace transform of the ABC derivative is defined by
(10)L0ABCDtβζt=Mβsβζ^s−Mβsβ−1ζ0sβ1−β+β.


## 1.1. Preliminaries

Definition 1.
The ABC fractional derivative of order0<β<1, of ζ∈S1a,b with base point a at t∈a,b, is defined as [46]
(4)0ABCDtβζt=Mβ1−β∫atζ′sEβ−β1−βt−sβds,where S1 is a Sobolev-space of first-order fitted with L2-norm defined over the Ω⊂R defined as follows:
(5)S1Ω=ζ∈L2Ω:ζ′∈L2Ω,where Mβ is defined as follows:
(6)Mβ=βΓβ+1−β;also, Eβt is a one-parameter Mittage-Leffler (ML) function defined as
(7)Eβt=∑m=0∞tmΓmβ+1,β>0,−∞<t<∞.Definition 2.
The ABC fractional integral of order0<β<1 of ζ∈S1a,b with base point a at t∈a,b is defined as [46]
(8)Itβζt=1−βMβζt+βMβΓβ∫atζst−sβ−1ds.Definition 3.
The Laplace transform of a piecewise continues functionζt is defined as
(9)Lζt=ζ^s=∫0∞e−stζtdt.Definition 4.
If0<β≤1, then the Laplace transform of the ABC derivative is defined by
(10)L0ABCDtβζt=Mβsβζ^s−Mβsβ−1ζ0sβ1−β+β.


## 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
(11)Mβsβζ^s−Mβsβ−1ζ0sβ+1−sββ=H^,which can be rearranged as
(12)ζ^s=G^ssβMβ,where
(13)G^s=Mβsβ−1α+H^sβ1−sββ.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 solutionζt of (1) as Bromwich integral:
(14)ζt=12πι∫Γestζ^sds,Res≥ϑ,where ϑ∈R and Γ is an initially appropriately chosen line Γ0 in a complex plane parallel to the y axis, with Ims→±∞. Then, in (14), ζt is the inverse transform of ζbs, satisfying the condition that all the singularities of ζbs lie to the left of Γ0. However, for our purposes, we assume that ζ^s may be continued analytically in an appropriate way; we shall want to take for Γ0 a deformed contour Γ in Σϕϑ=0∪s≠0:∣args∣<ϕ, which has asymptotic behavior in the left complex plane, with Res→−∞ when Ims→±∞, which force est to decay towards both ends of Γ. In our work, we choose Γ as
(15)sγ=ϑ+ρ−ρsinη−ιγ,γ∈R,where
(16)0<η<ϕ−π2,ρ>0,ϑ>0.Lettings=ς+ιζ, we see that (15) represents the left branch of the hyperbola given by
(17)ς−ϑ−ρρsinη2−ζρcosη2=1,where ζ=±ς−ϑ−ρcotη are the asymptotes for (17) and s=ϑ+ρ1−sinη is its x-intercept. It is confirmed from Eq. (16) that Γ lies in the sector Σϑφ=ϑ+Σφ⊂Σφ and grows into the left half plane. From (15) and (14), we get
(18)ζt=12πι∫−∞∞esγτζ^sγs′γdγ.The approximation of Eq. (18) using the trapezoidal rule with uniform step k is given as
(19)ζkt=k2πι∑j=−NNesjtζ^sjsj′,γj=jk,sj=sγj,s′j=s′γj.


### 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 Γ. In the process of numerical approximation of the Bromwich integral (18), the orders of convergence greatly depend on the step k of the trapezoidal rule and the temporal domain t0,T. 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).
Letζt be the solution of (1) with ζ^s analytic in Σϑφ. Let Γ⊂Ωr⊂Σϑφ and define coshb=θτ1sinη−1, for b>0, where 0<τ0<T, τ1=t0/T, 0<θ<1.0, and let β=θr¯N/bT. Then, for Eq. (19), with k=b/N≤r¯/log2, we have ∣ζt−ζkt∣≤CQeϑτ1l∥ζ0∥+∥H^∥ΣϕϑρrNe−μN, for μ=r¯1−θ/b, ρr=θr¯τ1sinη−r1/b, r¯=2πr1, r1>0, τ0≤t≤T,lx=max1,log1/x, and C=Cη,r1,β1. Hence, we have
(20)errorest=∣ζkt−ζt∣=Oe−μNlρrN.


## 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 Γ. In the process of numerical approximation of the Bromwich integral (18), the orders of convergence greatly depend on the step k of the trapezoidal rule and the temporal domain t0,T. 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).
Letζt be the solution of (1) with ζ^s analytic in Σϑφ. Let Γ⊂Ωr⊂Σϑφ and define coshb=θτ1sinη−1, for b>0, where 0<τ0<T, τ1=t0/T, 0<θ<1.0, and let β=θr¯N/bT. Then, for Eq. (19), with k=b/N≤r¯/log2, we have ∣ζt−ζkt∣≤CQeϑτ1l∥ζ0∥+∥H^∥ΣϕϑρrNe−μN, for μ=r¯1−θ/b, ρr=θr¯τ1sinη−r1/b, r¯=2πr1, r1>0, τ0≤t≤T,lx=max1,log1/x, and C=Cη,r1,β1. Hence, we have
(20)errorest=∣ζkt−ζt∣=Oe−μNlρrN.


## 4. Stehfest Method (SM)

In the Stehfest method, the approximate value ofζt is given as
(21)ζt=ln2t∑i=1Mwiζ^ln2ti,where the weights wi are given by
(22)wi=−1M/2+i∑h=i+1/2mini,M/2hM/22h!M/2−h!h!h−1!i−h!2h−i!.Solving (12) for the corresponding Laplace parameters s=ln2/ti,i=1,2,3,⋯,M.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 j significant digits are desired, then let M be the positive integer 1.1j. Given M, set the system precision at 2.2M. Given M and the system precision, calculate the weights wi, 1≤i≤2M, using (22). Then, given the transform ζ^s and the argument t, calculate ζt in (21).” According to these conclusions, the error is 10−j+1≤ζ^−ζ/ζ≤10−j, where M=1.1j [50].


## 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 j significant digits are desired, then let M be the positive integer 1.1j. Given M, set the system precision at 2.2M. Given M and the system precision, calculate the weights wi, 1≤i≤2M, using (22). Then, given the transform ζ^s and the argument t, calculate ζt in (21).” According to these conclusions, the error is 10−j+1≤ζ^−ζ/ζ≤10−j, where M=1.1j [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 areθ=0.1,τ=t0/T,η=0.15410,r¯=2πr,r=0.13870,t0,T=0.5,5,ϑ=2.0. The MATLAB command γ=−N:k:N is used to generate the quadrature nodes for CIM. The parameter used in our numerical computation for the Stehfest method is M.


### 5.1. Problem 1

Here, we consider the fractional linear Volterra integrodifferential equation [31]:
(23)0ABCDtβζt=1−∫0tζτdτ,t,β∈0,1.The exact solution of this problem isζt=sint. In Table 1, the approximate solutions of ζt=sint for various values of β and t∈0,1 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 N is shown in Figure 1(a), and the plot of the error function for t∈0,1 is shown in Figure 1(b). The graphs of approximate solutions for β are presented in Figure 2(b), and Figure 2(a) displays the plot absolute error vs. error estimate for β=1. The numerical results using the Stehfest method are obtained by using M=18. All the plots are obtained using the CIM. It is evident that these methods can solve such type of equations efficiently.Table 1
The approximate solutions for various values ofβ and t∈0,1 corresponding to Problem 1.
tβ=1β=0.9β=0.8β=0.7CIMSMCIMSMCIMSMCIMSM0.10.09980.09980.21710.21710.33490.33490.45190.45190.20.19870.19870.31630.31630.43120.43120.54160.54160.30.29550.29550.40700.40700.51310.51310.61180.61180.40.38940.38940.49050.49050.58390.58390.66800.66800.50.47940.47940.56680.56680.64470.64470.71200.71200.60.56460.56460.63550.63550.69590.69590.74500.74500.70.64420.64420.69650.69650.73760.73760.76730.76730.80.71740.71740.74920.74920.76980.76980.77940.77940.90.78330.78340.79340.79340.79250.79250.78160.781610.84150.84150.82860.82870.80570.80570.77400.7740Table 2
For various values ofβ,N,M the absolute errors corresponding to Problem 1.
(N,M)β=1β=0.9β=0.8CIMSMCIMSMCIMSM(30, 6)3.19×10−41.15×10−11.03×10−27.88×10−23.10×10−23.18×10−2(60, 8)5.21×10−62.56×10−21.28×10−21.64×10−23.58×10−24.30×10−3(90, 10)4.25×10−84.70×10−31.28×10−21.51×10−23.58×10−23.58×10−2(120, 12)6.69×10−111.90×10−31.28×10−21.46×10−23.58×10−23.75×10−2(150, 14)7.99×10−126.96×10−51.28×10−21.28×10−23.58×10−23.59×10−2(180, 16)1.13×10−136.41×10−51.28×10−21.28×10−23.58×10−23.57×10−2(210, 18)2.68×10−152.97×10−71.28×10−21.28×10−23.58×10−23.58×10−2(240, 20)2.55×10−155.45×10−61.28×10−21.28×10−23.58×10−23.58×10−2Figure 1
In (a), the error function for quadrature nodesN and β=1 corresponding to Problem 1 is shown. In (b), the absolute error is shown for β=1; a very high accuracy is evident.
(a)(b)Figure 2
In (a), the absolute error versus error estimate corresponding to Problem 1 forβ=1 is shown. From the figure, we see that the observed error agrees with the error estimate. In (b), the plots of approximate solutions for various β are shown.
(a)(b)


### 5.2. Problem 2

Here, we consider the fractional nonlinear Volterra integrodifferential equation [31]:
(24)0ABCDtβζt=−1+∫0tζ2τdτ,t,β∈0,1.The problem has exact solutionζt=1/28t4−t1/12t3+1−1. The Stehfest parameter used is M=18. The numerical results for various values of β and t∈0,1 are presented in Table 3. The graphs of numerical solutions for various values of β are shown in Figure 3(a), whereas Figure 3(b) presents the graph of error estimate vs. absolute error for β=1. For t∈0,1 and β=1, the error function is shown in Figure 4(a), and the error function for various N, and β=1 is shown in Figure 4(b). A similar performance is observed as was observed for Problem 1.Table 3
The approximate solutions for various values ofβ using the two numerical schemes corresponding to Problem 2.
tβ=1β=0.9β=0.8β=0.7CIMSMCIMSMCIMSMCIMSM0.1-0.0999-0.0999-0.2177-0.2177-0.3360-0.3360-0.4535-0.45350.2-0.1998-0.1998-0.3194-0.3194-0.4362-0.4362-0.5486-0.54860.3-0.2993-0.2993-0.4149-0.4149-0.5251-0.5251-0.6279-0.62790.4-0.3978-0.3978-0.5056-0.5056-0.6056-0.6056-0.6961-0.69610.5-0.4947-0.4947-0.5914-0.5914-0.6785-0.6785-0.7547-0.75470.6-0.5891-0.5891-0.6718-0.6718-0.7435-0.7435-0.8034-0.80340.7-0.6799-0.6799-0.7457-0.7457-0.7997-0.7997-0.8416-0.84160.8-0.7658-0.7658-0.8119-0.8119-0.8459-0.8459-0.8680-0.86800.9-0.8453-0.8453-0.8690-0.8690-0.8808-0.8808-0.8812-0.88121-0.9166-0.9166-0.9153-0.9153-0.9025-0.9025-0.8796-0.8796Figure 3
In (a), the graphs of numerical solutions for different values ofβ corresponding to Problem 2 are shown. In (b), the absolute error vs. error estimates for β=1 are shown.
(a)(b)Figure 4
In (a), absolute error versust for β=1 corresponding to Problem 2 is shown. In (b), the plot of error versus N for β=1 is shown.
(a)(b)


### 5.3. Problem 3

Here, we consider the fractional nonlinear Volterra integrodifferential equation [51]:
(25)0ABCDtβζt=et1+et−t2/41−e2t+2te2t−ζ2t+∫0tt2τζ2τdτ,t,β∈0,1.The exact solution of this problem isζt=et. The approximate solutions for different values of β and t∈0,1 are presented in Table 4. The Stehfest parameter used in this experiment is M=22. The graphs of approximate solutions for various values of β are depicted in Figure 5(a), whereas Figure 3(b) presents the plot of error estimate vs. observed error for β=1. In Figure 6(a), for β=1 and t∈0,1, the error function is shown, and in Figure 6(b) for different nodes N and for β=1 is shown.Table 4
The approximate solutions for different values ofβ using the two numerical schemes N=240, M=22, corresponding to Problem 3.
tβ=1β=0.9β=0.8β=0.7CIMSMCIMSMCIMSMCIMSM0.11.10521.10541.24911.25011.41141.41131.58941.59170.21.22141.22231.38931.38961.57441.57581.77191.77300.31.34991.35061.53831.53981.74261.74411.95581.95710.41.49181.49291.69941.69961.92131.92262.14922.14960.51.64871.64941.87491.87512.11392.11572.35602.35750.61.82211.82232.06672.06782.32292.32452.57952.57950.72.01382.01452.27712.27812.55082.55202.82262.82410.82.22552.22612.50822.50862.80032.80143.08803.08880.92.45962.45992.76232.76333.07383.07533.37863.377012.71832.71833.04223.04203.37423.37433.69743.6982Figure 5
In (a), the numerical solutions for various values ofβ corresponding to Problem 3 are shown. In (b), absolute error vs. error estimates for β=1 are shown.
(a)(b)Figure 6
In (a), the error function fort∈0,1 and β=1 and in (b), for different quadrature nodes N and β=1 for Problem 3 are shown.
(a)(b)


### 5.4. Problem 4

Here, we consider the fractional system of nonlinear Volterra integrodifferential equations [51]:
(26)0ABCDtβζ1t=1+t3+t4−∫0tt2ζ1τ+ζ2τdτ,t,β∈0,1,Dtβζ2t=1−t32−t43+∫0ttζ1τζ2τdτ,t,β∈0,1.The exact solutions of this system areζ1t=t and ζ2t=1+t. The approximate solution of ζ1t for various values of β and t∈0,1 and M=22 is using the two numerical schemes N=240,M=22, corresponding to Problem 3 presented in Table 5, and the approximate solution of ζ2t for various values of β and t∈0,1 and M=20 is presented in Table 6. In Table 7, the errors obtained using the two proposed schemes are shown. For β=1 and various nodes N for ζ1t, the error function is shown in Figure 7(a), and for ζ2t, the error function is shown in Figure 7(b). The plots of numerical solutions of ζ1t for different fractional-order β are displayed in Figure 8(a), and the plots of numerical solutions of ζ2t for various values of β are displayed in Figure 8(b).Table 5
The approximate solutions ofζ1 for various values of β using the two numerical schemes N=240,M=22, corresponding to Problem 4.
tβ=1β=0.9β=0.8β=0.7CIMSMCIMSMCIMSMCIMSM0.10.10.10.23120.23130.37890.37910.54060.54090.20.20.20.33950.33950.49260.49270.65500.65540.30.30.30.44220.44220.59500.59530.75260.75290.40.40.40.54150.54160.69060.69100.84080.84080.50.50.50.63840.63850.78150.78180.92250.92280.60.60.60.73330.73320.86880.86870.99940.99960.70.70.70.82660.82670.95330.95341.07261.07300.80.80.80.91860.91871.03541.03531.14261.14310.90.90.91.00951.00961.11541.11581.21011.21021111.09931.09951.19361.19391.27541.2756Table 6
The approximate solutions ofζ2 for various values of α using the two numerical schemes N=240,M=20, corresponding to Problem 4.
tβ=1β=0.9β=0.8β=0.7CIMSMCIMSMCIMSMCIMSM0.11.11.11.23121.23121.37891.37891.54061.54060.21.21.21.33951.33951.49261.49261.65501.65490.31.31.31.44221.44211.59501.59491.75261.75260.41.41.41.54151.54151.69061.69061.84081.84080.51.51.51.63841.63841.78151.78161.92251.92250.61.61.61.73331.73331.86881.86881.99941.99950.71.71.71.82661.82661.95331.95332.07262.07260.81.81.81.91861.91862.03542.03532.14262.14270.91.91.92.00952.00952.11542.11532.21012.21011222.09932.09932.19362.19372.27542.2754Table 7
Absolute errors forβ=1 using the two numerical schemes for different values of N and M corresponding to Problem 4.
(N,M)CIMSMζ1ζ2ζ1ζ2(30,6)3.19×10−42.06×10−22.10×10−32.10×10−3(60, 8)5.21×10−69.70×10−51.54×10−41.54×10−4(90, 10)4.25×10−82.60×10−63.47×10−53.47×10−5(120, 12)6.69×10−117.08×10−89.62×10−79.62×10−7(150, 14)7.99×10−125.92×10−103.60×10−73.63×10−7(180, 16)1.13×10−139.22×10−124.49×10−83.38×10−8(210, 18)3.88×10−153.38×10−132.01×10−73.88×10−7(240, 20)1.00×10−155.71×10−151.07×10−63.38×10−5Figure 7
In (a), absolute error versusN for fractional-order β=1 corresponding to Problem 4 for ζ1 is shown. In (b), the absolute error versus N for β=1 corresponding to Problem 4 for ζ2 is shown.
(a)(b)Figure 8
In (a), the numerical solutions corresponding to Problem 4 for fractional-orderβ for ζ1t are shown. In (b), the numerical solutions corresponding to Problem 4 for fractional-order β for ζ2t are shown.
(a)(b)


### 5.5. Problem 5

Here, we consider the fractional system of nonlinear Volterra integrodifferential equations [31]:
(27)0ABCDtβζ1t=1−ζ12t2+∫0tt−τζ2τ+ζ1τζ2τdτ,t,τ,β∈0,1,Dtβζ2t=2t+∫0tt−τζ1τ−1dτ,t,τ,β∈0,1.The exact solutions areζ1t=sinht, and ζ2t=cosht. The numerical solutions for various values of β and t∈0,1 are presented in Table 8.Table 8
The approximate solutions for various values ofβ using the CIM numerical schemes for N=240, corresponding to example 5.
tβ=1β=0.9β=0.8β=0.7ζ1ζ2ζ1ζ2ζ1ζ2ζ1ζ20.10.10021.00500.23191.01720.38031.03110.54281.04660.20.20131.02010.34331.04600.49931.07520.66481.10710.30.30451.04530.45251.08580.61181.13080.77661.17920.40.41081.08110.56291.13660.72381.19750.88681.26240.50.52111.12760.67641.19850.83831.27560.99931.35670.60.63671.18550.79471.27200.95751.36531.11711.46240.70.75861.25520.91941.35761.08351.46731.24231.58030.80.88811.33741.05201.45611.21791.58231.37681.71120.91.02651.43311.19401.56841.36251.71131.52261.856111.17521.54311.34691.69531.51901.85521.68132.0161The absolute errors forβ=1, and quadrature nodes N using the CIM are presented in Table 9. The plots of numerical solutions of ζ1t for different fractional-orders β are displayed in Figure 9(a), and the plots of numerical solutions of ζ2t for various values β are displayed in Figure 9(b). In Figure 10(a), the plots of error function of ζ1t for β=1, and t∈0,1 are displayed, and in Figure 10(b), the plots of error function of ζ2t for β=1, and t∈0,1 are displayed. The plots of error function of ζ1t for various nodes N and β=1 and are displayed in Figure 11(a), and the plots of error function of ζ2t for quadrature nodes N and β=1 are displayed in Figure 11(b).Table 9
The absolute error fractional-orderβ=1 and different quadrature nodes N using the CIM numerical scheme, corresponding to Problem 5.
Nζ1ζ2303.18×10−42.03×10−2605.21×10−69.17×10−5904.25×10−82.61×10−61206.69×10−117.07×10−81507.99×10−125.84×10−101801.13×10−139.34×10−122104.21×10−153.40×10−132401.99×10−156.65×10−15Figure 9
In (a), the numerical solutions corresponding to Problem 5 for fractional-orderβ for ζ1t are shown. In (b), the numerical solutions corresponding to Problem 5 for fractional-order β for ζ2t are shown.
(a)(b)Figure 10
In (a), forβ=1 and t∈0,1, the error function of ζ1t corresponding to Problem 5 is shown. In (b), for β=1 and t∈0,1, the error function of ζ2t corresponding to Problem 5 is shown.
(a)(b)Figure 11
In (a), the plot shows various nodesN along Γ and β=1 the error function of ζ1t corresponding to Problem 5. In (b), the plot shows various nodes N along Γ and β=1 the error function of ζ2t corresponding to Problem 5.
(a)(b)


## 5.1. Problem 1

Here, we consider the fractional linear Volterra integrodifferential equation [31]:
(23)0ABCDtβζt=1−∫0tζτdτ,t,β∈0,1.The exact solution of this problem isζt=sint. In Table 1, the approximate solutions of ζt=sint for various values of β and t∈0,1 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 N is shown in Figure 1(a), and the plot of the error function for t∈0,1 is shown in Figure 1(b). The graphs of approximate solutions for β are presented in Figure 2(b), and Figure 2(a) displays the plot absolute error vs. error estimate for β=1. The numerical results using the Stehfest method are obtained by using M=18. All the plots are obtained using the CIM. It is evident that these methods can solve such type of equations efficiently.Table 1
The approximate solutions for various values ofβ and t∈0,1 corresponding to Problem 1.
tβ=1β=0.9β=0.8β=0.7CIMSMCIMSMCIMSMCIMSM0.10.09980.09980.21710.21710.33490.33490.45190.45190.20.19870.19870.31630.31630.43120.43120.54160.54160.30.29550.29550.40700.40700.51310.51310.61180.61180.40.38940.38940.49050.49050.58390.58390.66800.66800.50.47940.47940.56680.56680.64470.64470.71200.71200.60.56460.56460.63550.63550.69590.69590.74500.74500.70.64420.64420.69650.69650.73760.73760.76730.76730.80.71740.71740.74920.74920.76980.76980.77940.77940.90.78330.78340.79340.79340.79250.79250.78160.781610.84150.84150.82860.82870.80570.80570.77400.7740Table 2
For various values ofβ,N,M the absolute errors corresponding to Problem 1.
(N,M)β=1β=0.9β=0.8CIMSMCIMSMCIMSM(30, 6)3.19×10−41.15×10−11.03×10−27.88×10−23.10×10−23.18×10−2(60, 8)5.21×10−62.56×10−21.28×10−21.64×10−23.58×10−24.30×10−3(90, 10)4.25×10−84.70×10−31.28×10−21.51×10−23.58×10−23.58×10−2(120, 12)6.69×10−111.90×10−31.28×10−21.46×10−23.58×10−23.75×10−2(150, 14)7.99×10−126.96×10−51.28×10−21.28×10−23.58×10−23.59×10−2(180, 16)1.13×10−136.41×10−51.28×10−21.28×10−23.58×10−23.57×10−2(210, 18)2.68×10−152.97×10−71.28×10−21.28×10−23.58×10−23.58×10−2(240, 20)2.55×10−155.45×10−61.28×10−21.28×10−23.58×10−23.58×10−2Figure 1
In (a), the error function for quadrature nodesN and β=1 corresponding to Problem 1 is shown. In (b), the absolute error is shown for β=1; a very high accuracy is evident.
(a)(b)Figure 2
In (a), the absolute error versus error estimate corresponding to Problem 1 forβ=1 is shown. From the figure, we see that the observed error agrees with the error estimate. In (b), the plots of approximate solutions for various β are shown.
(a)(b)


## 5.2. Problem 2

Here, we consider the fractional nonlinear Volterra integrodifferential equation [31]:
(24)0ABCDtβζt=−1+∫0tζ2τdτ,t,β∈0,1.The problem has exact solutionζt=1/28t4−t1/12t3+1−1. The Stehfest parameter used is M=18. The numerical results for various values of β and t∈0,1 are presented in Table 3. The graphs of numerical solutions for various values of β are shown in Figure 3(a), whereas Figure 3(b) presents the graph of error estimate vs. absolute error for β=1. For t∈0,1 and β=1, the error function is shown in Figure 4(a), and the error function for various N, and β=1 is shown in Figure 4(b). A similar performance is observed as was observed for Problem 1.Table 3
The approximate solutions for various values ofβ using the two numerical schemes corresponding to Problem 2.
tβ=1β=0.9β=0.8β=0.7CIMSMCIMSMCIMSMCIMSM0.1-0.0999-0.0999-0.2177-0.2177-0.3360-0.3360-0.4535-0.45350.2-0.1998-0.1998-0.3194-0.3194-0.4362-0.4362-0.5486-0.54860.3-0.2993-0.2993-0.4149-0.4149-0.5251-0.5251-0.6279-0.62790.4-0.3978-0.3978-0.5056-0.5056-0.6056-0.6056-0.6961-0.69610.5-0.4947-0.4947-0.5914-0.5914-0.6785-0.6785-0.7547-0.75470.6-0.5891-0.5891-0.6718-0.6718-0.7435-0.7435-0.8034-0.80340.7-0.6799-0.6799-0.7457-0.7457-0.7997-0.7997-0.8416-0.84160.8-0.7658-0.7658-0.8119-0.8119-0.8459-0.8459-0.8680-0.86800.9-0.8453-0.8453-0.8690-0.8690-0.8808-0.8808-0.8812-0.88121-0.9166-0.9166-0.9153-0.9153-0.9025-0.9025-0.8796-0.8796Figure 3
In (a), the graphs of numerical solutions for different values ofβ corresponding to Problem 2 are shown. In (b), the absolute error vs. error estimates for β=1 are shown.
(a)(b)Figure 4
In (a), absolute error versust for β=1 corresponding to Problem 2 is shown. In (b), the plot of error versus N for β=1 is shown.
(a)(b)


## 5.3. Problem 3

Here, we consider the fractional nonlinear Volterra integrodifferential equation [51]:
(25)0ABCDtβζt=et1+et−t2/41−e2t+2te2t−ζ2t+∫0tt2τζ2τdτ,t,β∈0,1.The exact solution of this problem isζt=et. The approximate solutions for different values of β and t∈0,1 are presented in Table 4. The Stehfest parameter used in this experiment is M=22. The graphs of approximate solutions for various values of β are depicted in Figure 5(a), whereas Figure 3(b) presents the plot of error estimate vs. observed error for β=1. In Figure 6(a), for β=1 and t∈0,1, the error function is shown, and in Figure 6(b) for different nodes N and for β=1 is shown.Table 4
The approximate solutions for different values ofβ using the two numerical schemes N=240, M=22, corresponding to Problem 3.
tβ=1β=0.9β=0.8β=0.7CIMSMCIMSMCIMSMCIMSM0.11.10521.10541.24911.25011.41141.41131.58941.59170.21.22141.22231.38931.38961.57441.57581.77191.77300.31.34991.35061.53831.53981.74261.74411.95581.95710.41.49181.49291.69941.69961.92131.92262.14922.14960.51.64871.64941.87491.87512.11392.11572.35602.35750.61.82211.82232.06672.06782.32292.32452.57952.57950.72.01382.01452.27712.27812.55082.55202.82262.82410.82.22552.22612.50822.50862.80032.80143.08803.08880.92.45962.45992.76232.76333.07383.07533.37863.377012.71832.71833.04223.04203.37423.37433.69743.6982Figure 5
In (a), the numerical solutions for various values ofβ corresponding to Problem 3 are shown. In (b), absolute error vs. error estimates for β=1 are shown.
(a)(b)Figure 6
In (a), the error function fort∈0,1 and β=1 and in (b), for different quadrature nodes N and β=1 for Problem 3 are shown.
(a)(b)


## 5.4. Problem 4

Here, we consider the fractional system of nonlinear Volterra integrodifferential equations [51]:
(26)0ABCDtβζ1t=1+t3+t4−∫0tt2ζ1τ+ζ2τdτ,t,β∈0,1,Dtβζ2t=1−t32−t43+∫0ttζ1τζ2τdτ,t,β∈0,1.The exact solutions of this system areζ1t=t and ζ2t=1+t. The approximate solution of ζ1t for various values of β and t∈0,1 and M=22 is using the two numerical schemes N=240,M=22, corresponding to Problem 3 presented in Table 5, and the approximate solution of ζ2t for various values of β and t∈0,1 and M=20 is presented in Table 6. In Table 7, the errors obtained using the two proposed schemes are shown. For β=1 and various nodes N for ζ1t, the error function is shown in Figure 7(a), and for ζ2t, the error function is shown in Figure 7(b). The plots of numerical solutions of ζ1t for different fractional-order β are displayed in Figure 8(a), and the plots of numerical solutions of ζ2t for various values of β are displayed in Figure 8(b).Table 5
The approximate solutions ofζ1 for various values of β using the two numerical schemes N=240,M=22, corresponding to Problem 4.
tβ=1β=0.9β=0.8β=0.7CIMSMCIMSMCIMSMCIMSM0.10.10.10.23120.23130.37890.37910.54060.54090.20.20.20.33950.33950.49260.49270.65500.65540.30.30.30.44220.44220.59500.59530.75260.75290.40.40.40.54150.54160.69060.69100.84080.84080.50.50.50.63840.63850.78150.78180.92250.92280.60.60.60.73330.73320.86880.86870.99940.99960.70.70.70.82660.82670.95330.95341.07261.07300.80.80.80.91860.91871.03541.03531.14261.14310.90.90.91.00951.00961.11541.11581.21011.21021111.09931.09951.19361.19391.27541.2756Table 6
The approximate solutions ofζ2 for various values of α using the two numerical schemes N=240,M=20, corresponding to Problem 4.
tβ=1β=0.9β=0.8β=0.7CIMSMCIMSMCIMSMCIMSM0.11.11.11.23121.23121.37891.37891.54061.54060.21.21.21.33951.33951.49261.49261.65501.65490.31.31.31.44221.44211.59501.59491.75261.75260.41.41.41.54151.54151.69061.69061.84081.84080.51.51.51.63841.63841.78151.78161.92251.92250.61.61.61.73331.73331.86881.86881.99941.99950.71.71.71.82661.82661.95331.95332.07262.07260.81.81.81.91861.91862.03542.03532.14262.14270.91.91.92.00952.00952.11542.11532.21012.21011222.09932.09932.19362.19372.27542.2754Table 7
Absolute errors forβ=1 using the two numerical schemes for different values of N and M corresponding to Problem 4.
(N,M)CIMSMζ1ζ2ζ1ζ2(30,6)3.19×10−42.06×10−22.10×10−32.10×10−3(60, 8)5.21×10−69.70×10−51.54×10−41.54×10−4(90, 10)4.25×10−82.60×10−63.47×10−53.47×10−5(120, 12)6.69×10−117.08×10−89.62×10−79.62×10−7(150, 14)7.99×10−125.92×10−103.60×10−73.63×10−7(180, 16)1.13×10−139.22×10−124.49×10−83.38×10−8(210, 18)3.88×10−153.38×10−132.01×10−73.88×10−7(240, 20)1.00×10−155.71×10−151.07×10−63.38×10−5Figure 7
In (a), absolute error versusN for fractional-order β=1 corresponding to Problem 4 for ζ1 is shown. In (b), the absolute error versus N for β=1 corresponding to Problem 4 for ζ2 is shown.
(a)(b)Figure 8
In (a), the numerical solutions corresponding to Problem 4 for fractional-orderβ for ζ1t are shown. In (b), the numerical solutions corresponding to Problem 4 for fractional-order β for ζ2t are shown.
(a)(b)


## 5.5. Problem 5

Here, we consider the fractional system of nonlinear Volterra integrodifferential equations [31]:
(27)0ABCDtβζ1t=1−ζ12t2+∫0tt−τζ2τ+ζ1τζ2τdτ,t,τ,β∈0,1,Dtβζ2t=2t+∫0tt−τζ1τ−1dτ,t,τ,β∈0,1.The exact solutions areζ1t=sinht, and ζ2t=cosht. The numerical solutions for various values of β and t∈0,1 are presented in Table 8.Table 8
The approximate solutions for various values ofβ using the CIM numerical schemes for N=240, corresponding to example 5.
tβ=1β=0.9β=0.8β=0.7ζ1ζ2ζ1ζ2ζ1ζ2ζ1ζ20.10.10021.00500.23191.01720.38031.03110.54281.04660.20.20131.02010.34331.04600.49931.07520.66481.10710.30.30451.04530.45251.08580.61181.13080.77661.17920.40.41081.08110.56291.13660.72381.19750.88681.26240.50.52111.12760.67641.19850.83831.27560.99931.35670.60.63671.18550.79471.27200.95751.36531.11711.46240.70.75861.25520.91941.35761.08351.46731.24231.58030.80.88811.33741.05201.45611.21791.58231.37681.71120.91.02651.43311.19401.56841.36251.71131.52261.856111.17521.54311.34691.69531.51901.85521.68132.0161The absolute errors forβ=1, and quadrature nodes N using the CIM are presented in Table 9. The plots of numerical solutions of ζ1t for different fractional-orders β are displayed in Figure 9(a), and the plots of numerical solutions of ζ2t for various values β are displayed in Figure 9(b). In Figure 10(a), the plots of error function of ζ1t for β=1, and t∈0,1 are displayed, and in Figure 10(b), the plots of error function of ζ2t for β=1, and t∈0,1 are displayed. The plots of error function of ζ1t for various nodes N and β=1 and are displayed in Figure 11(a), and the plots of error function of ζ2t for quadrature nodes N and β=1 are displayed in Figure 11(b).Table 9
The absolute error fractional-orderβ=1 and different quadrature nodes N using the CIM numerical scheme, corresponding to Problem 5.
Nζ1ζ2303.18×10−42.03×10−2605.21×10−69.17×10−5904.25×10−82.61×10−61206.69×10−117.07×10−81507.99×10−125.84×10−101801.13×10−139.34×10−122104.21×10−153.40×10−132401.99×10−156.65×10−15Figure 9
In (a), the numerical solutions corresponding to Problem 5 for fractional-orderβ for ζ1t are shown. In (b), the numerical solutions corresponding to Problem 5 for fractional-order β for ζ2t are shown.
(a)(b)Figure 10
In (a), forβ=1 and t∈0,1, the error function of ζ1t corresponding to Problem 5 is shown. In (b), for β=1 and t∈0,1, the error function of ζ2t corresponding to Problem 5 is shown.
(a)(b)Figure 11
In (a), the plot shows various nodesN along Γ and β=1 the error function of ζ1t corresponding to Problem 5. In (b), the plot shows various nodes N along Γ and β=1 the error function of ζ2t corresponding to Problem 5.
(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.



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