# Asymptotic Solutions of Time-Space Fractional Coupled Systems by Residual Power Series Method

Authors: Wenjin Li; Yanni Pang
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

This paper focuses on the asymptotic solutions to time-space fractional coupled systems, where the fractional derivative and integral are described in the sense of Caputo derivative and Riemann-Liouville integral. We introduce the Residual Power Series (for short RPS) method to construct the desired asymptotic solutions. Furthermore, we apply this method to some time-space fractional coupled systems. The simplicity and efficiency of RPS method are shown by the application.


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## Body

## 1. Introduction

Fractional derivative was mentioned in a letter from L’Hopital to Leibniz in 1695. In the letter, L’Hopital proposed a question “What is the result ofdny/dxn if n=1/2?” The answer of Leibniz was “d1/2x will be equal to xdx:x. This is an apparent paradox, from which, one day useful consequences will be drawn” [1, 2]. Furthermore, the generalization of this framework indicates that it is more appropriate to talk of integration and differentiation of, such as fractional order, real number order and even complex number order just as the development of number system. However, there is a basic question: “What is fractional integral and derivative?” Or “How to define the fractional integral and derivative?” More and more mathematicians focused on this problem, such as J. L. Lagrange, P. S. Laplace, and Joseph B. J. Fourier. Some different definitions of fractional integrals and derivatives have been defined according to different needs, like Riemann-Liouville integral, Caputo derivative, Weyl derivative, and so on [2, 3]. But there is no uniform definition of fractional integral and derivative, and the frequently used definition is Riemann-Liouville integral and Caputo derivative.Fractional differential equations, which involve fractional order derivatives, are applied in many engineering and scientific disciplines as the mathematical modelling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, and so on. Fractional differential equations also serve as an excellent tool for the description of hereditary properties of various materials and processes. An essential topic is to construct the solutions to fractional differential equations. And there are some effective methods to obtain different kinds of solutions, like Sumudu transform and variational iteration method [4], fractional Taylor vector approximate method [5], iterative method [6–8], Residual Power Series (RPS) method [9–13], and so on [14–16]. On the other hand, the study of coupled systems which involve fractional differential equations is also important because fractional coupled systems occur in many fields [17–21]. In this paper, we generalize the RPS method to time-space fractional coupled systems and obtain the asymptotic series solutions.The organization of this paper is as follows: In Section2, some concepts and lemmas on fractional calculus are presented. In Section 3, we introduce the algorithm of RPS method for time-space fractional coupled system. In Section 4, asymptotic solutions of some examples are solved via RPS method. In Section 5, some concluding remarks are presented.


## 2. Preliminaries

In this section, some concepts and main lemmas we need in this paper are presented [2, 3, 22–24]. And more details about fractional calculus can be found in [2, 23, 24].Definition 1.
A real functionfx is said to be in the space Cμ,μ∈R if there exists a real number ρ>μ such that fx=xρf1x, where f1x∈C0,∞. And it is said to be in the space Cμn if fnx∈Cμ,n∈N.Definition 2.
The Riemann-Liouville fractional integral operator of orderα⩾0 of a function f∈Cμ,μ⩾-1 is defined as (1)Itαft≔1Γα∫0tt-τα-1fτdτ,α>0,t>τ⩾0,ft,α=0,where the symbol Itα represents the αth Riemann-Liouville fractional integral of f.Definition 3.
The Caputo fractional derivative of orderα>0 of f∈C-1n,n∈N is defined as (2)Dtαft≔Itn-αfnt,n-1<α<n,t>0,dnftdtn,α=n,where the symbol Dtαft represents the αth Caputo fractional derivative of f.Definition 4.
The power series(3)∑n=0∞cnt-t0nα=c0+c1t-t0+c2t-t02+⋯is called a fractional power series about t=t0, where t is a variable and cnn=0,1,2…. are the coefficients of the series, α∈R+.Remark 5.
For convenience, we shall treatt0=0. In fact, the transformation T:t′=t-t0 reduces the fractional power series about t=t0 to the fractional power series about t=0 and meanwhile the transformation T is reversible.Definition 6.
A functionft is analytical at t=0 if f(t) can be written as a form of fractional power series.Lemma 7.
Suppose thatft is an analytic function at t=0; then ft can be written as follows:(4)ft=∑n=0∞cntnα,0<α≤1,t<R.Furthermore, if ft∈C-R,R and Dtnαft∈C-R,R for n=0,1,2… then the coefficients cn will take the form(5)cn=Dtnαftt=0Γnα+1,where Dtnα=Dtα·Dtα⋯Dtα︸n-times,n=0,1,2….Proof.
First of all, notice that if we putt=0 into (5), it yields (6)c0=f0=Dt0αftt=0Γ0α+1.Applying the operator Dtα one time on (4), (7)c1=Dtαftt=0Γα+1.Again, by applying the operator Dtα two times on (4), (8)c2=Dt2αftt=0Γ2α+1.Analogously (9)cn=Dtnαftt=0Γnα+1,n=0,1,2,….This completes the proof.


## 3. Algorithm of RPS Method for Coupled Systems

In this section, we consider the following system:(10)Dtpαu+Fx,t=0,Dtqαv+Gx,t=0,Dtrαw+Hx,t=0with initial values (11)Dtiαux,tt=0=aix,i=0,1,2,…,p-1,Dtjαvx,tt=0=bjx,j=0,1,2,…,q-1,Dtkαwx,tt=0=ckx,k=0,1,2,…,r-1as a generalized illustration for the main idea of RPS method, where the symbol Dt·α represents the ·th fractional derivative in the Caputo sense, max⁡p-1/p,q-1/q,r-1/r<α≤1p,q,r∈N, the functions u,v,w,F,G,H are analytic at t=0, and the initial functions ai,bj, and ck are infinitely many times differentiable for all i=0,1,2,…,p; j=0,1,2,…,q and k=0,1,2,…,r.Sinceu,v,w,F,G,H are analytic at t=0, then they can be expanded in the form of fractional power series at t=0 as follows: (12)ux,t=∑i=0∞uixΓiα+1tiα,Fx,t=∑i=0∞fixΓiα+1tiα;vx,t=∑j=0∞vjxΓjα+1tjα,Gx,t=∑j=0∞gjxΓjα+1tjα;wx,t=∑k=0∞wkxΓkα+1tkα,Hx,t=∑k=0∞hkxΓkα+1tkα,where x∈R,t∈-R,R, R is the minimum convergence radius of functions ux,t,vx,t,wx,t, Fx,t,Gx,t,Hx,t, and (13)mmmaix=Dtiαux,tt=0,mmmfix=DtiαFx,tt=0,mmmmmmmmmmmmmmmi=0,1,2,…;mmmbjx=Dtjαux,tt=0,mmmgjx=DtjαGx,tt=0,mmmmmmmmmmmmmmmj=0,1,2,…;mmmckx=Dtkαux,tt=0,mmmhkx=DtkαHx,tt=0,mmmmmmmmmmmmmmmk=0,1,2,….According to the initial conditions, (14)uix=aix,i=0,1,2,…,p-1;vjx=bjx,j=0,1,2,…,q-1;wkx=ckx,k=0,1,2,…,r-1.Thus the initial approximation of the solution u,v,w is as follows: (15)uinitx,t=∑i=0p-1uixΓiα+1tiα,vinitx,t=∑j=0q-1vjxΓjα+1tjα,winitx,t=∑k=0r-1wkxΓkα+1tkα.Then we calculate the coefficients uix, vjx, and wkx for i=p,p+1,…; j=q,q+1,…; k=r,r+1,…. Firstly, some symbols are given as follows: (16)Resux,t=Dtpαu+Fx,t,Resu,lx,t=Dtpαul+Fx,t;Resvx,t=Dtqαu+Gx,t,Resv,mx,t=Dtqαvm+Fx,t;Reswx,t=Dtrαu+Hx,t,Resw,nx,t=Dtrαun+Fx,t,where (17)ulx,t=∑i=0luixΓiα+1tiα=uinitx,t+∑i=pluixΓiα+1tiα,l=p,p+1,…;vmx,t=∑j=0mvjxΓjα+1tjα=vinitx,t+∑i=qmvixΓiα+1tiα,m=q,q+1,…;wnx,t=∑k=0nwkxΓkα+1tkα=winitx,t+∑i=rnwixΓiα+1tiα,n=r,r+1,….Hence there are two facts: (F1)(18)liml→∞⁡ulx,t=ux,t,limm→∞⁡vmx,t=ux,t,limn→∞⁡wnx,t=wx,t.(F2)(19)liml→∞⁡Resu,lx,t=Resu,∞x,t=Resux,t≡0;limm→∞⁡Resv,mx,t=Resv,∞x,t=Resvx,t≡0;limn→∞⁡Resw,nx,t=Resw,∞x,t=Reswx,t≡0.Furthermore (20)0=Dti-pαResu,∞x,tt=0=uix+Dti-pαFx,t,i=p,p+1,…⟹uix=fi-px≜aix,i=p,p+1,…;0=Dtj-qαResv,∞x,tt=0=vjx+Dtj-qαGx,t,j=q,q+1,…⟹vjx=gj-qx≜bjx,j=q,q+1,…;0=Dtk-rαResw,∞x,tt=0=wkx+Dtk-qαHx,t,k=r,r+1,…⟹wkx=hk-px≜ckx,k=r,p+1,….Thus the solutions of coupled system (10) are (21)ux,t=∑i=0∞aixΓiα+1tnα,vx,t=∑j=0∞bjxΓjα+1tnα,wx,t=∑k=0∞ckxΓkα+1tnα.Remark 8.
If(22)Fx,t=fx,t,u,v,w,Gx,t=gx,t,u,v,w,Hx,t=hx,t,u,v,wand even F,G,H include the term of space fractional derivative and the term of time-fractional derivative whose order is less than the order of the system, then RPS method is also effective in calculating the asymptotic solutions for coupled system (10). In fact, F,G,H could be expanded in the form of fractional power series about time variable t at the initial time t0, and facts (F1) and (F2) are reasonable as well; thus the coefficients appearing in the asymptotic solutions could be obtained successfully.


## 4. Application of RPS Method to Time-Space Fractional Coupled Systems

### 4.1. The Time-Space Fractional Coupled KdV System

KdV equation plays an important role in nonlinear evolution equation for its wide application in physics and engineering. Coupled KdV system was introduced by Hirota and Satsuma [25] to describe the iterations of water waves and they claimed that the system exits a soliton solution. In [26], Fan and Zhang got several kinds of solutions by an improved homogeneous method. In [20], Bhrawy et al. reduced the time-fractional coupled KdV equations into a problem consisting of a system of algebraic equations that greatly simplifies the problem via the shifted Legendre polynomials. The time-fractional coupled KdV equation is a generalization of the classical coupled KdV equation and in this subsection we generalize time-fractional coupled KdV system to time-space fractional coupled system (23) and obtain the asymptotic solution using RPS method.Consider the time-space fractional coupled KdV system:(23)Dtαu-aDx3βu-6auDxγu-2bvDxδv=0,Dtαv+Dx3λv+3uDxτv=0with initial values(24)ux,0=a0x,vx,0=b0x,where 0<α,γ,δ,τ≤1,2/3<β,λ≤1, u=ux,t,v=vx,t,x,t∈R×R.Ifux,t and vx,t are analytic at t=0, then they can be expanded in the form of fractional power series (25)ux,t=∑i=0∞uixΓiα+1tiα,vx,t=∑j=0∞vjxΓjα+1tjα,Fx,t=∑n=0∞tnαΓnα+1-aDx3βunx-∑s=0nΓnα+1Γsα+1Γn-sα+16ausxDxγun-sx+2bvsxDxδvn-sx,Gx,t=∑n=0∞tnαΓnα+1Dx3λvn+∑s=0nΓnα+1Γsα+1Γn-sα+1·3usxDxτvi-1-sx.Under the initial conditions,(26)u0x=a0x,v0x=b0x;that is, the initial approximate solutions are (27)uinitialx,t=a0x,vinitialx,t=b0x.Set (28)Resux,t=Dtαu-aDx3βu-6auDxγu-2bvDxδv,Resvx,t=Dtαv+Dx3λv+3uDxτv,Resu,lx,t=Dtαul-aDx3βu-6aulDxγul-2bvmDxδvm,Resv,mx,t=Dtαvm+Dx3λvm+3ulDxτvm,where (29)ulx,t=a0x+∑i=1luixΓiα+1tiα,vmx,t=b0x+∑j=1mvjxΓjα+1tjα.Using RPS method (30)uix=aDx3βui-1x+∑s=0i-1Γi-1α+1Γsα+1Γi-1-sα+1·6ausxDxγui-1-sx+2bvsxDxδvi-1-sx≜aix,i=1,2,…,vjx=-Dx3λvj-1x-∑s=0j-1Γj-1α+1Γsα+1Γj-1-sα+1·3usxDxτvj-1-sx≜bjx,j=1,2,….Thus the fractional power series solutions of coupled system (23) are(31)ux,t=∑i=0∞aixΓiα+1tiα,vx,t=∑j=0∞bjxΓjα+1tjα.


### 4.2. The Time-Space Fractional Coupled KdV System of Generalized Hirota-Satsuma Type

In this subsection, we consider the time-space fractional coupled KdV system of generalized Hirota-Satsuma type(32)Dtαu-12Dx3βu+3uDxγu-3Dxδvw=0,Dtαv+Dx3λv-3uDxτv=0,Dtαw+Dx3θw-3uDxσw=0,ux,0=a0x,vx,0=b0x,wx,0=c0x,where 0<α,γ,δ,λ,τ≤1,2/3<β,σ,θ≤1,u=ux,t,v=vx,t,w=wx,t,x,t∈R×R. The equation describes an interaction of two long waves with different dispersion relations.Ifu,v,w are analytic at t=0, then u,v,w can be written as the form of fractional power series (33)ux,t=∑i=0∞uixΓiα+1tiα,vx,t=∑j=0∞vjxΓjα+1tjα,wx,t=∑k=0∞wkxΓkα+1tkα,Fx,t=-12Dx3βu+3uDxγu-3Dxδvw=∑n=0∞tnαΓnα+1·-12Dx3βunx+3∑s=0nΓnα+1Γsα+1Γn-sα+1usxDxγun-sx-vnxwn-sx,Gx,t=Dx3λv-3uDxτv=∑n=0∞tnαΓnα+1·Dx3λvnx-3∑s=0nΓnα+1Γsα+1Γn-sα+1usxDxτvn-sx,Hx,t=Dx3θw-3uDxσw=∑n=0∞tnαΓnα+1·Dx3θwnx-3∑s=0nΓnα+1Γsα+1Γn-sα+1usxDxσwn-sx.With the initial values(34)u0x=a0x,v0x=b0x,w0x=c0x,the initial approximate solutions are (35)uinitialx,t=a0x,vinitialx,t=b0x,winitialx,t=c0x.Set (36)Resux,t=Dtαu-12Dx3βu+3uDxγu-3Dxδvw=0,Resvx,t=Dtαv+Dx3λv-3uDxτv=0,Reswx,t=Dtαw+Dx3θw-3uDxσw=0,Resu,lx,t=Dtαul-12Dx3βul+3ulDxγul-3Dxδvmwn,Resv,mx,t=Dtαvm+Dx3λv-3ulDxτvm,Resw,nx,t=Dtαwn+Dx3θwn-3ulDxσwn,where (37)ulx,t=a0x+∑i=1luixΓiα+1tα,vmx,t=b0x+∑j=1mvixΓjα+1,wnx,t=c0x+∑k=1nwkxΓkα+1tkα,with the results of RPS method: (38)uix=12Dx3βui-1x-3∑s=0i-1Γi-1α+1Γsα+1Γi-1-sα+1·usxDxγui-1-sx-Dxδvsxwi-1-sx≜aix,i=1,2,…,vjx=-Dx3σvj-1x+3∑s=0j-1Γj-1α+1Γjα+1Γj-1-sα+1usxDxλvj-1-sx≜bjx,j=1,2,…,wkx=-Dx3θwk-1x+3∑s=0k-1Γk-1α+1Γjα+1Γk-1-sα+1usxDxτwk-1-jx≜ckx,k=1,2,….So the fractional power series solutions of coupled system (32) are (39)ux,t=∑i=0∞aixΓiα+1tiα,vx,t=∑j=0∞bjxΓjα+1tjα,wx,t=∑k=0∞ckxΓkα+1tkα.


### 4.3. The Time-Space Fractional Coupled Whitham-Broer-Kaup (WBK) System

Whitham [27], Broer [28], and Kaup [29] obtained nonlinear WBK system. In [30], Rashidi et al. obtained the approximate traveling wave solutions of the coupled WBK system in shallow water using homotopy analysis method. In [31], Kadem and Baleanu applied the homotopy perturbation method to find an analytical approximate solution for the coupled WBK system. In this subsection, we consider the time-space fractional coupled WBK system and construct the approximate solution by RPS method.Consider the time-space fractional coupled WBK system:(40)Dtαu+uDxβu+Dxγv+aDx2δu=0,Dtαv+Dxλuv-aDx2τv+bDx3θu=0,ux,0=a0x,vx,0=b0x,where 0<α,σ,τ,λ≤1,1/2<δ,η≤1,2/3<θ≤1,x,t∈R×R,a,b∈R represent different dispersive power, u=ux,t is the field of horizontal velocity, and v=vx,t is the height deviating equilibrium position of liquid. And this is a very good model to describe dispersive wave.Ifu,v are analytic at t=0, then u,v can be expanded in the form of fractional power series (41)ux,t=∑i=0∞uixΓiα+1tiα,vx,t=∑j=0∞vjxΓjα+1tjα,Fx,t=∑n=0∞tnαΓnα+1∑s=0nΓnα+1Γsα+1Γn-sα+1usxDxβun-sx+Dxγvnx+aDx2δunx,Gx,t=∑n=0∞tnαΓnα+1∑s=0nΓnα+1Γsα+1Γn-sα+1Dxλusxvn-sx+aDx2τvnx+bDx3θunx.Under the initial conditions (42)u0x=a0x,v0x=b0x,and the initial approximate solutions are(43)uinitialx,t=a0x,vinitialx,t=b0x,t.Set (44)Resux,t=Dtαu+uDxβu+Dxγv+aDx2δu,Resvx,t=Dtαv+Dxλuv-aDx2τv+bDx3θu,Resu,lx,t=Dtαul+ulDxβul+Dxγvm+aDx2δul,Resv,mx,t=Dtαvm+Dxλulvm-aDx2τvm+bDx3θul,where (45)ulx,t=a0x+∑i=1luixΓiα+1tα,vmx,t=b0x+∑j=1vjxΓjα+1tjα.With RPS method (46)uix=-∑s=0i-1Γi-1α+1Γsα+1Γi-1-sα+1usxDxβui-1-sx+Dxγvi-1x+aDx2δui-1x≜aix,i=1,2,…,vjx=-∑s=0j-1Γj-1α+1Γsα+1Γj-1-sα+1Dxλusxvj-1-sx+aDx2τvj-1x-bDx3θuj-1x≜bjx,j=1,2,….So the fractional power series solution of coupled system (40) is (47)ux,t=∑i=0∞aixΓiα+1tiα,vx,t=∑j=0∞bjxΓjα+1tjα.Remark 9.
Whenα=σ=τ=δ=λ=η=θ=1, β≠0, γ=0, (40) reduces to the classical long-wave equations that describe the shallow water wave with diffusion.Remark 10.
Whenα=σ=τ=δ=λ=η=θ=1, β=0, γ=1, (40) reduces to the variant Boussinesq equation.


### 4.4. The Time-Space Fractional Coupled Shallow Water System

Shallow water systems are widely used in predicting hydrodynamics of surface flows such as water flows in rivers, channels, flood plains, and coastal regions. It is well known that the shallow water systems can accurately predict the hydraulic parameters under conditions of slow erosion and low sediment concentration of the time-space fractional coupled shallow water system [32]. In this subsection, consider the time-space fractional coupled shallow water system (48)Dtαu+uDxβu+Dxγv+aDx2δu=0,Dtαv+vDxλu+uDxτv-aDx2θv+bDx3σu=0,with initial values(49)ux,0=a0x,vx,0=b0x,where 0<α,β,γ,λ,τ≤1,1/2<δ,θ≤1, 2/3<σ≤1,u=ux,t,v=vx,t,x,t∈R×R.Ifu,v are analytic at t=0, then u,v can be written as the form of fractional power series: (50)ux,t=∑i=0∞uixΓiα+1tiα,vx,t=∑j=0∞vjxΓjα+1tjα,Fx,t=∑n=0∞tnαΓnα+1·∑s=0nΓnα+1Γsα+1Γn-sα+1usxDxβun-sx+Dxγvnx+aDx2δunx,Gx,t=∑n=0∞tnαΓnα+1·∑s=0nΓnα+1Γsα+1Γn-sα+1vsxDxλun-sx+usxDxτvn-sx-aDx2θvnx+bDx3σunx.With the initial conditions (51)u0x=a0x,v0x=b0x,and the initial approximate solutions are (52)uinitialx,t=a0x,vinitialx,t=b0x.Set (53)Resux,t=Dtαu+uDxβu+Dxγv+aDx2δu,Resvx,tDtαv+vDxλu+uDxτv-aDx2θv+bDx3σu,Resu,lx,t=Dtαul+ulDxβul+Dxγvm+aDx2δul,Resv,mx,t=Dtαvm+vmDxλul+ulDxτvm-aDx2θvm+bDx3σul,where (54)ulx,t=a0x+∑i=1luixΓiα+1tiα,vmx,t=b0x+∑j=1mvjxΓjα+1tjα.Using the RPS method (55)uix=-∑s=0i-1Γi-1α+1Γsα+1Γi-1-sα+1usxDxβui-1-sx-Dxγvi-1x+aDx2δui-1x≜aix,i=1,2,…,vjx=-∑s=0i-1Γi-1α+1Γsα+1Γi-1-sα+1vsxDxλuj-1-s+usxDxτvj-1-sx+aDx2θvj-1x-bDx3σui-1x≜bjx,j=1,2,….So the solutions of coupled system (48) are (56)ux,t=∑n=0∞anxΓnα+1tnα,vx,t=∑n=0∞bnxΓnα+1tnα.


## 4.1. The Time-Space Fractional Coupled KdV System

KdV equation plays an important role in nonlinear evolution equation for its wide application in physics and engineering. Coupled KdV system was introduced by Hirota and Satsuma [25] to describe the iterations of water waves and they claimed that the system exits a soliton solution. In [26], Fan and Zhang got several kinds of solutions by an improved homogeneous method. In [20], Bhrawy et al. reduced the time-fractional coupled KdV equations into a problem consisting of a system of algebraic equations that greatly simplifies the problem via the shifted Legendre polynomials. The time-fractional coupled KdV equation is a generalization of the classical coupled KdV equation and in this subsection we generalize time-fractional coupled KdV system to time-space fractional coupled system (23) and obtain the asymptotic solution using RPS method.Consider the time-space fractional coupled KdV system:(23)Dtαu-aDx3βu-6auDxγu-2bvDxδv=0,Dtαv+Dx3λv+3uDxτv=0with initial values(24)ux,0=a0x,vx,0=b0x,where 0<α,γ,δ,τ≤1,2/3<β,λ≤1, u=ux,t,v=vx,t,x,t∈R×R.Ifux,t and vx,t are analytic at t=0, then they can be expanded in the form of fractional power series (25)ux,t=∑i=0∞uixΓiα+1tiα,vx,t=∑j=0∞vjxΓjα+1tjα,Fx,t=∑n=0∞tnαΓnα+1-aDx3βunx-∑s=0nΓnα+1Γsα+1Γn-sα+16ausxDxγun-sx+2bvsxDxδvn-sx,Gx,t=∑n=0∞tnαΓnα+1Dx3λvn+∑s=0nΓnα+1Γsα+1Γn-sα+1·3usxDxτvi-1-sx.Under the initial conditions,(26)u0x=a0x,v0x=b0x;that is, the initial approximate solutions are (27)uinitialx,t=a0x,vinitialx,t=b0x.Set (28)Resux,t=Dtαu-aDx3βu-6auDxγu-2bvDxδv,Resvx,t=Dtαv+Dx3λv+3uDxτv,Resu,lx,t=Dtαul-aDx3βu-6aulDxγul-2bvmDxδvm,Resv,mx,t=Dtαvm+Dx3λvm+3ulDxτvm,where (29)ulx,t=a0x+∑i=1luixΓiα+1tiα,vmx,t=b0x+∑j=1mvjxΓjα+1tjα.Using RPS method (30)uix=aDx3βui-1x+∑s=0i-1Γi-1α+1Γsα+1Γi-1-sα+1·6ausxDxγui-1-sx+2bvsxDxδvi-1-sx≜aix,i=1,2,…,vjx=-Dx3λvj-1x-∑s=0j-1Γj-1α+1Γsα+1Γj-1-sα+1·3usxDxτvj-1-sx≜bjx,j=1,2,….Thus the fractional power series solutions of coupled system (23) are(31)ux,t=∑i=0∞aixΓiα+1tiα,vx,t=∑j=0∞bjxΓjα+1tjα.


## 4.2. The Time-Space Fractional Coupled KdV System of Generalized Hirota-Satsuma Type

In this subsection, we consider the time-space fractional coupled KdV system of generalized Hirota-Satsuma type(32)Dtαu-12Dx3βu+3uDxγu-3Dxδvw=0,Dtαv+Dx3λv-3uDxτv=0,Dtαw+Dx3θw-3uDxσw=0,ux,0=a0x,vx,0=b0x,wx,0=c0x,where 0<α,γ,δ,λ,τ≤1,2/3<β,σ,θ≤1,u=ux,t,v=vx,t,w=wx,t,x,t∈R×R. The equation describes an interaction of two long waves with different dispersion relations.Ifu,v,w are analytic at t=0, then u,v,w can be written as the form of fractional power series (33)ux,t=∑i=0∞uixΓiα+1tiα,vx,t=∑j=0∞vjxΓjα+1tjα,wx,t=∑k=0∞wkxΓkα+1tkα,Fx,t=-12Dx3βu+3uDxγu-3Dxδvw=∑n=0∞tnαΓnα+1·-12Dx3βunx+3∑s=0nΓnα+1Γsα+1Γn-sα+1usxDxγun-sx-vnxwn-sx,Gx,t=Dx3λv-3uDxτv=∑n=0∞tnαΓnα+1·Dx3λvnx-3∑s=0nΓnα+1Γsα+1Γn-sα+1usxDxτvn-sx,Hx,t=Dx3θw-3uDxσw=∑n=0∞tnαΓnα+1·Dx3θwnx-3∑s=0nΓnα+1Γsα+1Γn-sα+1usxDxσwn-sx.With the initial values(34)u0x=a0x,v0x=b0x,w0x=c0x,the initial approximate solutions are (35)uinitialx,t=a0x,vinitialx,t=b0x,winitialx,t=c0x.Set (36)Resux,t=Dtαu-12Dx3βu+3uDxγu-3Dxδvw=0,Resvx,t=Dtαv+Dx3λv-3uDxτv=0,Reswx,t=Dtαw+Dx3θw-3uDxσw=0,Resu,lx,t=Dtαul-12Dx3βul+3ulDxγul-3Dxδvmwn,Resv,mx,t=Dtαvm+Dx3λv-3ulDxτvm,Resw,nx,t=Dtαwn+Dx3θwn-3ulDxσwn,where (37)ulx,t=a0x+∑i=1luixΓiα+1tα,vmx,t=b0x+∑j=1mvixΓjα+1,wnx,t=c0x+∑k=1nwkxΓkα+1tkα,with the results of RPS method: (38)uix=12Dx3βui-1x-3∑s=0i-1Γi-1α+1Γsα+1Γi-1-sα+1·usxDxγui-1-sx-Dxδvsxwi-1-sx≜aix,i=1,2,…,vjx=-Dx3σvj-1x+3∑s=0j-1Γj-1α+1Γjα+1Γj-1-sα+1usxDxλvj-1-sx≜bjx,j=1,2,…,wkx=-Dx3θwk-1x+3∑s=0k-1Γk-1α+1Γjα+1Γk-1-sα+1usxDxτwk-1-jx≜ckx,k=1,2,….So the fractional power series solutions of coupled system (32) are (39)ux,t=∑i=0∞aixΓiα+1tiα,vx,t=∑j=0∞bjxΓjα+1tjα,wx,t=∑k=0∞ckxΓkα+1tkα.


## 4.3. The Time-Space Fractional Coupled Whitham-Broer-Kaup (WBK) System

Whitham [27], Broer [28], and Kaup [29] obtained nonlinear WBK system. In [30], Rashidi et al. obtained the approximate traveling wave solutions of the coupled WBK system in shallow water using homotopy analysis method. In [31], Kadem and Baleanu applied the homotopy perturbation method to find an analytical approximate solution for the coupled WBK system. In this subsection, we consider the time-space fractional coupled WBK system and construct the approximate solution by RPS method.Consider the time-space fractional coupled WBK system:(40)Dtαu+uDxβu+Dxγv+aDx2δu=0,Dtαv+Dxλuv-aDx2τv+bDx3θu=0,ux,0=a0x,vx,0=b0x,where 0<α,σ,τ,λ≤1,1/2<δ,η≤1,2/3<θ≤1,x,t∈R×R,a,b∈R represent different dispersive power, u=ux,t is the field of horizontal velocity, and v=vx,t is the height deviating equilibrium position of liquid. And this is a very good model to describe dispersive wave.Ifu,v are analytic at t=0, then u,v can be expanded in the form of fractional power series (41)ux,t=∑i=0∞uixΓiα+1tiα,vx,t=∑j=0∞vjxΓjα+1tjα,Fx,t=∑n=0∞tnαΓnα+1∑s=0nΓnα+1Γsα+1Γn-sα+1usxDxβun-sx+Dxγvnx+aDx2δunx,Gx,t=∑n=0∞tnαΓnα+1∑s=0nΓnα+1Γsα+1Γn-sα+1Dxλusxvn-sx+aDx2τvnx+bDx3θunx.Under the initial conditions (42)u0x=a0x,v0x=b0x,and the initial approximate solutions are(43)uinitialx,t=a0x,vinitialx,t=b0x,t.Set (44)Resux,t=Dtαu+uDxβu+Dxγv+aDx2δu,Resvx,t=Dtαv+Dxλuv-aDx2τv+bDx3θu,Resu,lx,t=Dtαul+ulDxβul+Dxγvm+aDx2δul,Resv,mx,t=Dtαvm+Dxλulvm-aDx2τvm+bDx3θul,where (45)ulx,t=a0x+∑i=1luixΓiα+1tα,vmx,t=b0x+∑j=1vjxΓjα+1tjα.With RPS method (46)uix=-∑s=0i-1Γi-1α+1Γsα+1Γi-1-sα+1usxDxβui-1-sx+Dxγvi-1x+aDx2δui-1x≜aix,i=1,2,…,vjx=-∑s=0j-1Γj-1α+1Γsα+1Γj-1-sα+1Dxλusxvj-1-sx+aDx2τvj-1x-bDx3θuj-1x≜bjx,j=1,2,….So the fractional power series solution of coupled system (40) is (47)ux,t=∑i=0∞aixΓiα+1tiα,vx,t=∑j=0∞bjxΓjα+1tjα.Remark 9.
Whenα=σ=τ=δ=λ=η=θ=1, β≠0, γ=0, (40) reduces to the classical long-wave equations that describe the shallow water wave with diffusion.Remark 10.
Whenα=σ=τ=δ=λ=η=θ=1, β=0, γ=1, (40) reduces to the variant Boussinesq equation.


## 4.4. The Time-Space Fractional Coupled Shallow Water System

Shallow water systems are widely used in predicting hydrodynamics of surface flows such as water flows in rivers, channels, flood plains, and coastal regions. It is well known that the shallow water systems can accurately predict the hydraulic parameters under conditions of slow erosion and low sediment concentration of the time-space fractional coupled shallow water system [32]. In this subsection, consider the time-space fractional coupled shallow water system (48)Dtαu+uDxβu+Dxγv+aDx2δu=0,Dtαv+vDxλu+uDxτv-aDx2θv+bDx3σu=0,with initial values(49)ux,0=a0x,vx,0=b0x,where 0<α,β,γ,λ,τ≤1,1/2<δ,θ≤1, 2/3<σ≤1,u=ux,t,v=vx,t,x,t∈R×R.Ifu,v are analytic at t=0, then u,v can be written as the form of fractional power series: (50)ux,t=∑i=0∞uixΓiα+1tiα,vx,t=∑j=0∞vjxΓjα+1tjα,Fx,t=∑n=0∞tnαΓnα+1·∑s=0nΓnα+1Γsα+1Γn-sα+1usxDxβun-sx+Dxγvnx+aDx2δunx,Gx,t=∑n=0∞tnαΓnα+1·∑s=0nΓnα+1Γsα+1Γn-sα+1vsxDxλun-sx+usxDxτvn-sx-aDx2θvnx+bDx3σunx.With the initial conditions (51)u0x=a0x,v0x=b0x,and the initial approximate solutions are (52)uinitialx,t=a0x,vinitialx,t=b0x.Set (53)Resux,t=Dtαu+uDxβu+Dxγv+aDx2δu,Resvx,tDtαv+vDxλu+uDxτv-aDx2θv+bDx3σu,Resu,lx,t=Dtαul+ulDxβul+Dxγvm+aDx2δul,Resv,mx,t=Dtαvm+vmDxλul+ulDxτvm-aDx2θvm+bDx3σul,where (54)ulx,t=a0x+∑i=1luixΓiα+1tiα,vmx,t=b0x+∑j=1mvjxΓjα+1tjα.Using the RPS method (55)uix=-∑s=0i-1Γi-1α+1Γsα+1Γi-1-sα+1usxDxβui-1-sx-Dxγvi-1x+aDx2δui-1x≜aix,i=1,2,…,vjx=-∑s=0i-1Γi-1α+1Γsα+1Γi-1-sα+1vsxDxλuj-1-s+usxDxτvj-1-sx+aDx2θvj-1x-bDx3σui-1x≜bjx,j=1,2,….So the solutions of coupled system (48) are (56)ux,t=∑n=0∞anxΓnα+1tnα,vx,t=∑n=0∞bnxΓnα+1tnα.


## 5. Concluding Remarks

This paper introduced a new analytical iterative technique to construct asymptotic solutions to time-space fractional coupled systems, which is based on the general Residual Power Series method. Furthermore, we apply this method to some specific time-space fractional coupled systems to obtain asymptotic solutions with respect to initial values, which shows that this method is efficient and does not require linearization or perturbation.



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*Source: 7695924-2017-07-10.xml*