COMPLEXITYComplexity1099-05261076-2787Hindawi10.1155/2020/10103821010382Research ArticleApproximate Analytical Solution for a Coupled System of Fractional Nonlinear Integrodifferential Equations by the RPS Methodhttps://orcid.org/0000-0002-9384-7133Al e’damatAyedXuHongleiDepartment of MathematicsFaculty of ScienceAl-Hussein Bin Talal UniversityP.O. Box 20Ma’anJordanahu.edu.jo202013820202020030220201807202013820202020Copyright © 2020 Ayed Al e’damat.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this work, a modified residual power series method is implemented for providing efficient analytical and approximate solutions for a class of coupled system of nonlinear fractional integrodifferential equations. The proposed algorithm is based on the concept of residual error functions and generalized power series formula. The fractional derivative is described under the Caputo concept. To illustrate the potential, accuracy, and efficiency of the proposed method, two numerical applications of the coupled system of nonlinear fractional integrodifferential equations are tested. The numerical results confirm the theoretical predictions and depict that the suggested scheme is highly convenient, is quite effective, and practically simplifies computational time. Consequently, the proposed method is simple, accurate, and convenient in handling different types of fractional models arising in the engineering and physical systems.

1. Introduction

The theory of fractional calculus is indeed a generalization of the standard calculus that deals with differentiation and integration of a noninteger order, which is utilized to describe various real-world phenomena arising in natural sciences, applied mathematics, and engineering fields with great applications for these tools, including nonlinear oscillation of earthquakes, fractional fluid-dynamic traffic, economics, solid mechanics, viscoelasticity, and control theory . The major cause behind this is that modeling of a specific phenomenon does not depend only at the time instant but also the historical state, so the differential and integral operators for integer and fractional cases are found to be a superb tool in describing the hereditary and memory properties for different engineering and physical phenomena . However, several mathematical forms of the abovementioned issues contain nonlinear fractional integrodifferential equations (FIDEs), and other nonlinear models can be found in . Since most fractional differential and integrodifferential equations cannot be solved analytically, it is necessary to find an accurate numerical and analytical method to deal with the complexity of fractional operators involving such equations .

This paper aims to introduce a recent analytical as well as numerical method based on the use of fractional residual power series (RPS) technique for obtaining the approximate solution for a class of coupled system of fractional integrodifferential equations in the following form:(1)Dβv1t=λ1abK1t,ξJ1v1ξ,v2ξdξ+λ2atK2t,ξJ2v1ξ,v2ξdξ+f1t,Dβv2t=λ3abH1t,ξT1v1ξ,v2ξdξ+λ4atH2t,ξT2v1ξ,v2ξdξ+f2t.

This is subject to the following initial conditions:(2)v10=v1,0,v20=v2,0,where 0<β1.

The residual power series method (RPSM) has a wide range of applications, especially in simulating nonlinear issues in a fractional meaning, which has been developed and modified over recent years as a powerful mathematical treatment indispensable in dealing with the emerging realistic system in physics, engineering, and natural sciences . More specifically, it is a modern analytical and approximation technique that relies on the expansion of the fractional power series and residual error functions, which was first proposed in 2013 to provide analytical series solutions to fuzzy differential equations of the first and second orders and minimize the residual errors. This method has many advantages and properties as follows: it is an accurate alternative instrument, it requires less effort to achieve results, it provides a rapid convergence rate to the exact solution, it deals directly with different types of nonlinear terms and complex functions, and it has the ability to choose any point in the integration domain, making the approximate solution applicable. It has also excellent estimating characteristics that reflect high reliability. Furthermore, it is a systematic tool to find sequential solutions for several types of nonlinear differential equations and integrodifferential equations of fractional order without having to discretize, linearize, or even expose to perturbation. Therefore, it attracted the attention of many researchers.

Freihet et al.  have used the fractional power series for solving the fractional stiff system and introduced some basic theorems related to RPS generalization in the sense of Caputo fractional derivative. The (2 + 1)-dimensional time-fractional Burgers–Kadomtsev–Petviashvili equation has been solved by the RPS method . In , analytic-approximate solutions for nonlinear coupled fractional Jaulent–Miodek equations with energy-dependent Schrödinger potential have been obtained using the RPS and q-homotopy analysis methods. Moreover, this method was successfully applied for solving both linear and nonlinear ordinary, partial, and fuzzy differential equations . Therefore, such adaptives can be used as an alternative technique in solving several nonlinear problems arising in engineering and sciences.

The outline of this paper is organized as follows: In the next section, we review some basic definitions and theories related to fractional differentiation and fractional power series representations. In Section 3, the solution by the RPS technique is provided. In Section 4, numerical application is performed to show accuracy and efficiency of the RPS method. Finally, we give concluding remarks in Section 5.

2. Basic Mathematical Concepts

In this section, basic definitions and results related to fractional calculus are given and fractional power series concept is also represented.

Definition 1.

The Riemann–Liouville fractional integral operator of order β, over the interval a,b for a function vL1a,b, is defined by(3)Ja+βvx=1Γβaxvεxε1βdε,0<ε<x,β>0,vt,β=0.

For β,α0 and μ1, the operator Ja+β has the following basic properties:

Ja+βxaμ=Γμ+1/Γμ+1+βxaμ+β

Ja+βJa+αvx=Ja+β+αvx

Ja+βJa+αvx=Ja+αJa+βvx

Definition 2.

For β>0,a,t,β. The Caputo fractional derivative of order β is given by(4)Da+βvx=1Γnβaxvnεxεβn+1 dε.

For n1<β<n,n. In case β=n and n, then Da+βvx=dn/dxnvx.

The following are some interesting properties of the operator Da+β:

For any constant c, then Da+βc=0.

Da+βxaμ=Γμ+1/Γμ+1βxaμβ,n1<βn,μ>n1,nN,qR,0,otherwise.

Da+βJa+βvx=vx.

Ja+βDa+βvx=vxm=0n1vma+/k!xam.

Definition 3.

A fractional power series (FPS) representation at t=a has the following form:(5)m=0vmxamβ=v0+v1xaβ+v2xa2β+,where 0n1<βn and xa and vm is the coefficient of the series.

Theorem 1.

Suppose that vx has the following FPS representation at t=a:(6)vx=m=0vmxamβ,where n1<βn,a<x<a+R,φxCa,a+R, and Da+mβφxCa,a+R for m=0,1,2,. Then, the coefficients vm will be in the form vm=Da+mβva/Γmβ+1 such that Da+mβ=Da+βDa+βDa+β (m-times).

3. Fractional RPS Method for the Coupled System of IDEs

The purpose of this section is to construct FPS solution for the coupled system of nonlinear fractional integrodifferential equations (1) and (2) by substituting the FPS expansion among the truncated residual functions.

The RPS algorithm proposing the solution of equations (1) and (2) about x0=0 gives the following FPS expansion:(7)vix=m=0vi,mxmβΓmβ+1,i=1,2.

The truncated series form of equation (7) can be given by the following kth-FPS approximate solution:(8)vi,kx=m=0kvi,mxmβΓmβ+1,i=1,2.

Clearly, if vi0=vi,0,i=1,2, then expansion (8) can be written as(9)vi,kx=vi,0+m=1kvi,mxmβΓmβ+1,i=1,2.

Define the residual function for equations (1) and (2) as follows:(10)Res1x=D0+αv1tλ1abK1t,ξJ1v1ξ,v2ξdξλ2atK2t,ξJ2v1ξ,v2ξdξf1t,Res2x=D0+αv2tλ3abH1t,ξT1v1ξ,v2ξdξλ4atH2t,ξT2v1ξ,v2ξdξf2t.

Also, define the kth-residual function as(11)Res1,kx=D0+αv1,ktλ1abK1t,ξJ1v1,kξ,v2,kξdξλ2atK2t,ξJ2v1,kξ,v2,kξdξf1t,Res2,kx=D0+αv2,ktλ3abH1t,ξT1v1,kξ,v2,kξdξλ4atH2t,ξT2v1,kξ,v2,kξdξf2t.

According to the RPS algorithm , we have the following relations:

limkResi,kx=Resx=0, for each x0,1,i=1,2.

D0+mβRes0=D0+mβResi,k0=0, for each m=0,1,2,,k,i=1,2.

For obtaining the coefficients vi,m, m=0,1,2,,k,i=1,2, one can solve the solution of the following relation:(12)D0+k1βResi,k0=0,k=1,2,3,,i=1,2.

Algorithm 1.

To find the coefficients vi,m,m=1,2,3,,k,i=1,2, in equation (9), perform the following steps:

Step 1: substitute expansion (9) function vi,kx,i=1,2, into k-th residual function (11) such that Res1,kt=D0+αv1,0+m=1kv1,mtmβ/Γmβ+1λ1abK1t,ξJ1v1,0+m=1kv1,mξmβ/Γmβ+1,v2,0+m=1kv2,mξmβ/Γmβ+1dξλ2atK2t,ξJ2v1,0+m=1kv1,mξmβ/Γmβ+1,v2,0+m=1kv2,mξmβ/Γmβ+1dξf1t,Res2.kt=D0+αv2,0+m=1kv2,mtmβ/Γmβ+1λ3abH1t,ξT1v1,0+m=1kv1,mξmβ/Γmβ+1,v2,0+m=1kv2,mξmβ/Γmβ+1dξλ4atH2t,ξT2v1,0+m=1kv1,mξmβ/Γmβ+1,v2,0+m=1kv2,mξmβ/Γmβ+1dξf2t.

Step 2: find the relation of fractional formula Dx0k1β of Resi,kx at x=x0,i=1,2.

Step 3: for k=1, obtain the relation through the fact Resi,1xx=0=0,i=1,2. For k=2, obtain the relation through the fact D0+βResi,2xx=0=0,i=1,2. For k=3, obtain the relation through the fact D0+2βResi,3xx=0=0,i=1,2,. For k=m, obtain the relation through the fact D0+m1βResi,mxx=0=0,i=1,2.

Step 4: solve the obtained algebraic fractional system Dx0k1βResi,kx0,k=1,2,3,,i=1,2.

Step 5: substitute the values of vi,m back into equation (8) and then stop.

4. Numerical Applications and Simulation

This section aims to test two applications of the system of nonlinear fractional IDEs to show the efficiency, accuracy, and applicability of the proposed method. In this section, all calculations are preformed using Wolfram-Mathematica 10.

Example 1.

Consider the following nonlinear fractional integrodifferential equation:(13)Dαv1x+0xtv1tv2tdt+01tx5v1t+v2tdt=exx+2x52x5e+ex1+x,Dαv2x+0xtv1t+v2tdt+01tx5v1tv2tdt=2+ex1+x+2x5eex2+x.

This is subject to the following initial conditions:(14)v10=v20=1.

The exact solution of this coupled system is v1x=ex and v2x=ex.

Using the RPS algorithm, the k-th residual functions Res1,kx and Res2,kx are given by(15)Res1,kx=Dαv1,mx+0xtv1,mtv2,mtdt+01tx5v1,mt+v2,mtdtexx+2x52x5e+ex1+x,Res2,kx=Dαv2,mx+0xtv1,mt+v2,mtdt+01tx5v1,mtv2,mtdt2+ex1+x+2x5eex2+x,where vi,mx has the following form:(16)vi,mx=1+m=1kvi,mxmβΓmβ+1,i=1,2.

Consequently,(17)Res1,kx=Dα1+m=1kv1,mxmβΓmβ+1+0xtm=1kv1,mv2,mtmβΓmβ+1dt+01tx52+m=1kv1,m+v2,mtmβΓmβ+1dtexx+2x52x5e+ex1+x,Res2,kx=Dα1+m=1kv2,mxmβΓmβ+1+0xt2+m=1kv1,m+v2,mtmβΓmβ+1dt+01tx5m=1kv1,mv2,mtmβΓmβ+1dt2+ex1+x+2x5eex2+x.

Numerical simulation specializes in advanced numeric or approximate methods for finding digital or approximate solutions and estimating errors of these approximations. For such purpose, a few decimal numbers are often recorded to calculate the absolute error, which produces some vague estimates related to significance and units, as well as does not provide clear and explicit evidence regarding the subject matter of the study. Therefore, the comparison of absolute error with the exact value leads to the determination of the relative error as a ratio between the value of absolute error and the exact, which gives some importance and reduces ambiguity for a deeper understanding of the behavior of the approximate solutions. By using the RPS method of Example 1, the numerical results of v1t and v2t are shown in Tables 1 and 2 at β=1 and k=8. The results obtained in Tables 1 and 2 show that the error estimate using the proposed method is very small and that the solutions correspond well to each other. In general, it should be noted that increasing the number of iterations k will lead to an improvement in numerical solutions and approaching the exact value.

Tables 3 and 4 show the sixth approximate solutions of Example 1 at different values of β such that β1,0.9,0.8,0.7 with step size 0.1 and k=6. From these tables, it can be concluded that the RPS algorithm and the approximate solutions are consistent with each other and with the exact solutions for all values of t in 0,1. Here, it is worth noting that the closer the value of the fractional derivative approaching the integer case β=1, the closer the approximate solution is to the exact solution.

Numerical results of exact v1t and approximate v1,8t solutions for Example 1 at β=1 and k=8 over the interval [0, 1].

tExact solution v1tApproximate solution v1,8tAbsolute error v1tv1,8t
01.01.00.0
0.11.10517091.10517088.4700×108
0.21.22140281.22140002.7581×106
0.31.34985881.34983752.1307×105
0.41.49182471.49173339.1364×105
0.51.64872131.64843752.8377×104
0.61.82211881.82140007.1880×104
0.72.01375272.01217081.5819×103
0.82.22554092.22240003.1409×103
0.92.45960312.45383755.7656×103
1.02.71828182.70833339.9485×103

Numerical results of exact v2t and approximate v2,8t solutions for Example 1 at β=1 and k=8 over the interval [0, 1].

tExact solution v2tApproximate solution v2,8tAbsolute error v2tv2,8t
01.01.00.0
0.10.90483740.90483758.1900×108
0.20.81873080.81873332.5802×106
0.30.74081820.74083751.9279×105
0.40.670332010.67040007.9954×105
0.50.60653070.60677082.4017×104
0.60.54881160.54940005.8836×104
0.70.49658530.49783751.2522×103
0.80.44932900.45173332.4043×103
0.90.40656970.41083754.2678×103
1.00.36787940.37500007.1206×103

Numerical results of sixth approximate solution v1,6t for Example 1 at different values of fractional order β, k=6, and t0,1.

tβ=1β=0.9β=0.8β=0.7
0.01.01.01.01.0
0.11.05108331.07339611.10738841.1601510
0.21.12466671.17168231.23921291.3365098
0.31.22125001.29563811.39742291.5359285
0.41.34133331.44461541.58015781.7560644
0.51.48541671.61817481.78607761.9951767
0.61.65400001.81603452.01420212.2519736
0.71.84758332.03802172.26378742.5254605
0.82.06666672.28404012.53425262.8148458
0.92.31175002.55404952.82513333.1194830
1.02.58333332.84805183.13605143.4388329

Numerical results of sixth approximate solution v2,6t for Example 1 at different values of fractional order β, k=6, and t0,1.

tβ=1β=0.9β=0.8β=0.7
0.01.01.01.01.0
0.10.90483330.87807790.84606940.8088467
0.20.81866660.78554780.74993950.7123611
0.30.74049990.70717360.67345200.6391004
0.40.66933330.63862100.60841740.5772768
0.50.60416660.57720550.55052850.5214466
0.60.54399990.52091630.49696430.4684158
0.70.48783330.46810270.44565630.4160595
0.80.43466660.41733750.39498000.3628482
0.90.38350000.36734490.34359910.3076204
1.00.33333330.31695920.29037840.2494581
Example 2.

Consider the following fractional integrodifferential equation:(18)Dαv1x+0xv2ξdξ+01ξv1ξdξ=cos1+cosx+sin1+sinx,Dαv2x+0xv1ξdξ+01ξv2ξdξ=cos1cosx+sin1sinx.

This is subject to the following initial conditions:(19)v10=v20=1.

The exact solution of this coupled system is v1x=sinx and v2x=cosx. Using the RPS algorithm, the k-th residual functions Res1,kx and Res2,kx are given by(20)Res1,kx=Dαv1,mx+0xv2,mtdt+01v1,mtdtcosx+sinxcos1+sin1,Res2,kx=Dαv2,mx+0xv1,mtdt+01v2,mtdtcosxsinx+sin1+cos1,where v1,mx and v2,mx have the following form:(21)v1,mx=m=1kv1,mxmβΓmβ+1,v2,mx=1+m=1kv2,mxmβΓmβ+1.

The absolute errors are listed in Tables 5 and 6. The results obtained by the RPS method show that the exact solutions are in good agreement with approximate solutions at β=1, k=6, and step size 0.1. Tables 7 and 8 show approximate solutions at different values of β such that β0.9,0.8,0.7 and k=6 with step size 0.1. From these tables, one can find that the RPS method provides us with an accurate approximate solution, which is in good agreement with the exact solutions for all values of t in 0,1. Also, it is worth noting that the closer the value of the fractional derivative approaching the integer case β=1, the closer the approximate solution is to the exact solution.

Numerical results of exact v1t and approximate v1,6t solutions for Example 2 at β=1 and k=6 over the interval [0, 1].

tExact solution v1tApproximate solution v1,6tAbsolute error v1tv1,6t
0.10.09983340.09982577.7205×106
0.20.19866930.19866246.8911×106
0.30.29552020.29552161.3995×106
0.40.38941830.38941523.1144×106
0.50.47942550.47942322.3173×106
0.60.56464250.56464926.7460×106
0.70.64421770.64421992.2046×106
0.80.71735610.71735332.8095×106
0.90.78332690.78332861.7189×106
1.00.84147100.84144752.3458×105

Numerical results of exact v2t and approximate v2,6t solutions for Example 2 at β=1 and k=6 over the interval [0, 1].

tExact solution v1tApproximate solution v2,6tAbsolute error v2tv2,6t
0.10.99500420.99501036.1745×106
0.20.98006660.98005976.9024×106
0.30.95533650.95532709.4863×106
0.40.92106100.92104109.6762×106
0.50.87758260.87757715.4999×106
0.60.82533560.82532698.7210×106
0.70.76484220.76456222.2974×105
0.80.69670670.69669848.3385×106
0.90.62161000.62159861.1407×105
1.00.54030230.54026693.5406×105

Numerical results of approximate solution v1,6t for Example 2 for different values of β,k=6, and t0,1.

tβ=0.9β=0.8β=0.7
0.00.00.00.0
0.10.07495010.10278650.1422048
0.20.15536180.20210550.2644892
0.30.24514620.30866290.3897369
0.40.34455770.42310600.5195960
0.50.45358020.54543020.6544017
0.60.57214340.67549410.7941689
0.70.70016950.81312910.9388129
0.80.83758540.95816931.0882176
0.90.98432691.11046091.2422602
1.01.14033821.26986351.4008207

Numerical results of approximate solution v2,6t of Example 2 for different values of β,k=6, and t0,1.

tβ=0.9β=0.8β=0.7
0.01.01.01.0
0.11.00368390.99957980.9902709
0.20.99181770.97706930.9526430
0.30.96781310.94094530.9018037
0.40.93296890.89384120.8413882
0.50.88811390.83727770.7733064
0.60.83385920.77229220.6987749
0.70.77069010.69965780.6186510
0.80.69901080.61998320.5335787
0.90.61916850.53376640.4440635
1.00.53146840.44142620.3505153
5. Concluding Remarks

In this work, a class of a coupled system of nonlinear fractional integrodifferential equations of fractional order has been discussed by using the RPS method under the Caputo fractional derivative. The RPS algorithm has been given to optimize the approximate solution by minimizing a residual error function with the help of generalized Taylor formula. To demonstrate the consistency with the theoretical framework, two illustrative examples have been provided. The obtained results indicated that the approximate solutions are coinciding with the exact solution and with each other for different values of the fractional order over the selected nods and parameters. From our results, we can conclude that the RPS method is a systematic and suitable scheme to address many fractional initial value problems with great potential in scientific applications. The calculations have been performed by using Wolfram-Mathematica 10.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The author declares that there are no conflicts of interest.