# Application of ARA-Residual Power Series Method in Solving Systems of Fractional Differential Equations

Authors: Ahmad Qazza; Aliaa Burqan; Rania Saadeh
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this research, systems of linear and nonlinear differential equations of fractional order are solved analytically using the novel interesting method: ARA- Residual Power Series (ARA-RPS) technique. This approach technique is based on the combination of the residual power series scheme with the ARA transform to establish analytical approximate solutions in a fast convergent series representation using the concept of the limit. The proposed method needs less time and effort compared with the residual power series technique. To prove the simplicity, applicability, and reliability of the presented method, three numerical examples are proposed and simulated. The obtained results show that the ARA-RPS technique is applicable, simple, and effective to get solutions to linear and nonlinear engineering and physical problems.


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## Body

## 1. Introduction

Numerous phenomena in various fields of science can be fruitfully formulated by the use of fractional derivatives. This is because the sensible modeling for a physical phenomenon depends on instantaneous time as well as on prior time history; hence, we may use fractional calculus to deal with these problems [1–10].Many physical and engineering problems can be formulated by fractional differential equations (FDEs) and obtaining the solutions of these equations have been the theme of many interesting investigations and have attracted the attention of researchers. Recently, there have been a large number of techniques dedicated to get solutions of FDEs [11–16]. These techniques can be classified into two categories, approximate and analytical, such as transform methods, homotopy analysis methods, and residual power series methods. [17–25].On the other hand, there are a lot of contributions related to solving systems of FDEs and since the finding of the analytical solution for a system of FDEs sometimes required a complex computation, analytical, and numerical techniques were established and developed to search for solutions of linear and nonlinear FDEs, see [26–36].In the literature, various integral transforms such as Laplace, Fourier, Sumudu, and ARA transforms have been introduced to deal with differential equations. ARA transform was introduced first in 2020, which is a powerful technique in solving differential equations and systems. One advantage of this transform is that it can solve differential equations with singular points around zero. Also, ARA transform is applicable for some functions that the Laplace transform cannot be applied to; see [37].In this paper, we employ the ARA-RPS method to investigate analytical and approximate solutions of linear and nonlinear systems of FDEs. The ARA-RPS technique that has been introduced in [39] is a combination of the ARA transform [37–39] and the residual power series method [40–47].To achieve our goal of finding the solutions of systems of FDEs, we operate the ARA transform on the original system and implement appropriate series expansions to solve the new system. The expansion coefficients are determined based on the concepts of limits at infinity, unlike the fractional power series (FPS) method which may take more time in the solution procedure.This work is organized as follows: in the second section, we introduce some basic definitions and theorems of ARA transform and fractional derivatives. The methodology of the ARA-RPS method is presented in section three. Numerical examples and simulations are introduced in section four.


## 2. Basic Definitions and Theorems

In this section, we introduce the definition of the fractional operators Riemann–Liouville and Caputo. Definition of the ARA transform and some properties and theorems of the fractional ARA-Tayler’s series representations are also revisited.Definition 1.
The Riemann–Liouville fractional integral of the functionyt of order α>0 is defined by(1)Itαyt=1Γα∫att−τα−1yτdτ,where Γα is the gamma function.Definition 2.
The Caputo fractional derivative of the functionyt of order α>0 is defined by(2)Dαyt=1Γm−α∫0tymτt−τα+1−mdτ,m−1<α<m,ymt,α=m,where m is a nonnegative integer.Definition 3.
[37] The ARA transform of order n of the continuous function yt on the interval 0,∞, is defined by(3)Gnyts=s∫0∞tn−1e−stytdt,s>0.
The inverse of the ARA transform is given by(4)yt=−1n2πi∫c−i∞c+i∞est−1n1sΓn−1∫0ss−xn−1Gn+1ytxdx+∑k=0n−1skk!∂kY0∂skds,where c=Res and(5)Ys=∫0∞e−stytdt.
In the following arguments, we present some basic properties of the ARA transform [39] that are essential in our research.
Letyt and ht be two continuous functions on the interval 0,∞ for which the ARA transform exists. Then we have for s>0(1)
Gnαyt+βhts=αGnyts+βGnhts, where α and β are nonzero constants.(2)
lims⟶∞G1yts=y0.(3)
G1y′ts=sG1yts−sy0.(4)
G2yts=−sd/dsG1yts/s.(5)
G2y′ts=−sd/dsG1yts.(6)
G2tαs=Γα+2/sα+1,s>0,α>0.(7)
G1Dαyts=1/sm−αG1ymts,m−1<α≤m.(8)
G2Dαyts=sαG2yts−αsα−1G1yts+α−1sα−1y0,0<α≤1.(9)
lims⟶∞sG2yt=y0.Theorem 1.
[26] Suppose that the FPS representation of the function yt at t=0 has the form(6)yt=∑n=0∞antnα,m−1<α≤m,m=1,2,…,0≤t≤β.
Ifyt and Dnαyt are continuous on 0,β, for each n=0,1,…, then the coefficients an′s have the form
an=Dnαy0/Γnα+1, for n=0,1,2,…,
whereDnα=Dα⋅Dα…Dα (n− times).Theorem 2.
[39] Let yt be a continuous function on every finite interval 0,β in which the ARA transform of order two exists and has the FPS representation:(7)G2yts=∑n=0∞hnsnα+1;0<α≤1s>0.
Then(8)hn=nα+1Dnαy0.Remarks(1)
The inverse ARA transform of order two for the FPS representation in Theorem2 has the following form(9)yt=∑n=0∞Dnαy0tnαΓnα+1.(2)
The ARA transform of order one of the functionyt has the following presentation(10)G1yts=∑n=0∞hnnα+1snα.For the convergence analysis and conditions of the previous theorem, we propose the following results.Theorem 3.
[39] Let yt be a continuous function on every finite interval 0,β in which the ARA transform exists. Assume that G1yts has the following series representation:(11)G1yts=∑n=0∞Cnsnα,where(12)Cn=Dnαy0.IfG1Dn+1αyts≤M on 0<s≤d, then the remainder Rns, of the above series, when we consider only the first n terms, satisfies the following inequality:(13)Rns≤Msn+1α,0<s≤d.


## 3. Methodology of the ARA-RPSM

Many phenomena in the fields of engineering and physics could be regulated by fractional initial value problems to simulate and describe some features of these phenomena. Analytical methods are used to get approximate solutions for some fractional initial value problems that we could not find out their exact solutions for. One of these techniques is the ARA-RPS method that depends on coupling the residual power series method with the ARA transform.To perform the ARA-RPS method, consider the following system of FDEs in the sense of the Caputo derivative(14)Dαyit=fit,y1t,y2t,…,ymt,t≥0,0<α≤1,with the initial conditions (ICs)(15)yi0=ci,i=1,…,m,where fi are continuous real valued functions defined on 0,∞×ℝm and yit are unknown analytical functions to be determined.Firstly, we apply the ARA transform of order twoG2 on both sides of each equation in system (14)(16)G2Dαyits=G2fit,y1t,y2t,…,ymts.Using property 8 and the ICs in (15), we get(17)sαG2yits−αsα−1G1yits+α−1sα−1ci=Fis,where Fis=G2fit,y1t,y2t,…,ymts.Hence (17) can be written as(18)G2yits=αsG1yits−α−1sci+Fissα,fori=1,…,m.Assuming that the ARA-RPS solutions of equations in (18) have the series expansions(19)G1yits=∑n=0∞hi,nnα+1snα,(20)G2yits=∑n=0∞hi,nsnα+1,fori=1,2,…,m.Using the fact that(21)lims⟶∞sG2yits=yi0=ci,we have hi,0=ci for i=1,…m and so the kth truncated series expansions of (19) and (20) have the form(22)G1yitks=∑n=0khi,nnα+1snα=ci+∑n=1khi,nnα+1snα,(23)G2yitks=∑n=0khi,nsnα+1=cis+∑n=1khi,nsnα+1.To find the coefficients of the series expansions in (22) and (23), we define the ARA-residual functions of the (18) as follows:(24)G2Resis=G2yits−αsG1yits+α−1sci−Fissα,fori=1,…,m.and the kth ARA-residual functions are(25)G2Reskis=G2yitks−αsG1yitks+α−1sci−Fissα,fori=1,…,m,k=1,2,⋯.The following facts are necessary to obtain the ARA-RPS solutions:(i)
G2Resis=0.(ii)
limk⟶∞G2Reskis=G2Resis, for s>0.(iii)
lims⟶∞sG2Resis=0 and lims⟶∞sG2Reskis=0.(iv)
lims⟶∞skα+1G2Resis=lims⟶∞skα+1G2Reskis=0, for 0<α≤1,i=1,2,…,m, k=1,2,….To find the coefficientshi,n’s in the series expansion (24) and (25), we solve the system(26)lims⟶∞skα+1G2Reskis=0,fori=1,…,m and k=1,2,….Then the obtained coefficientshi,n are substituted in the series solution (23) and then apply the inverse ARA transform of order two G2−1 to get the solution of system (14) in the original space.


## 4. Numerical Examples

In this section, we introduce three interesting examples on systems of linear and nonlinear FDEs. These examples demonstrate the applicability and efficiency of the presented method.Example 1.
Consider the linear system of FDEs(27)Dαy1t=y1t+y2t,Dαy2t=−y1t+y2t,0≤t≤1,0<α≤1,with the ICs(28)y10=0,y20=1.
Solution: Applying the ARA transform of order twoG2, on both sides of the equations in system (27), we have(29)G2Dαy1ts=G2y1ts+G2y2ts,G2Dαy2ts=G2−y1ts+G2y2ts.
Running the ARA transform on the equations in system (29), we have(30)sαG2y1ts−αsα−1G1y1ts+α−1sα−1y10=G2y1ts+G2y2ts,sαG2y2ts−αsα−1G1y2ts+α−1sα−1y20=−G2y1ts+G2y2ts.
Substituting the ICs (28) and simplifying the equations in system (30), we get(31)G2y1ts−αsG1y1ts−1sαG2y1ts−1sαG2y2t=0,G2y2ts−αsG1y2ts+α−1s+1sαG2y1ts−1sαG2y2ts=0.
Assume that the ARA-RPS solutions of (31) have the following series representations:(32)G1y1ts=∑n=0∞h1,nnα+1snα,G2y1ts=∑n=0∞h1,nsnα+1,G1y2ts=∑n=0∞h2,nnα+1snα,G2y2ts=∑n=0∞h2,nsnα+1.
Using the fact in property 9(33)lims⟶∞sG2yits=yi0,i=1,2,…,
we get the coefficientsh1,0=0,h2,0=1, and so the kth ARA-RPS solutions of system (31) have the form(34)G1y1tks=∑n=1kh1,nnα+1snα,G2y1tks=∑n=1kh1,nsnα+1,G1y2tks=1+∑n=1kh2,nnα+1snα,G2y2tks=1s+∑n=1kh2,nsnα+1.
Define the ARA-residual functions of the equations in system (31) as follows:(35)G2Res1s=G2y1ts−αsG1y1ts−1sαG2y1ts−1sαG2y2ts,G2Res2s=G2y2ts−αsG1y2ts+α−1s+1sαG2y1ts−1sαG2y2ts,and the kth ARA-residual functions of system (35)(36)G2Resk1s=G2y1tks−αsG1y1tks−1sαG2y1tks−1sαG2y2tks,G2Resk2s=G2y2tks−αsG1y2tks+α−1s+1sαG2y1tks−1sαG2y2tks.
To find the first unknown coefficientsh1,1 and h2,1 of the series expansions (34), we substitute the first truncated expansions G1y1t1s, G1y2t1s, G2y1t1s, and G2y2t1s into the first ARA-residual functions G2Res11s and G2Res12s in system (36) to get(37)G2Res11s=h1,1sα+1−αh1,1α+1sα+1−h1,1s2α+1−1sα+1−h2,1s2α+1,G2Res12s=1s+h2,1sα+1−αs−αh2,1α+1sα+1+α−1s+h1,1s2α+1−1sα+1−h2,1s2α+1,
which can be written as(38)G2Res11s=1sα+1h1,1−αh1,1α+1−1−1s2α+1h1,1−h2,1,G2Res12s=1sα+1h2,1−αh2,1α+1−1+1s2α+1h1,1−h2,1.
Simplifying the equations in system (38) and multiplying the resulting equations by sα+1, we obtain(39)sα+1G2Res11s=h1,1−αh1,1α+1−1−1sαh1,1−h2,1,sα+1G2Res12s=h2,1−αh2,1α+1−1+1sαh1,1−h2,1.
Taking the limit ass⟶∞ to both sides of equations in system (39), we have(40)h1,1=α+1,h2,1=α+1.
To find the second unknown coefficientsh1,2 and h2,2 of the series expansions (34), we substitute the second truncated expansion G1y1t2s, G1y2t2s, G2y1t2s, and G2y2t2s into the second ARA-residual functions G2Res21s and G2Res22s in system (39) to get(41)G2Res21s=h1,1sα+1+h1,2s2α+1−αh1,1α+1sα+1−αh1,22α+1s2α+1−h1,1s2α+1−h1,2s3α+1−1sα+1−h2,1s2α+1−h2,2s3α+1,G2Res22s=h2,1sα+1+h2,2s2α+1−αh2,1α+1sα+1−αh2,22α+1s2α+1+h1,1s2α+1+h1,2s3α+1−1sα+1−h2,1s2α+1−h2,2s3α+1.
Simplifying the equations in system (41) and multiplying both sides by s2α+1, we get(42)s2α+1G2Res21s=sαh1,1−αh1,1α+1−1+h1,2−αh1,22α+1−h1,1−h2,1−h1,2sα−h2,2sα,s2α+1G2Res22s=sαh2,1−αh2,1α+1−1+h2,2−αh2,22α+1+h1,1−h2,1+h1,2sα−h2,2sα.
Substituting the values ofh1,1 and h2,1 then taking the limit as s⟶∞ to both sides of equations in system (42), we have(43)h1,2=22α+1,h2,2=0.
By following the same steps as before, and based on the fact
lims⟶∞skα+1G2Reskis=0 for k=1,2,…,i=1,2.
We can conclude that(44)h1,3=23α+1,h2,3=−23α+1,h1,4=0,h2,4=−44α+1,h1,5=−45α+1,h2,5=−45α+1,h1,6=−86α+1,h2,6=0,h1,7=−87α+1,h2,7=87α+1.
Consequently, the ARA-RSP solutions of system (31) are(45)G2y1ts=α+1sα+1+22α+1s2α+1+23α+1s3α+1−45α+1s5α+1−86α+1s6α+1−87α+1s7α+1+…,G2y2ts=1s+α+1sα+1−23α+1s3α+1−44α+1s4α+1−45α+1s5α+1+87α+1s7α+1+…..
Furthermore, to get the series solutions in the original space of system (27), we apply the inverse ARA transform of order two G2−1 to both sides of equations in system (45) to get(46)y1t=tαΓα+1+2t2αΓ2α+1+2t3αΓ3α+1−4t5αΓ5α+1−8t6αΓ6α+1−8t7αΓ7α+1+…,y2t=1+tαΓα+1−2t3αΓ3α+1−4t4αΓ4α+1−4t5αΓ5α+1+8t7αΓ7α+1+…..
Forα=1, the series expansions (46) have the form, y1t=etsint and y2t=etcost which are coincide with the exact solutions for the ordinary system (27) and (28).
Table1, shows a comparison of the ARA-RPS solution and the exact solution of Example 1 with α=1 at various values of t and k=8. The absolute error is also presented in Table 1. Numerical results of Example 1 with k=8, and different values of α are listed in Table 2. In addition, the comparison of the results obtained for the exact solution corresponding to, α=1 and the numerical solutions given by the ARA-RPS method for different values of α, α=0.6,α=0.7,α=0.8, and α=0.9 are plotted in Figure 1. Figure 2 portrays a very precise agreement of the exact solutions y1t,y2t and the ARA-RPS solutions y18t,y28t of Example 1 at different time levels with fixed α. The obtained results show that the ARA-RPS method is efficient in obtaining approximate solutions of systems of FDEs. This is obvious in the following tables, which illustrate the small errors between the ARA-RPS solutions and the exact ones.Table 1
Numerical results ofy1t and y2t for Example 1 at α=1 and k=8.
tiNumerical results ofy1tExact solutionARA-RPSy1t−y18t0.160.18696164740873700.186961647405608643.1283586832×10−120.320.43319834435612560.43319834270261641.65350916292×10−90.480.74626950505832910.74626943944528456.56130446641×10−80.641.13256973575185801.13256883391640809.01835450096×10−70.801.59650534060025121.59649840761904786.93298120336×10−60.962.13947972998967062.13944282753082553.69024588451×10−5tiNumerical results ofy2tExact solutionARA-RPSy2t−y28t0.161.15852194918101061.15852194917798213.0284663666×10−120.321.30721844913237931.30721844758408651.54829282728×10−90.481.43344978957866331.43344973021052915.93681341865×10−80.641.52115926819917481.52115848049068087.87708493988×10−70.8 01.55054929680742241.55054345752380955.83928361286×10−60.961.49786012506311051.49783018956072072.99355023898×10−5Table 2
Approximate solutions ofy1t and y2t at k=8, with different values of α, for Example 1.
tiApproximate solution ofy1tα=1α=0.95α=0.75α=0.550.250.3176729720.3596605420.6057639041.0462071100.500.7904389880.8748652611.3180037991.8340297270.751.4430254261.5696241752.1313542122.2935493291.002.2873015872.4428447052.9398344652.158261999tiApproximate solution ofy2tα=1α=0.95α=0.75α=0.550.251.2441081761.2645026351.3383033441.2927803990.501.4468889511.4546740611.3822642120.8916433910.751.5489820751.5122671291.1164869850.0752149091.001.4686507941.3484872780.461189999−1.010101627867Figure 1
(a) Plots of the exact solutiony1t and the approximate solutions y18t at various α values and (b) plots of the exact solution y2t and the approximate solutions y28t at various α values, for Example 1.
(a)(b)Figure 2
Plots of the exact solutionsy1t,y2t and the ARA-RPS solutions y18t,y28t at α=1 of Example 1.Example 2.
Consider the nonlinear system of FDEs(47)Dαy1t=−1002y1t+1000y22t,Dαy2t=y1t−y2t−y22t,0≤t≤1,0<α≤1,
with the ICs(48)y10=1,y20=1.
Solution: Applying the ARA transform of order twoG2, on both sides of the equations in system (47)(49)G2Dαy1ts=−1002G2y1ts+1000G2y22ts,G2Dαy2ts=G2y1ts−G2y2ts−G2y22ts.
Running the ARA transform on equations in system (49), we get(50)sαG2y1ts−αsα−1G1y1ts+α−1sα−1y10=−1002G2y1ts+1000G2y22ts,sαG2y2ts−αsα−1G1y2ts+α−1sα−1y20=G2y1ts−G2y2ts−G2y22ts.
Substituting the ICs (48) and simplifying system (50), to get(51)G2y1ts−αsG1y1ts+α−1s+1002sαG2y1ts−1000sαG2G2−1G2y2t2s=0,G2y2ts−αsG1y2ts+α−1s−G2y1tssα+G2y2tssα+G2G2−1G2y2t2ssα=0.
That the ARA-RPS solutions of system (51) have the series representations(52)G1y1ts=∑n=0∞h1,nnα+1snα,G2y1ts=∑n=0∞h1,nsnα+1,G1y2ts=∑n=0∞h2,nnα+1snα,G2y2ts=∑n=0∞h2,nsnα+1.
Using the fact that(53)lims⟶∞sG2yits=yi0,i=1,2,we get the coefficients h1,0=1,h2,0=1, and so the kth ARA-RPS solutions of system (51) have the form(54)G1y1tks=1+∑n=1kh1,nnα+1snα,G2y1tks=1s+∑n=1kh1,nsnα+1,G1y2tks=1+∑n=1kh2,nnα+1snα,G2y2tks=1s+∑n=1kh2,nsnα+1.
Define the ARA-residual functions of the equations in system (51) as follows:(55)G2Res1s=G2y1ts−αsG1y1ts+α−1s+1002sαG2y1ts−1000sαG2G2−1G2y2ts2s,G2Res2s=G2y2ts−αsG1y2ts+α−1s−G2y1tssα+G2y2tssα+G2G2−1G2y2ts2ssα,and the kth ARA-residual functions of the equations in system (55)(56)G2Resk1s=G2y1tks−αsG1y1tks+α−1s+1002sαG2y1tks−1000sαG2G2−1G2y2tks2ks,G2Resk2s=G2y2tks−αsG1y2tks+α−1s−G2y1tkssα+G2y2tkssα+G2G2−1G2y2tks2kssα.
To find the first unknown coefficientsh1,1 and h2,1 of the series expansions (54), we substitute the first truncated expansions G1y1t1s, G1y2t1s, G2y1t1s, and G2y2t1s into the first ARA-residual functions G2Res11s and G2Res12s in system (56) to get(57)G2Res11s=1s+h1,1sα+1−αs1+h1,1α+1sα+α−1s+1002sα1s+h1,1sα+1−1000sαG2G2−11s+h2,1sα+121s,G2Res12s=1s+h2,1sα+1−αs1+h2,1α+1sα+α−1s−1sα1s+h1,1sα+1+1sα1s+h2,1sα+1+1sαG2G2−11s+h2,1sα+121s,
which can be written as(58)G2Res11s=h1,1sα+1−αh1,1α+1sα+1+1002sα+1+1002h1,1s2α+1−1000sαG21+2tαh2,1Γα+1+t2αh2,12Γ2α+11s,G2Res12s=h2,1sα+1−αh2,1α+1sα+1−h1,1s2α+1+h2,1s2α+1+1sαG2t+2tαh2,1Γα+1+t2αh2,12Γ2α+11s.
Multiplying both sides of equations in system (58) by sα+1 and taking the limit as s⟶∞, one can get(59)h1,1=−2α+1,h2,1=−1α+1.
To find out the second unknown coefficientsh1,2 and h2,2 of the series expansions (54), we substitute the second truncated expansions G1y1t2s, G1y2t2s, G2y1t2s, and G2y2t2s into the second ARA-residual functions G2Res21s and G2Res22s in system (56) to get(60)G2Res21s=1s+h1,1sα+1+h1,2s2α+1−αs1+h1,1α+1sα+h1,22α+1s2α+α−1s+1002sα1s+h1,1sα+1+h1,2s2α+1−1000sαG2G2−11s+h2,1sα+1+h2,2s2α+121s,G2Res22s=1s+h2,1sα+1+h2,2s2α+1−αs1+h2,1α+1sα+h2,22α+1s2α+α−1s−1sα1s+h1,1sα+1+h1,2s2α+1+1sα1s+h2,1sα+1+h2,2s2α+1+1sαG2G2−11s+h2,1sα+1+h2,2s2α+121s.
Simplifying equations in system (60), we have(61)G2Res21s=1sα+1h1,1−h1,1α+1+2+h1,2s2α+1−αh1,22α+1s2α+1+1002h1,1s2α+1+1002h1,2s3α+1−1000sαG2t2+2t1+αh2,1Γ1+α+2t1+2αh2,1Γ1+2α+t2αh2,12Γ21+α+t4αh2,12Γ21+2α+2t3αh2,12Γ1+αΓ1+2α1s,G2Res22s=1sα+1h2,1−αh2,1α+1+1+1s2α+1h2,2−αh2,22α+1−h1,1+h2,1+1s3α+1h1,2+h2,2+1sαG2t2+2t1+αh2,1Γ1+α+2t1+2αh2,1Γ1+2α+t2αh2,12Γ21+α+t4αh2,12Γ21+2α+2t3αh2,12Γ1+αΓ1+2α1s.
Multiplying both sides of equations in system (61) by s2α+1 and taking the limit as s⟶∞, we have(62)h1,2=42α+1,h2,2=2α+1.
We obtain the coefficientsh1,n and h2,n,n=3,4,…, in the expansions (54), by solving the system(63)lims⟶∞skα+1G2Resk1s=0,and
lims⟶∞skα+1G2Resk2s=0, for k=3,4,…,
which lead to(64)h1,3=−83α+1251−125Γ2α+1Γ2α+1,h2,3=3α+11−Γ2α+1Γ2α+1,h1,4=4α+12014016−1004000Γ2α+1Γ2α+1−2000Γ3α+1Γα+1Γ2α+1,h2,4=4α+12011+1003Γ2α+1Γ2α+1+2Γ3α+1Γα+1Γ2α+1,h1,5=5α+1−2022066032+1008014000Γ2α+1Γ2α+1+2008000Γ3α+1Γα+1Γ2α+1+1000Γ4α+1Γ22α+1−2000Γ4α+1Γα+1Γ3α+1+2000Γ2α+1Γ4α+1Γ3α+1Γ3α+1,h2,5=5α+12020049−1007009Γ2α+1Γ2α+1−2006Γ3α+1Γα+1Γ2α+1−Γ4α+1Γ22α+1+2Γ4α+1Γα+1Γ3α+1−2Γ2α+1Γ4α+1Γ3α+1Γ3α+1.
Consequently, the ARA-RPS solutions of system (51) can be expressed as(65)G2y1ts=1s−2α+1sα+1+42α+1s2α+1−83α+1s3α+1251−125Γ2α+1Γ2α+1+…,G2y2ts=1s−α+1sα+1−2α+1s2α+1+3α+1s3α+11−Γ2α+1Γ2α+1+…..
To get the series solutions in the original space of system (47), we apply the inverse ARA transform of order two G2−1 to both sides of equations in system (65) to get(66)y1t=1−2tαΓα+1+4t2αΓ2α+1+1000Γ2α+1Γ2α+1Γ3α+1−2008Γ3α+1t3α+⋯,y2t=1−tαΓα+1+t2αΓ2α+1+1Γ3α+1+Γ2α+1Γ2α+1Γ3α+1t3α+…..
Forα=1, the series expansions (66) have the forms, y1t=e−2t, y2t=e−t which are coincide with the exact solutions for the ordinary system (47) and (48).
Numerical results of Example2 with k=6, and different values of α are listed in Table 3. In addition, the comparison of the results obtained for the exact solutions corresponding to α=1 and the numerical solutions given by the ARA-RPS method for different values of α: α=0.6,α=0.7 and α=0.8 are plotted in Figure 3. Figure 4 portrays a very precise agreement of the exact solutions y1t,y2t and the ARA-RPS solutions y16t,y26t of Example 2 at different time levels with fixed α. The obtained results show that the ARA-RPS method is efficient in obtaining approximate solutions of systems of FDEs. This is obvious in the following tables, which illustrate the small errors between the ARA-RPS solutions and the exact ones.Table 3
Numerical results ofy1t and y2t in Example 2 at α=1 and k=6.
tiNumerical results ofy1tExact solutionARA-RPSy1t−y16t0.150.74081822068171790.740818262499999904.18182820594×10−80.300.54881163609402650.548816800000000005.16390597349×10−60.450.40656965974059900.406654862500000058.52027594009×10−50.600.30119421191220200.301811200000000066.16988087798×10−40.750.22313016014842980.225976562499999982.84640235157×10−30.900.16529888822158600.175175200000000039.87631177841×10−3tiNumerical results ofy2tExact solutionARA-RPSy2t−y26t0.150.86070797642505780.86070797675781243.3275460165×10−100.300.74081822068171790.74081826249999994.18182820594×10−80.450.63762815162177330.63762885332031257.01698539163×10−70.600.54881163609402650.54881680000000005.16390597349×10−60.750.47236655274101470.47239074707031252.41943292978×10−50.900.40656965974059910.40665486250000008.52027594009×10−5Figure 3
(a) Plots of the exact solutiony1t and the approximate solution y16t at various α values and (b) plots of the exact solution y2t and the approximate solution y26t at various α values, for Example 2.
(a)(b)Figure 4
Plots of the exact solutionsy1t,y2t and the ARA-RPS solutions y16t,y26t at α=1 of Example 2.Example 3.
Consider the following nonlinear system of ordinary FDEs(67)Dαy1t=y1t,Dαy2t=2y12t,Dαy3t=3y1ty2t,with the ICs(68)y10=y20=y30=1.
Solution. For eacht∈0,1, 0<α≤1. The exact solution of the nonlinear system (67) and (68), when α=1, is y1t=et,y2t=e2t and y3t=e3t.
Applying the ARA transform of order twoG2 on both sides of equations in system (67)(69)G2Dαy1ts=G2y1ts,G2Dαy2ts=2G2y12ts,G2Dαy3ts=3G2y1ty2ts.
Running the ARA transform on the equations in system (69), we have(70)sαG2y1ts−αsα−1G1y1ts+α−1sα−1y10=G2y1ts,sαG2y2ts−αsα−1G1y2ts+α−1sα−1y20=2G2G2−1G2y1t2s,sαG2y3ts−αsα−1G1y3ts+α−1sα−1y30s=2G3G2−1G2y1tG2−1G2y2t.
Substituting the ICs (67) and simplifying system (69), we get(71)G2y1ts−αsG1y1ts+α−1s+1sαG2y1ts=0,G2y2ts−αsG1y2ts+α−1s+2G2G2−1G2y1t2ssα=0,G2y3ts−αsG1y3ts+α−1s+3G2G2−1G2y1tG2−1G2y2tssα=0.
Assuming that the ARA-RPS solutions of system (71) have the series representations(72)G1y1ts=∑n=0∞h1,nnα+1snα,G2y1ts=∑n=0∞h1,nsnα+1,G1y2ts=∑n=0∞h2,nnα+1snα,G2y2ts=∑n=0∞h2,nsnα+1G1y3ts=∑n=0∞h3,nnα+1snα,G2y3ts=∑n=0∞h2,nsnα+1.,
Using the fact that(73)lims⟶∞sG2yits=yi0,i=1,2,3,we get the coefficients h1,0=1,h2,0=1,h3,0=1, and so the kth ARA-RPS solutions of system (71) have the form(74)G1y1tks=1+∑n=1kh1,nnα+1snα,G2y1tks=1s+∑n=1kh1,nsnα+1,G1y2tks=1+∑n=1kh2,nnα+1snα,G2y2tks=1s+∑n=1kh2,nsnα+1,G1y3ts=1+∑n=1kh3,nnα+1snα,G2y3ts=1s+∑n=1kh3,nsnα+1,where the coefficients h1,n,h2,n, and h3,n could be obtained by considering the following ARA-residual functions of the equations in system (71), we get(75)ς2Res1s=ς2y1ts−αsς1y1ts+α−1s−1sας2y1ts,ς2Res2s=ς2y1ts−αsς1y2ts+α−1s−2ς2ς2ς2−1ς2y1ts2ssα,ς2Res3s=ς3y1ts−αsς1y2ts+α−1s−3ς2ς2−1ς2y1tsς2−1ς2y1tsssα,and the kth ARA-residual functions of the equations in system (75)(76)ς2Resk1s=ς2y1tks−αsς1y1tks+α−1s−1sας2y1tks,ς2Resk1s=ς2y1tks−αsς1y2tks+α−1s−2ς2ς2−1ς2y1tks2kssα,ς2Resk3s=ς3y1tks−αsς1y2tks+α−1s−3ς2ς2−1ς2y1tksς2−1ς2y1tkskssα.
Using the fact thatlims⟶∞skα+1G2Reskis=0 for k=1,2,…andi=1,2,3.
We can find out the coefficients,h1,n,h2,n, and h3,n as follows:(77)h1,1=α+1,h2,1=2α+1,h3,1=3α+1,h1,2=2α+1,h2,2=42α+1,h3,2=92α+1,h1,3=3α+1,h2,3=43α+1,h3,3=43α+115+6Γ2α+1Γ2α+1,h1,4=4α+1,h2,4=4α+14+4Γ3α+1Γα+1Γ2α+1,h3,4=4α+115+6Γ2α+1Γ2α+1+18Γ2α+1Γα+1Γ2α+1,h1,5=5α+1,h2,5=5α+14+2Γ3α+1Γ22α+1+4Γ3α+1Γα+1Γ3α+1,h3,5=5α+115+12Γ3α+1Γα+1Γ2α+1+12Γ4α+1Γ22α+1,+18Γ4α+1Γα+1Γ3α+1+6Γ2α+1Γ4α+1Γ2α+1Γ3α+1.
Consequently, the ARA-RPS solutions of system (71) can be expressed as(78)G2y1ts=1s+α+1sα+1+2α+1s2α+1+3α+1s3α+1+…,G2y2ts=1s+2α+1sα+1+42α+1s2α+1+3α+1s3α+14+2Γ2α+1Γ2α+1+…,G2y3ts=1s+2α+1sα+1+42α+1s2α+1+3α+1s3α+115+6Γ2α+1Γ2α+1+…..
To get the series solutions in the original space of system (67), we apply the inverse ARA transform of order two G2−1 to both sides of equations in system (78) to get(79)y1t=1+tαΓα+1+t2αΓ2α+1+t3αΓ3α+1+⋯,y2t=1+2tαΓα+1+4t2αΓ2α+1+4Γ3α+1+2Γ2α+1Γ2α+1Γ3α+1t3α+…,y2t=1+3tαΓα+1+9t2αΓ2α+1+15Γ3α+1+6Γ2α+1Γ2α+1Γ3α+1t3α+…..
Forα=1, the series expansions (79) have the form, y1t=et, y2t=e2t, and y3t=e3t which are coincide with the exact solutions for the ordinary system (67) and (68).
Numerical results of Example3 with k=6, and different values of α are listed in Table 4. In addition, the comparison of the results obtained for the exact solution corresponding to α=1 and the numerical solutions given by the ARA-RPS method for different values of α: α=0.2,α=0.5,α=0.7 and α=0.9 are plotted in Figure 5. Figure 6 portrays a very precise agreement of the exact solutions y1t,y2t,y3t and the ARA-RPS solutions y16t,y26t,y26t of Example 3 at different time levels with fixed α. The obtained results show that the ARA-RPS method is efficient in obtaining approximate solutions of systems of FDEs. This is obvious in the following tables which illustrate the small errors between the ARA-RPS solutions and the exact ones.Table 4
Numerical results ofy1t,y2t and y3t in Example 3 at α=1 and k=6.
tiNumerical results ofy1tExact solutionARA-RPSy1t−y16t0.11.1051709180756481.1051709180555562.0091928121×10−110.21.2214027581601691.2214027555555562.60461430202×10−90.31.3498588075760031.3498587625000004.50760029302×10−80.41.4918246976412701.4918243555555563.42085714866×10−70.51.6487212707001281.6487196180555551.65264457275×10−60.61.8221188003905091.8221128000000006.00039050899×10−6tiNumerical results ofy2tExact solutionARA-RPSy2t−y26t0.11.2214027581601691.2214027555555562.60461430202×10−90.21.4918246976412701.4918243555555563.42085714866×10−70.31.8221188003905091.8221128000000006.00039050913×10−60.42.2255409284924682.2254947555555564.61729369121×10−50.52.7182818284590452.7180555555555552.26272903489×10−40.63.3201169227365473.3192832000000008.33722736547×10−4tiNumerical results ofy3tExact solutionARA-RPSy3t−y36t0.11.3498588075760031.3498587625000004.50760029302×10−70.21.8221188003905091.8221128000000006.00039050913×10−60.32.4596031111569502.4594963625000001.06748656949×10−50.43.3201169227365483.3192832000000008.33722736548×10−40.54.4816890703380654.4775390625000004.15000783806×10−30.66.0496474644129456.0341032000000001.55442644129×10−2Figure 5
(a) Plots of the exact solutiony1t and the approximate solution y16t at various α values; (b) plots of the exact solution y2t and the approximate solution y26t at various α values; and (c) plots of the exact solution y3t and approximate solution y36t at various α values, for Example 3.
(a)(b)(c)Figure 6
Plots of the exact solutionsy1t,y2t,y3t and the ARA-RPS solutions y16t,y26t,y26t at α=1 of Example 3.


## 5. Conclusion

Many numerical techniques have been established to investigate numerical solutions of FDEs. Some methods may not provide exact solutions. This research emphasized that the proposed technique, ARA-RPS, is an effective method to construct series solutions that may match with the exact for some systems of FDEs with suitable ICs. The proposed scheme provides us the solutions in a new space “ARA transform space” in which it is easy to determine the unknown constants of the suggested series representations using the concept of the limit. The ARA-RPS solutions are easy to compute with fewer terms. We mention that our results are obtained by using the Mathematica software package 12. As future work, we plan to solve fractional integral equations and systems of fractional partial differential equations with different orders ofα using the proposed method [48–50].



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