# Homotopy Series Solutions to Time-Space Fractional Coupled Systems

Authors: Jin Zhang; Ming Cai; Bochao Chen; Hui Wei
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

We apply the homotopy perturbation Sumudu transform method (HPSTM) to the time-space fractional coupled systems in the sense of Riemann-Liouville fractional integral and Caputo derivative. The HPSTM is a combination of Sumudu transform and homotopy perturbation method, which can be easily handled with nonlinear coupled system. We apply the method to the coupled Burgers system, the coupled KdV system, the generalized Hirota-Satsuma coupled KdV system, the coupled WBK system, and the coupled shallow water system. The simplicity and validity of the method can be shown by the applications and the numerical results.


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## Body

## 1. Introduction

Fractional calculus, compared to integer calculus, was mentioned in a letter from L’Hospital to Leibniz in 1695. In the letter, L’Hospital raised 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]. Furthermore, the generalization of this framework indicates that it is more appropriate to talk about integration and differentiation of arbitrary order, such as fractional order, real number order, and even complex number order just as the development of number system. Thus, there is a basic question: “what are the definitions of fractional integral and derivative?” Or “how to define the fractional integral and derivative?” More and more mathematicians focused on this problem, like Lagrange, Laplace, Fourier, and so on. Some different fractional integrals and derivatives have been given according to different needs, like Riemann-Liouville fractional integral, Caputo fractional derivative, Weyl fractional derivative, and so on [2]. But there are no uniform definitions of fractional integral and derivative, and the frequently used definitions are Riemann-Liouville integral and Caputo derivative.Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, and so on, which involve derivatives of fractional order. Fractional differential equations also serve as an excellent tool for the description of hereditary properties of various materials and processes. In consequence, the subject of fractional differential equations is gaining much more attention. For example, in electromagnetism, Sebaa et al. [3] studied ultrasonic wave propagation in human cancellous bone by using fractional calculus to describe the viscous interactions between fluid and solid structure. In signal processing, Assaleh and Ahmad [4] proposed a new approach for speech signal modeling through using fractional calculus. Magin and Ovadia [5] molded the cardiac tissue electrode interface using fractional calculus. In control theory, Suarez et al. [6] applied fractional controllers to the path-tracking problem in an autonomous electric vehicle. In fluid mechanics, Kulish and Lage [7] applied fractional calculus to the solution of time-dependent, viscous-diffusion fluid mechanics problems.In this paper, we intend to construct the approximate solutions to the nonlinear time-space fractional coupled systems. There are many effective methods to solve this problem, like Adomian decomposition method [8–10], variation iteration method [11], differential transform method [12], residual power series method [13, 14], iteration method [15], homotopy perturbation method [16], homotopy analysis method [17], and so on. Furthermore, for the nonlinear problem, the multiple exp-function method [18, 19], the transformed rational function method [20–22], and invariant subspace method [23, 24] are three systematical approaches to handle the nonlinear terms. The first one is to propose the exact solution of nonlinear partial differential equations by using rational function transformations. Its key point is to search for rational solutions to variable-coefficient ordinary differential equations transformed from given partial differential equations. The second one is to consider the form of solution as rational exponential functions with unknown coefficients whose advantage is direct applicability to underlying equation. The invariant subspace method is refined to present more unity and more diversity of exact solutions to evolution equations. The key idea is to take subspaces of solutions to linear ordinary differential equations as invariant subspaces that evolution equations admit. Motivated by these fruitful results, Singh et al. [25] proposed the homotopy perturbation Sumudu transform method based on the homotopy perturbation method and Sumudu transform method and applied it to nonlinear partial differential equations. The HPSTM was extended to the time-fractional PDEs in [26, 27]. It is worth mentioning that the HPSTM is applied without any using of Adomian polynomials, over restrictive assumption or linearization, and is capable of reducing the volume of computational work as compared to the classical numerical methods while still maintaining the high accuracy of the result. Meanwhile, it is appropriate not only for strongly nonlinear system but also for weakly nonlinear system.The rest of the paper is organized as follows. In Section2, we introduce some concepts on fractional calculus and the Sumudu transform. In Section 3, we illustrate the basic idea of HPSTM which is applied to the time-space fractional coupled systems. In Section 4, we apply HPSTM to obtain fractional power series solutions of nonlinear time-space fractional coupled systems with initial values, and some numerical results are presented as well.


## 2. Preliminaries

Definition 1 (see [2]).
A real functionfx, x>0 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 that fx∈Cμn, if fnx∈Cμ, n∈N.Definition 2 (see [2]).
The fractional integral offt in the Riemann-Liouville (left-sided) sense is defined as(1)Itαft≔1Γα∫0tt-τα-1fτdτ,α>0,t>τ⩾0;ft,α=0,where α⩾0, f∈Cμ, μ⩾-1, and Γ is the Gamma function.Definition 3 (see [2]).
The fractional integral offx in the Riemann-Liouville sense is defined as(2)Ixαfx≔1Γα∫-∞xx-τα-1fτdτ,α>0,-∞<x<∞;fx,α=0,where α⩾0, f∈Cμ, μ⩾-1, and Γ is the Gamma function.Definition 4 (see [2]).
The Caputo (left-sided) fractional derivative operator of orderα≥0, of a function f∈Cμnμ⩾-1,n∈N, is defined as(3)Dtαft≔1Γn-α∫0tt-τn-α-1fnτdτ,n-1<α<n,t>0;dnftdtn,α=n.Definition 5 (see [2]).
The Caputo fractional derivative operator of orderα≥0, of a function f∈Cμnμ⩾-1,n∈N, is defined as(4)Dxαfx≔1Γn-α∫-∞xx-τn-α-1fnτdτ,n-1<α<n,-∞<x<∞;dnfxdxn,α=n.Lemma 6 (see [28]).
Ifm-1<α≤m, m∈N, and f∈Cμm, μ≥-1, one has(5)DtαItαft=ft,ItαDtαft=ft-∑k=0m-1fk0tkk!.In 1998, a new integral transform, named Sumudu transform, was introduced by Watugala [29] to study solutions of ordinary differential equations in control engineering problems. The Sumudu transform is defined over the set of functions A=ft:∃M,τ1,τ2>0,s.t.ft<Met/τj,ift∈-1j×0,+∞ by the following formula:(6)Fu=Sft=∫0∞fute-tdt,u∈-τ1,τ2.Property 7 (see [30]).
(i) The Sumudu transform satisfies linear property; that is,(7)Saft+bht=aSft+bSht,where a, b are constants.
(ii)(8)Stn=unΓn+1,n∈N.Lemma 8 (see [31]).
The Sumudu transform of the Caputo fractional derivative is(9)SDtαft=u-αSft-∑k=0mu-α+kfk0,m<α≤m+1,m∈N.


## 3. Homotopy Perturbation Sumudu Transform Method

In this section, to illustrate the basic idea of this method, we consider a general nonhomogeneous fractional partial differential coupled system(10)DtαU1x,t+R1U1,U2,U3+N1U1,U2,U3=g1x,t,DtβU2x,t+R2U1,U2,U3+N2U1,U2,U3=g2x,t,DtγU3x,t+R3U1,U2,U3+N3U1,U2,U3=g3x,t,with the initial conditions(11)U1x,0=f1x,U2x,0=f2x,U3x,0=f3x,where α,β,γ∈0,1, Ri, Ni, i=1,2,3, denote linear differential operators and nonlinear differential operators, respectively, and gix,t are the source terms. Applying the Sumudu transform on both sides of (10) yields (12)SDtαU1x,t+SR1U1,U2,U3+SN1U1,U2,U3=Sg1x,t,SDtβU2x,t+SR2U1,U2,U3+SN2U1,U2,U3=Sg2x,t,SDtγU3x,t+SR3U1,U2,U3+SN3U1,U2,U3=Sg3x,t.It follows from the property of the Sumudu transform in (9) that(13)SU1x,t=f1x-uαSR1U1,U2,U3+SN1U1,U2,U3+uαSg1x,t,SU2x,t=f2x-uβSR2U1,U2,U3+SN2U1,U2,U3+uβSg2x,t,SU3x,t=f3x-uγSR3U1,U2,U3+SN3U1,U2,U3+uγSg3x,t.Furthermore, applying the inverse Sumudu transform S-1 on both sides of (13) yields (14)U1x,t=M1x,t-S-1uαSR1U1,U2,U3+N1U1,U2,U3,U2x,t=M2x,t-S-1uβSR2U1,U2,U3+N2U1,U2,U3,U3x,t=M3x,t-S-1uγSR3U1,U2,U3+N3U1,U2,U3,where Mix,t, i=1,2,3, represent the terms arising from the source terms and prescribed initial conditions; that is,(15)M1x,t=S-1f1x+uαSg1x,t,M2x,t=S-1f2x+uβSg2x,t,M3x,t=S-1f3x+uγSg3x,t.Let us construct the homotopy perturbation equations(16)U1x,t=M1x,t-p×S-1uαSR1U1,U2,U3+N1U1,U2,U3,U2x,t=M2x,t-p×S-1uβSR2U1,U2,U3+N2U1,U2,U3,U3x,t=M3x,t-p×S-1uγSR3U1,U2,U3+N3U1,U2,U3,where homotopy parameter p∈0,1. Suppose that Uix,t and the nonlinear terms NjUix,t can be written as(17)Uix,t=∑n=0∞pnUnix,t,i=1,2,3,NjU1,U2,U3=∑n=0∞pnHnjU1,U2,U3,j=1,2,3,where the coefficient polynomials Uni can be determined below and HnjU1,U2,U3 are given by the following formulae:(18)HnjU1,U2,U3=1n!∂n∂pnNj∑k=0∞pkUk1,∑k=0∞pkUk2,∑k=0∞pkUk3p=0=1n!∂n∂pnNj∑k=0npkUk1,∑k=0npkUk2,∑k=0npkUk3+Nj∑k=n+1∞pkUk1,∑k=n+1∞pkUk2,∑k=n+1∞pkUk3p=0=1n!∂n∂pnNj∑k=0npkUk1,∑k=0npkUk2,∑k=0npkUk3p=0,j=1,2,3.Substituting (17) into (16) gives (19)∑n=0∞pnUn1x,t=M1x,t-p×S-1uαSR1∑k=0∞pkUk1,∑k=0∞pkUk2,∑k=0∞pkUk3+∑n=0∞pnHn1∑k=0npkUk1,∑k=0npkUk2,∑k=0npkUk3,∑n=0∞pnUn2x,t=M2x,t-p×S-1uβSR2∑k=0∞pkUk1,∑k=0∞pkUk2,∑k=0∞pkUk3+∑n=0∞pnHn2∑k=0npkUk1,∑k=0npkUk2,∑k=0npkUk3,∑n=0∞pnUn3x,t=M3x,t-p×S-1uγSR3∑k=0∞pkUk1,∑k=0∞pkUk2,∑k=0∞pkUk3+∑n=0∞pnHn3∑k=0npkUk1,∑k=0npkUk2,∑k=0npkUk3.Comparing the coefficients of p, we obtain the following recurrence equations:(20)p0:U01x,t=M1x,t,U02x,t=M2x,t,U03x,t=M3x,t,⋮pn:Un1x,t=-S-1uαSR1∑r=0n-1∑s=0n-1-rUr1Us2Un-1-r-s3+Hn-11∑k=0npkUk1,∑k=0npkUk2,∑k=0npkUk3,Un2x,t=-S-1uβSR2∑r=0n-1∑s=0n-1-rUr1Us2Un-1-r-s3+Hn-12∑k=0npkUk1,∑k=0npkUk2,∑k=0npkUk3,Un3x,t=-S-1uγSR3∑r=0n-1∑s=0n-1-rUr1Us2Un-1-r-s3+Hn-13∑k=0npkUk1,∑k=0npkUk2,∑k=0npkUk3,where n=1,2,….According to the homotopy perturbation method, we assume that the solution of (10)-(11) can be written as(21)Vix,t=limn→∞⁡U0i+U1ip+U2ip2+⋯+Unipn,i=1,2,3.Setting p→1, the approximate solution to (10)-(11) is(22)Uix,t=limp→1⁡Vix1,…,xm,t=limn→∞⁡U0i+U1i+U2i+⋯+Uni,i=1,2,3.The convergence of series (21) depends on the nonlinear differential operator N. Generally, the derivative with respect to U of the nonlinear part in the splitting must be sufficiently small, since the parameter p may be relatively large; in fact we take p→1. The series is convergent for most cases [32].Remark 9.
HPSTM is applied to construct homotophy series solutions for fractional coupled systems, which has not too many overstrict assumptions compared to some classical methods.


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

In this section, we apply HPSTM to nonlinear time-space fractional coupled systems with initial conditions.


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

The Burgers equation is one of the most important partial differential equations from fluid mechanics, which not only describes many phenomena, for example, modeling the motion of turbulence [33], but also has many applications in science and engineering [34]. Here we apply HPSTM to solve the following nonlinear time-space fractional coupled Burgers system:(23)DtαU-Dx2U-2UDxβU+DxUV=0,DtγV-Dx2V-2VDxδV+DxUV=0,with the initial conditions (24)Ux,0=sin⁡x,Vx,0=sin⁡x,where 0<α,β,δ,γ≤1, x,t∈R×0,∞.Applying the Sumudu transform on both sides of (23) with the initial conditions, we can obtain(25)SUx,t=sin⁡x+uαSDx2Ux,t+2Ux,tDxβUx,t-DxUx,tVx,t,SVx,t=sin⁡x+vγSDx2Vx,t+2Vx,tDxδVx,t-DxUx,tVx,t.The inverse Sumudu transform of (25) implies that(26)Ux,t=sin⁡x+S-1uαSDx2Ux,t+2Ux,tDxβUx,t-DxUx,tVx,t,Vx,t=sin⁡x+S-1vγSDx2Vx,t+2Vx,tDxδVx,t-DxUx,tVx,t.Now applying the homotopy perturbation method gives (27)∑n=0∞pnUnx,t=sin⁡x+p×S-1uαSDx2∑n=0∞pnUnx,t+∑n=0∞pnHnUx,t,∑n=0∞pnVnx,t=sin⁡x+p×S-1vγSDx2∑n=0∞pnVnx,t+∑n=0∞pnHnVx,t,where HnUx,t and HnVx,t are polynomials which denote the homotopy coefficients of the nonlinear term and are given by (28)∑n=0∞pnHnUx,t=2Ux,tDxβUx,t-DxUx,tVx,t,∑n=0∞pnHnVx,t=2Vx,tDxδVx,t-DxUx,tVx,t.Set (29)Unx,t≔U~nxtnαΓnα+1,Vnx,t≔V~nxtnγΓnγ+1,and then (30)HnUx,t=2∑i=0nU~ixDxβU~n-ixtnαΓiα+1Γn-iα+1-∑i=0nDxU~ixV~n-ixtnαΓiα+1Γn-iα+1,HnVx,t=2∑i=0nV~ixDxδV~n-ixtnγΓiγ+1Γn-iγ+1-∑i=0nDxU~ixV~n-ixtnγΓiγ+1Γn-iγ+1.Comparing the coefficients of p, this gives (31)p0:U0x,t=U~0x=sin⁡x,V0x,t=V~0x=sin⁡x,p1:U1x,t=S-1uαSDx2U0x,t+H0ux,t=S-1uαSDx2U~0x+2U~0xDxβU~0x-DxU~0xV~0x=tαΓα+1Dx2U~0x+2U~0xDxβU~0x-DxU~0xV~0x=tαΓα+1U~1x,V1x,t=S-1uγSDx2V0x,t+H0vx,t=S-1uγSDx2V~0x+2V~0xDxδV~0x-DxU~0xV~0x=tγΓγ+1Dx2V~0x+2V~0xDxδU~0x-DxU~0xV~0x=tγΓγ+1V~1x.Generally, we have(32)pk:Ukx,t=tkαΓkα+1Dx2U~k-1x+∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+12U~ixDxβU~k-1-ix-DxU~ixV~k-1-ix=tkαΓkα+1U~kx,Vkx,t=tkγΓkγ+1Dx2V~k-1x+∑i=0k-1Γk-1γ+1Γiγ+1Γk-i-1γ+12V~ixDxδV~k-1-ix-DxU~ixV~k-1-ix=tkγΓkγ+1V~kx,where (33)U~kx=Dx2U~k-1x+∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+12U~ixDxβU~k-1-ix-DxU~ixV~k-1-ix,V~kx=Dx2V~k-1x+∑i=0k-1Γk-1γ+1Γiγ+1Γk-i-1γ+12V~ixDxδV~k-1-ix-DxU~ixV~k-1-ix.Hence, the series solution of (23) is (34)Ux,t=∑k=0∞Ukx,t=∑k=0∞U~kxtkαΓkα+1,Vx,t=∑k=0∞Vkx,t=∑k=0∞V~kxtkγΓkγ+1.Particularly, when α=γ=β=δ=1, the exact solution of (23) is (35)Ux,t=e-tsin⁡x,Vx,t=e-tsin⁡x.Using HPSTM, when α=γ=β=δ=1, the third approximate solution of (23) is (36)Uappx,t=sin⁡x+t2!-sin⁡x+t23!sin⁡x+t34!-sin⁡x=sin⁡x1+-t2!+-t23!+-t34!,Vappx,t=sin⁡x+t2!-sin⁡x+t23!sin⁡x+t34!-sin⁡x=sin⁡x1+-t2!+-t23!+-t34!.In general, the limit of the approximate solution is (37)Ux,t=sin⁡x+t2!-sin⁡x+t23!sin⁡x+t34!-sin⁡x+⋯=sin⁡x1+-t2!+-t23!+-t34!+⋯=e-tsin⁡x,Vx,t=sin⁡x+t2!-sin⁡x+t23!sin⁡x+t34!-sin⁡x+⋯=sin⁡x1+-t2!+-t23!+-t34!+⋯=e-tsin⁡x,which is as same as the exact solution. However, if the initial values are too complex to find the limit of the approximated solution, then we replace the exact solution by the approximated solution within a certain scale, which is useful in the application of engineering.Thus we plot the images of the approximate solution (see Figure1(a)), the exact solution (see Figure 1(b)), and the error function (see Figures 1(c) and 1(d)). It is clear that the error function Uapp-Uex depends on time t. When time t is small (e.g., t=0.01), the error function is in the scale of 10-3 (see Figure 1(c)), which indicates that this is a good approximation in the neighbour of time 0 for system (23) with some explicit parameters. However, when time becomes large (e.g., t=0.1), the error function tends to be large as well (see Figure 1(d)); that is to say, this method is only suitable for constructing the approximated solution around the initial data.Figure 1
(a)
The third-order approximate solution ofU(x,t)
(b)
The exact solution ofU(x,t)
(c)
The errorUapp-Uex of U(x,t) at t=0.01
(d)
The errorUapp-Uex of U(x,t) at t=0.1


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

In this subsection, consider the time-space fractional generalization of the Hirota-Satsuma coupled KdV system(38)DtαU-12Dx3βU+3UDxγU-3DxδVW=0,DtαV+Dx3λV-3UDxτV=0,DtαW+Dx3θW-3UDxσW=0,with respect to the initial conditions (39)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×0,∞. The Hirota-Satsuma coupled KdV equation describes the unidirectional propagation of shallow water waves, which was initiated by Wu et al. [35]. Further (38) becomes a generalized fractional KdV equation for U=0 and a fractional MKdV equation for V=0.Applying the Sumudu transform on both sides of (38) with the initial conditions, we obtain(40)SUx,t=a0x+uαS12Dx3βUx,t-3Ux,tDxγUx,t+3DxδVx,tWx,t,SVx,t=b0x-vαSDx3λVx,t-3Ux,tDxτVx,t,SWx,t=c0x-wαSDx3θWx,t-3Ux,tDxσWx,t.The inverse Sumudu transform of (40) implies that(41)Ux,t=a0x+S-1uαS12Dx3βUx,t-3Ux,tDxγUx,t+3DxδVx,tWx,t,Vx,t=b0x-S-1vαSDx3λVx,t-3Ux,tDxτVx,t,Wx,t=c0x-S-1wαSDx3θWx,t-3Ux,tDxσWx,t.Via the homotopy perturbation method, it gives(42)∑n=0∞pnUnx,t=a0x+p×S-1uαS12Dx3β∑n=0∞pnUnx,t+∑n=0∞pnHnUx,t,∑n=0∞pnVnx,t=b0x+p×S-1vαS-Dx3λ∑n=0∞pnVnx,t+∑n=0∞pnHnVx,t,∑n=0∞pnWnx,t=c0x+p×S-1wαS-Dx3θ∑n=0∞Wnx,t+∑n=0∞pnHnWx,t,where HnUx,t, HnVx,t, and HnWx,t are polynomials which denote the nonlinear term, and they are given by (43)∑n=0∞pnHnUx,t=3DxδVx,tWx,t-3Ux,tDxγUx,t,∑n=0∞pnHnVx,t=3Ux,tDxτVx,t,∑n=0∞pnHnWx,t=3Ux,tDxσWx,t.Set (44)Unx,t≔U~nxtnαΓnα+1,Vnx,t≔V~nxtnαΓnα+1,Wnx,t≔W~nxtnαΓnα+1,and then (45)HnU=3Dxδ∑i=0nV~ixW~n-ixtnαΓiα+1Γn-iα+1-3∑i=0nU~ixDxγU~n-ixtnαΓiα+1Γn-iα+1,HnV=3∑i=0nU~ixDxτV~n-ixtnαΓiα+1Γn-iα+1,HnW=3∑i=0nU~ixDxσW~n-ixtnαΓiα+1Γn-iα+1.Comparing the coefficients of p shows(46)p0:U0x,t=U~0x=a0x,V0x,t=V~0x=b0x,W0x,t=W~0x=c0x,p1:U1x,t=S-1uαS12Dx3βU0x,t+H0Ux,t=S-1uαS12Dx3βU~0x+3DxδV~0xW~0x-3U~0xDxγU~0x=tαΓα+112Dx3βU~0x+3DxδV~0xW~0x-3U~0xDxγU~0x=tαΓα+1U~1x,V1x,t=S-1vαS-Dx3λV0x,t+H0Vx,t=S-1vαS-Dx3λV~0x+3U~0xDxτV~0x=tαΓα+1-Dx3λV~0x+3U~0xDxγU~0x=tαΓα+1V~1x,W1x,t=S-1wαS-Dx3θW0x,t+H0Wx,t=S-1wαS-Dx3θW~0x+3U~0xDxσW~0x=tαΓα+1-Dx3θW~0x+3U~0xDxσW~0x=tαΓα+1W~1x.Generally, one has(47)pk:Ukx,t=tkαΓkα+112Dx3βU~k-1x+∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+13DxδV~k-1-ixW~ix-3U~ixDxγU~k-1-ix=tkαΓkα+1U~kx,Vkx,t=tkαΓkα+1-Dx3λV~k-1x+∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+13U~ixDxτV~k-1-ix=tkαΓkα+1V~kx,Wkx,t=tkαΓkα+1-Dx3θW~k-1x+∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+13U~ixDxσW~k-1-ix=tkαΓkα+1W~kx,where(48)U~kx=12Dx3βU~k-1x+∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+13DxδV~k-1-ixW~ix-3U~ixDxγU~k-1-ix,V~kx=-Dx3λV~k-1x+∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+13U~ixDxτV~k-1-ix,W~kx=-Dx3θW~k-1x+∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+13U~ixDxσW~k-1-ix.Therefore, the approximate series solution of (38) is (49)Ux,t=∑k=0∞Ukx,t=∑k=0∞U~kxtkαΓkα+1,Vx,t=∑k=0∞Vkx,t=∑k=0∞V~kxtkαΓkα+1,Wx,t=∑k=0∞Wkx,t=∑k=0∞W~kxtkαΓkα+1.Particularly, when α=γ=δ=λ=1, τ=β=σ=θ=2/3, and the special initial value of (38) is (50)Ux,0=-2+4tanh2⁡x,Vx,0=-4+4tanh2⁡x,Wx,0=1+tanh2⁡x,then the exact solutions are (51)Ux,t=-2+4tanh2⁡x+2t,Vx,t=-4+4tanh2⁡x+2t,Wx,t=1+tanh2⁡x+2t.Under these special conditions, via HPSTM, the first approximate solution of (38) is (52)Uappx,t=-2+4tanh2⁡x+16tanh⁡x-tanh3⁡xt,Vappx,t=-4+4tanh2⁡x+16tanh⁡x-tanh3⁡xt,Wappx,t=1+tanh2⁡x+4tanh⁡x-tanh3⁡xt.Similarly, we obtain the following numerical results: see Figures 2(a), 2(b), 2(c), 2(d), 3(a), 3(b), 3(c), 3(d), 4(a), 4(b), 4(c), and 4(d).Figure 2
(a)
The first-order approximate solution ofU(x,t)
(b)
The exact solution ofU(x,t)
(c)
The errorUapp-Uex of U(x,t) at t=0.01
(d)
The errorUapp-Uex of U(x,t) at t=0.1Figure 3
(a)
The first-order approximate solution ofV(x,t)
(b)
The exact solution ofV(x,t)
(c)
The errorVapp-Vex of V(x,t) at t=0.01
(d)
The errorVapp-Vex of V(x,t) at t=0.1Figure 4
(a)
The first-order approximate solution ofW(x,t)
(b)
The exact solution ofW(x,t)
(c)
The errorWapp-Wex of W(x,t) at t=0.01
(d)
The errorWapp-Wex of W(x,t) at t=0.1


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

Shallow water systems are widely used in many areas of fluid dynamics, such as multiphase flows [36], turbulence [37], and viscoelasticity [38]. It is well known that the shallow water systems can accurately predict both the hydraulic parameters under conditions of slow erosion and low sediment concentration. Let us consider the time-space fractional coupled shallow water system(53)DtαU+UDxβU+DxγV+aDx2δU=0,DtαV+VDxλU+UDxτV-aDx2θV+bDx3σU=0with initial values(54)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×0,1/3.Applying the Sumudu transform on both sides of (53) with the initial conditions, we obtain(55)SUx,t=a0x+uαS-Ux,tDxβUx,t-DxγVx,t-aDx2δUx,t,SVx,t=b0x+vαS-Vx,tDxλUx,t-Ux,tDxτVx,t+aDx2θVx,t-bDx3σUx,t.The inverse Sumudu transform of (55) implies that(56)Ux,t=a0x+S-1uαS-Ux,tDxβUx,t-DxγVx,t-aDx2δUx,t,Vx,t=b0x+S-1vαS-Vx,tDxλUx,t-Ux,tDxτVx,t+aDx2θVx,t-bDx3σUx,t.According to homotopy perturbation method, we have(57)∑n=0∞pnUnx,t=a0x+p×S-1uαS-Dxγ∑n=0∞pnVnx,t-aDx2δ∑n=0∞pnUnx,t-∑n=0∞pnHnUx,t,∑n=0∞pnVnx,t=b0x+p×S-1vαSaDx2θ∑n=0∞pnVnx,t-bDx3σ∑n=0∞pnUnx,t-∑n=0∞pnHnVx,t,where HnUx,t and HnVx,t are polynomials of the nonlinear term and are given by (58)∑n=0∞pnHnUx,t=Ux,tDxβUx,t,∑n=0∞pnHnVx,t=Vx,tDxλUx,t+Ux,tDxτVx,t.Setting (59)Unx,t≔U~nxtnαΓnα+1,Vnx,t≔V~nxtnαΓnα+1,then we arrive at (60)HnUx,t=∑i=0nU~ixDxβU~n-ixtnαΓiα+1Γn-iα+1,HnVx,t=∑i=0nV~ixDxλU~n-ix+∑i=0nU~ixDxτV~n-ixtnαΓiα+1Γn-iα+1.Comparing the coefficients of p yields(61)p0:U0x,t=U~0x=a0x,V0x,t=V~0x=b0x,p1:U1x,t=S-1uαS-DxγV0x,t-aDx2δU0x,t-H0Ux=S-1uαS-DxγU~0x-aDx2δU~0x-U~0xDxβU~0x=tαΓα+1-DxγU~0x-aDx2δU~0x-U~0xDxβU~0x=tαΓα+1U~1x,V1x,t=S-1vαS-bDx3σV0x,t+aDx2θV0x,t-H0Vx=S-1vαS-bDx3σV~0x+aDx2θV~0x-V~0xDxλU~0x-U~0xDxτV~0x=tαΓα+1-bDx3σV~0x+aDx2θV~0x-V~0xDxλU~0x-U~0xDxτV~0x=tαΓα+1V~1x.Generally, we have (62)pk:Ukx,t=tkαΓkα+1-DxγV~k-1x-aDx2δU~k-1x-∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+1U~ixDxβU~k-1-ix=tkαΓkα+1U~kx,Vkx,t=tkαΓkα+1-bDx3σV~k-1x+aDx2θV~k-1x-∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+1U~ixDxλV~k-1-ix+V~ixDxτU~k-1-ix=tkαΓkα+1V~kx,where(63)U~kx=-DxγV~k-1x-aDx2δU~k-1x-∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+1U~ixDxβU~k-1-ix,V~kx=-bDx3σV~k-1x+aDx2θV~k-1x-∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+1U~ixDxλV~k-1-ix+V~ixDxτU~k-1-ix.Hence, the series solution is (64)Ux,t=∑k=0∞Ukx,t=∑k=0∞U~kxtkαΓkα+1,Vx,t=∑k=0∞Vkx,t=∑k=0∞V~kxtkαΓkα+1.


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

KdV equation plays an important role in nonlinear equations for wide applications in physics and engineering. Hirota and Satsuma [39] firstly found coupled KdV system to describe the iterations of water waves; meanwhile, they claimed that the system exists with a soliton solution. In [40], Fan and Zhang settled several kinds of solutions by an improved homogeneous method. The time-space fractional coupled KdV equation is a generalization of the classical coupled KdV equation. In this subsection, we consider the following time-space fractional coupled KdV system:(65)DtαU-aDx3βU-6aUDxγU-2bVDxδV=0,DtαV+Dx3λV+3UDxτV=0,with respect to initial values(66)Ux,0=a0x,Vx,0=b0x,where 0<α,γ,δ,τ≤1, 2/3<β, λ≤1, (x,t)∈R×0,∞, and the coefficients a, b are constants.Applying the Sumudu transform on both sides of (65) with the initial conditions, we obtain(67)SUx,t=a0x+uαSaDx3βUx,t+6aUx,tDxγUx,t+2bVx,tDxδVx,t,SVx,t=b0x-vαSDx3λVx,t+3Ux,tDxτVx,t.The inverse Sumudu transform of (67) implies that(68)Ux,t=a0x+S-1uαSaDx3βUx,t+6aUx,tDxγUx,t+2bVx,tDxδVx,t,Vx,t=b0x-S-1vαSDx3λVx,t+3Ux,tDxτVx,t.Analogously, using homotopy perturbation method gives (69)∑n=0∞pnUnx,t=a0x+p×S-1uαSaDx3β∑n=0∞pnUnx,t+∑n=0∞pnHnUx,t,∑n=0∞pnVnx,t=b0x+p×S-1vαS-Dx3λ∑n=0∞pnVnx,t+∑n=0∞pnHnVx,t,where HnUx,t and HnVx,t are homotopy polynomials coefficients of the nonlinear term, which are given by (70)∑n=0∞pnHnUx,t=6aUx,tDxγUx,t+2bVx,tDxδVx,t,∑n=0∞pnHnVx,t=-3Ux,tDxτVx,t.Setting (71)Unx,t≔U~nxtnαΓnα+1,Vnx,t≔V~nxtnαΓnα+1,then (72)HnUx,t=6a∑i=0nU~ixDxγU~n-ixtnαΓiα+1Γn-iα+1+2b∑i=0ntnαΓiα+1Γn-iα+1V~ixDxδV~n-ix,HnVx,t=-3∑i=0nU~ixDxτV~n-ixtnαΓiα+1Γn-iα+1.Comparing the coefficients of p, (73)p0:U0x,t=U~0x=a0x,V0x,t=V~0x=b0x,p1:U1x,t=S-1uαSaDx3βU0x,t+H0Ux,t=S-1uαSaDx3βU~0x+6aU~0xDxγU~0x+2bV~0xDxδV~0x=tαΓα+1aDx3βU~0x+6aU~0xDxγU~0x+2bV~0xDxδV~0x=tαΓα+1U~1x,V1x,t=S-1vαS-Dx3λV0x,t+H0Vx,t=S-1vαS-Dx3λV~0x-3U~0DxτV~0x=tαΓα+1-Dx3λV~0x-3U~0DxτV~0x=tαΓα+1V~1x.Generally, we get (74)pk:Ukx,t=tkαΓkα+1aDx3βU~k-1x+∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+16aU~ixDxγU~k-1-ix+2bV~ixDxδV~k-1-ix=tkαΓkα+1U~kx,Vkx,t=tkαΓkα+1-Dx3λV~k-1x-∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+13U~ixDxτV~k-1-ix=tkαΓkα+1V~kx,where(75)U~kx=aDx3βU~k-1x+∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+16aU~ixDxγU~k-1-ix+2bV~ixDxδV~k-1-ix,V~kx=-Dx3λV~k-1x-∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+13U~ixDxτV~k-1-ix.Thus, the series solution of (65) is (76)Ux,t=∑k=0∞Ukx,t=∑k=0∞U~kxtkαΓkα+1,Vx,t=∑k=0∞Vkx,t=∑k=0∞V~kxtkαΓkα+1.


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

Under the Boussinesq approximation, Whitham [41], Broer [42], and Kaup [43] obtained the following nonlinear WBK system. In this subsection, we construct the approximate solution by the HPST method to the time-space fractional coupled WBK system.Consider the time-space fractional coupled WBK system(77)DtαU+UDxβU+DxγV+aDx2δU=0,DtαV+DxλUV-aDx2τV+bDx3θU=0with respect to the initial conditions (78)Ux,0=a0x,Vx,0=b0x,where 0<α,β,γ,λ≤1, 1/2<δ,τ≤1, 2/3<θ≤1, x,t∈R×0,∞, a, b∈R denote different dispersive power, U=Ux,t is the field of horizontal velocity, and V=Vx,t is the height deviating equilibrium position of liquid.Applying the Sumudu transform on both sides of (77) with the initial conditions, we obtain(79)SUx,t=a0x+uαS-Ux,tDxβUx,t-DxγVx,t-aDx2δUx,t,SVx,t=b0x+vαS-DxλUx,tVx,t+aDx2τVx,t-bDx3θUx,t.The inverse Sumudu transform of (79) implies that(80)Ux,t=a0x+S-1uαS-Ux,tDxβUx,t-DxγVx,t-aDx2δUx,t,Vx,t=b0x+S-1vαS-DxλUx,tVx,t+aDx2τVx,t-bDx3θUx,t.Using homotopy perturbation method, it leads to(81)∑n=0∞pnUnx,t=a0x+p×S-1uαS-Dxγ∑n=0∞pnVnx,t-aDx2δ∑n=0∞pnUnx,t-∑n=0∞pnHnUx,t,∑n=0∞pnVnx,t=b0x+p×S-1vαSaDx2τ∑n=0∞pnVnx,t-bDx3θ∑n=0∞pnUnx,t-∑n=0∞pnHnVx,t,where HnUx,t and HnVx,t are polynomials of the nonlinear term and are given by (82)∑n=0∞pnHnUx,t=Ux,tDxβUx,t,∑n=0∞pnHnVx,t=DxλVx,tUx,t.Setting (83)Unx,t≔U~nxtnαΓnα+1,Vnx,t≔V~nxtnαΓnα+1,then (84)HnUx,t=∑i=0nU~ixDxβU~n-ixtnαΓiα+1Γn-iα+1,HnVx,t=Dxλ∑i=0nU~ixV~n-ixtnαΓiα+1Γn-iα+1.Comparing the coefficients of p, this gives(85)p0:U0x,t=U~0x=a0x,V0x,t=V~0x=b0x,p1:U1x,t=S-1uαS-DxγV0x,t-aDx2δU0x,t-H0Ux,t=S-1uαS-DxγV~0x-aDx2δU~0x-U~0xDxβU~0x=tαΓα+1-DxγV~0x-aDx2δU~0x-U~0xDxβU~0x=tαΓα+1U~1x,V1x,t=S-1vαSaDx2τV0x,t-bDx3θU0x,t-H0Vx,t=S-1vαSaDx2τV~0x-bDx3θU~0x-DxλU~0xV~0x=tαΓα+1aDx2τV~0x-bDx3θU~0x-DxλU~0xV~0x=tαΓα+1V~1x.Generally, we have (86)pk:Ukx,t=tkαΓkα+1-DxγV~k-1x-aDx2δU~k-1x-∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+1U~ixDxβU~k-1-ix=tkαΓkα+1U~kx,Vkx,t=tkαΓkα+1aDx2τV~k-1x-bDx3θU~k-1x-∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+1DxλU~ixV~k-1-ix=tkαΓkα+1V~kx,where (87)U~kx=-DxγV~k-1x-aDx2δU~k-1x-∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+1U~ixDxβU~k-1-ix,V~kx=aDx2τV~k-1x-bDx3θU~k-1x-∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+1DxλU~ixV~k-1-ix.Hence, the series solution of (77) is (88)Ux,t=∑k=0∞Ukx,t=∑k=0∞U~kxtkαΓkα+1,Vx,t=∑k=0∞Vkx,t=∑k=0∞V~kxtkαΓkα+1.Remark 10.
Whenalpha=τ=δ=λ=θ=1, β≠0, and γ=0, (77) reduces to the classical long-wave system that describes the shallow water wave with diffusion.Remark 11.
Whenα=τ=δ=λ=θ=1, β=0, and γ=1, (77) reduces to the variant Boussinesq system.


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

The Burgers equation is one of the most important partial differential equations from fluid mechanics, which not only describes many phenomena, for example, modeling the motion of turbulence [33], but also has many applications in science and engineering [34]. Here we apply HPSTM to solve the following nonlinear time-space fractional coupled Burgers system:(23)DtαU-Dx2U-2UDxβU+DxUV=0,DtγV-Dx2V-2VDxδV+DxUV=0,with the initial conditions (24)Ux,0=sin⁡x,Vx,0=sin⁡x,where 0<α,β,δ,γ≤1, x,t∈R×0,∞.Applying the Sumudu transform on both sides of (23) with the initial conditions, we can obtain(25)SUx,t=sin⁡x+uαSDx2Ux,t+2Ux,tDxβUx,t-DxUx,tVx,t,SVx,t=sin⁡x+vγSDx2Vx,t+2Vx,tDxδVx,t-DxUx,tVx,t.The inverse Sumudu transform of (25) implies that(26)Ux,t=sin⁡x+S-1uαSDx2Ux,t+2Ux,tDxβUx,t-DxUx,tVx,t,Vx,t=sin⁡x+S-1vγSDx2Vx,t+2Vx,tDxδVx,t-DxUx,tVx,t.Now applying the homotopy perturbation method gives (27)∑n=0∞pnUnx,t=sin⁡x+p×S-1uαSDx2∑n=0∞pnUnx,t+∑n=0∞pnHnUx,t,∑n=0∞pnVnx,t=sin⁡x+p×S-1vγSDx2∑n=0∞pnVnx,t+∑n=0∞pnHnVx,t,where HnUx,t and HnVx,t are polynomials which denote the homotopy coefficients of the nonlinear term and are given by (28)∑n=0∞pnHnUx,t=2Ux,tDxβUx,t-DxUx,tVx,t,∑n=0∞pnHnVx,t=2Vx,tDxδVx,t-DxUx,tVx,t.Set (29)Unx,t≔U~nxtnαΓnα+1,Vnx,t≔V~nxtnγΓnγ+1,and then (30)HnUx,t=2∑i=0nU~ixDxβU~n-ixtnαΓiα+1Γn-iα+1-∑i=0nDxU~ixV~n-ixtnαΓiα+1Γn-iα+1,HnVx,t=2∑i=0nV~ixDxδV~n-ixtnγΓiγ+1Γn-iγ+1-∑i=0nDxU~ixV~n-ixtnγΓiγ+1Γn-iγ+1.Comparing the coefficients of p, this gives (31)p0:U0x,t=U~0x=sin⁡x,V0x,t=V~0x=sin⁡x,p1:U1x,t=S-1uαSDx2U0x,t+H0ux,t=S-1uαSDx2U~0x+2U~0xDxβU~0x-DxU~0xV~0x=tαΓα+1Dx2U~0x+2U~0xDxβU~0x-DxU~0xV~0x=tαΓα+1U~1x,V1x,t=S-1uγSDx2V0x,t+H0vx,t=S-1uγSDx2V~0x+2V~0xDxδV~0x-DxU~0xV~0x=tγΓγ+1Dx2V~0x+2V~0xDxδU~0x-DxU~0xV~0x=tγΓγ+1V~1x.Generally, we have(32)pk:Ukx,t=tkαΓkα+1Dx2U~k-1x+∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+12U~ixDxβU~k-1-ix-DxU~ixV~k-1-ix=tkαΓkα+1U~kx,Vkx,t=tkγΓkγ+1Dx2V~k-1x+∑i=0k-1Γk-1γ+1Γiγ+1Γk-i-1γ+12V~ixDxδV~k-1-ix-DxU~ixV~k-1-ix=tkγΓkγ+1V~kx,where (33)U~kx=Dx2U~k-1x+∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+12U~ixDxβU~k-1-ix-DxU~ixV~k-1-ix,V~kx=Dx2V~k-1x+∑i=0k-1Γk-1γ+1Γiγ+1Γk-i-1γ+12V~ixDxδV~k-1-ix-DxU~ixV~k-1-ix.Hence, the series solution of (23) is (34)Ux,t=∑k=0∞Ukx,t=∑k=0∞U~kxtkαΓkα+1,Vx,t=∑k=0∞Vkx,t=∑k=0∞V~kxtkγΓkγ+1.Particularly, when α=γ=β=δ=1, the exact solution of (23) is (35)Ux,t=e-tsin⁡x,Vx,t=e-tsin⁡x.Using HPSTM, when α=γ=β=δ=1, the third approximate solution of (23) is (36)Uappx,t=sin⁡x+t2!-sin⁡x+t23!sin⁡x+t34!-sin⁡x=sin⁡x1+-t2!+-t23!+-t34!,Vappx,t=sin⁡x+t2!-sin⁡x+t23!sin⁡x+t34!-sin⁡x=sin⁡x1+-t2!+-t23!+-t34!.In general, the limit of the approximate solution is (37)Ux,t=sin⁡x+t2!-sin⁡x+t23!sin⁡x+t34!-sin⁡x+⋯=sin⁡x1+-t2!+-t23!+-t34!+⋯=e-tsin⁡x,Vx,t=sin⁡x+t2!-sin⁡x+t23!sin⁡x+t34!-sin⁡x+⋯=sin⁡x1+-t2!+-t23!+-t34!+⋯=e-tsin⁡x,which is as same as the exact solution. However, if the initial values are too complex to find the limit of the approximated solution, then we replace the exact solution by the approximated solution within a certain scale, which is useful in the application of engineering.Thus we plot the images of the approximate solution (see Figure1(a)), the exact solution (see Figure 1(b)), and the error function (see Figures 1(c) and 1(d)). It is clear that the error function Uapp-Uex depends on time t. When time t is small (e.g., t=0.01), the error function is in the scale of 10-3 (see Figure 1(c)), which indicates that this is a good approximation in the neighbour of time 0 for system (23) with some explicit parameters. However, when time becomes large (e.g., t=0.1), the error function tends to be large as well (see Figure 1(d)); that is to say, this method is only suitable for constructing the approximated solution around the initial data.Figure 1
(a)
The third-order approximate solution ofU(x,t)
(b)
The exact solution ofU(x,t)
(c)
The errorUapp-Uex of U(x,t) at t=0.01
(d)
The errorUapp-Uex of U(x,t) at t=0.1


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

In this subsection, consider the time-space fractional generalization of the Hirota-Satsuma coupled KdV system(38)DtαU-12Dx3βU+3UDxγU-3DxδVW=0,DtαV+Dx3λV-3UDxτV=0,DtαW+Dx3θW-3UDxσW=0,with respect to the initial conditions (39)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×0,∞. The Hirota-Satsuma coupled KdV equation describes the unidirectional propagation of shallow water waves, which was initiated by Wu et al. [35]. Further (38) becomes a generalized fractional KdV equation for U=0 and a fractional MKdV equation for V=0.Applying the Sumudu transform on both sides of (38) with the initial conditions, we obtain(40)SUx,t=a0x+uαS12Dx3βUx,t-3Ux,tDxγUx,t+3DxδVx,tWx,t,SVx,t=b0x-vαSDx3λVx,t-3Ux,tDxτVx,t,SWx,t=c0x-wαSDx3θWx,t-3Ux,tDxσWx,t.The inverse Sumudu transform of (40) implies that(41)Ux,t=a0x+S-1uαS12Dx3βUx,t-3Ux,tDxγUx,t+3DxδVx,tWx,t,Vx,t=b0x-S-1vαSDx3λVx,t-3Ux,tDxτVx,t,Wx,t=c0x-S-1wαSDx3θWx,t-3Ux,tDxσWx,t.Via the homotopy perturbation method, it gives(42)∑n=0∞pnUnx,t=a0x+p×S-1uαS12Dx3β∑n=0∞pnUnx,t+∑n=0∞pnHnUx,t,∑n=0∞pnVnx,t=b0x+p×S-1vαS-Dx3λ∑n=0∞pnVnx,t+∑n=0∞pnHnVx,t,∑n=0∞pnWnx,t=c0x+p×S-1wαS-Dx3θ∑n=0∞Wnx,t+∑n=0∞pnHnWx,t,where HnUx,t, HnVx,t, and HnWx,t are polynomials which denote the nonlinear term, and they are given by (43)∑n=0∞pnHnUx,t=3DxδVx,tWx,t-3Ux,tDxγUx,t,∑n=0∞pnHnVx,t=3Ux,tDxτVx,t,∑n=0∞pnHnWx,t=3Ux,tDxσWx,t.Set (44)Unx,t≔U~nxtnαΓnα+1,Vnx,t≔V~nxtnαΓnα+1,Wnx,t≔W~nxtnαΓnα+1,and then (45)HnU=3Dxδ∑i=0nV~ixW~n-ixtnαΓiα+1Γn-iα+1-3∑i=0nU~ixDxγU~n-ixtnαΓiα+1Γn-iα+1,HnV=3∑i=0nU~ixDxτV~n-ixtnαΓiα+1Γn-iα+1,HnW=3∑i=0nU~ixDxσW~n-ixtnαΓiα+1Γn-iα+1.Comparing the coefficients of p shows(46)p0:U0x,t=U~0x=a0x,V0x,t=V~0x=b0x,W0x,t=W~0x=c0x,p1:U1x,t=S-1uαS12Dx3βU0x,t+H0Ux,t=S-1uαS12Dx3βU~0x+3DxδV~0xW~0x-3U~0xDxγU~0x=tαΓα+112Dx3βU~0x+3DxδV~0xW~0x-3U~0xDxγU~0x=tαΓα+1U~1x,V1x,t=S-1vαS-Dx3λV0x,t+H0Vx,t=S-1vαS-Dx3λV~0x+3U~0xDxτV~0x=tαΓα+1-Dx3λV~0x+3U~0xDxγU~0x=tαΓα+1V~1x,W1x,t=S-1wαS-Dx3θW0x,t+H0Wx,t=S-1wαS-Dx3θW~0x+3U~0xDxσW~0x=tαΓα+1-Dx3θW~0x+3U~0xDxσW~0x=tαΓα+1W~1x.Generally, one has(47)pk:Ukx,t=tkαΓkα+112Dx3βU~k-1x+∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+13DxδV~k-1-ixW~ix-3U~ixDxγU~k-1-ix=tkαΓkα+1U~kx,Vkx,t=tkαΓkα+1-Dx3λV~k-1x+∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+13U~ixDxτV~k-1-ix=tkαΓkα+1V~kx,Wkx,t=tkαΓkα+1-Dx3θW~k-1x+∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+13U~ixDxσW~k-1-ix=tkαΓkα+1W~kx,where(48)U~kx=12Dx3βU~k-1x+∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+13DxδV~k-1-ixW~ix-3U~ixDxγU~k-1-ix,V~kx=-Dx3λV~k-1x+∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+13U~ixDxτV~k-1-ix,W~kx=-Dx3θW~k-1x+∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+13U~ixDxσW~k-1-ix.Therefore, the approximate series solution of (38) is (49)Ux,t=∑k=0∞Ukx,t=∑k=0∞U~kxtkαΓkα+1,Vx,t=∑k=0∞Vkx,t=∑k=0∞V~kxtkαΓkα+1,Wx,t=∑k=0∞Wkx,t=∑k=0∞W~kxtkαΓkα+1.Particularly, when α=γ=δ=λ=1, τ=β=σ=θ=2/3, and the special initial value of (38) is (50)Ux,0=-2+4tanh2⁡x,Vx,0=-4+4tanh2⁡x,Wx,0=1+tanh2⁡x,then the exact solutions are (51)Ux,t=-2+4tanh2⁡x+2t,Vx,t=-4+4tanh2⁡x+2t,Wx,t=1+tanh2⁡x+2t.Under these special conditions, via HPSTM, the first approximate solution of (38) is (52)Uappx,t=-2+4tanh2⁡x+16tanh⁡x-tanh3⁡xt,Vappx,t=-4+4tanh2⁡x+16tanh⁡x-tanh3⁡xt,Wappx,t=1+tanh2⁡x+4tanh⁡x-tanh3⁡xt.Similarly, we obtain the following numerical results: see Figures 2(a), 2(b), 2(c), 2(d), 3(a), 3(b), 3(c), 3(d), 4(a), 4(b), 4(c), and 4(d).Figure 2
(a)
The first-order approximate solution ofU(x,t)
(b)
The exact solution ofU(x,t)
(c)
The errorUapp-Uex of U(x,t) at t=0.01
(d)
The errorUapp-Uex of U(x,t) at t=0.1Figure 3
(a)
The first-order approximate solution ofV(x,t)
(b)
The exact solution ofV(x,t)
(c)
The errorVapp-Vex of V(x,t) at t=0.01
(d)
The errorVapp-Vex of V(x,t) at t=0.1Figure 4
(a)
The first-order approximate solution ofW(x,t)
(b)
The exact solution ofW(x,t)
(c)
The errorWapp-Wex of W(x,t) at t=0.01
(d)
The errorWapp-Wex of W(x,t) at t=0.1


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

Shallow water systems are widely used in many areas of fluid dynamics, such as multiphase flows [36], turbulence [37], and viscoelasticity [38]. It is well known that the shallow water systems can accurately predict both the hydraulic parameters under conditions of slow erosion and low sediment concentration. Let us consider the time-space fractional coupled shallow water system(53)DtαU+UDxβU+DxγV+aDx2δU=0,DtαV+VDxλU+UDxτV-aDx2θV+bDx3σU=0with initial values(54)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×0,1/3.Applying the Sumudu transform on both sides of (53) with the initial conditions, we obtain(55)SUx,t=a0x+uαS-Ux,tDxβUx,t-DxγVx,t-aDx2δUx,t,SVx,t=b0x+vαS-Vx,tDxλUx,t-Ux,tDxτVx,t+aDx2θVx,t-bDx3σUx,t.The inverse Sumudu transform of (55) implies that(56)Ux,t=a0x+S-1uαS-Ux,tDxβUx,t-DxγVx,t-aDx2δUx,t,Vx,t=b0x+S-1vαS-Vx,tDxλUx,t-Ux,tDxτVx,t+aDx2θVx,t-bDx3σUx,t.According to homotopy perturbation method, we have(57)∑n=0∞pnUnx,t=a0x+p×S-1uαS-Dxγ∑n=0∞pnVnx,t-aDx2δ∑n=0∞pnUnx,t-∑n=0∞pnHnUx,t,∑n=0∞pnVnx,t=b0x+p×S-1vαSaDx2θ∑n=0∞pnVnx,t-bDx3σ∑n=0∞pnUnx,t-∑n=0∞pnHnVx,t,where HnUx,t and HnVx,t are polynomials of the nonlinear term and are given by (58)∑n=0∞pnHnUx,t=Ux,tDxβUx,t,∑n=0∞pnHnVx,t=Vx,tDxλUx,t+Ux,tDxτVx,t.Setting (59)Unx,t≔U~nxtnαΓnα+1,Vnx,t≔V~nxtnαΓnα+1,then we arrive at (60)HnUx,t=∑i=0nU~ixDxβU~n-ixtnαΓiα+1Γn-iα+1,HnVx,t=∑i=0nV~ixDxλU~n-ix+∑i=0nU~ixDxτV~n-ixtnαΓiα+1Γn-iα+1.Comparing the coefficients of p yields(61)p0:U0x,t=U~0x=a0x,V0x,t=V~0x=b0x,p1:U1x,t=S-1uαS-DxγV0x,t-aDx2δU0x,t-H0Ux=S-1uαS-DxγU~0x-aDx2δU~0x-U~0xDxβU~0x=tαΓα+1-DxγU~0x-aDx2δU~0x-U~0xDxβU~0x=tαΓα+1U~1x,V1x,t=S-1vαS-bDx3σV0x,t+aDx2θV0x,t-H0Vx=S-1vαS-bDx3σV~0x+aDx2θV~0x-V~0xDxλU~0x-U~0xDxτV~0x=tαΓα+1-bDx3σV~0x+aDx2θV~0x-V~0xDxλU~0x-U~0xDxτV~0x=tαΓα+1V~1x.Generally, we have (62)pk:Ukx,t=tkαΓkα+1-DxγV~k-1x-aDx2δU~k-1x-∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+1U~ixDxβU~k-1-ix=tkαΓkα+1U~kx,Vkx,t=tkαΓkα+1-bDx3σV~k-1x+aDx2θV~k-1x-∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+1U~ixDxλV~k-1-ix+V~ixDxτU~k-1-ix=tkαΓkα+1V~kx,where(63)U~kx=-DxγV~k-1x-aDx2δU~k-1x-∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+1U~ixDxβU~k-1-ix,V~kx=-bDx3σV~k-1x+aDx2θV~k-1x-∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+1U~ixDxλV~k-1-ix+V~ixDxτU~k-1-ix.Hence, the series solution is (64)Ux,t=∑k=0∞Ukx,t=∑k=0∞U~kxtkαΓkα+1,Vx,t=∑k=0∞Vkx,t=∑k=0∞V~kxtkαΓkα+1.


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

KdV equation plays an important role in nonlinear equations for wide applications in physics and engineering. Hirota and Satsuma [39] firstly found coupled KdV system to describe the iterations of water waves; meanwhile, they claimed that the system exists with a soliton solution. In [40], Fan and Zhang settled several kinds of solutions by an improved homogeneous method. The time-space fractional coupled KdV equation is a generalization of the classical coupled KdV equation. In this subsection, we consider the following time-space fractional coupled KdV system:(65)DtαU-aDx3βU-6aUDxγU-2bVDxδV=0,DtαV+Dx3λV+3UDxτV=0,with respect to initial values(66)Ux,0=a0x,Vx,0=b0x,where 0<α,γ,δ,τ≤1, 2/3<β, λ≤1, (x,t)∈R×0,∞, and the coefficients a, b are constants.Applying the Sumudu transform on both sides of (65) with the initial conditions, we obtain(67)SUx,t=a0x+uαSaDx3βUx,t+6aUx,tDxγUx,t+2bVx,tDxδVx,t,SVx,t=b0x-vαSDx3λVx,t+3Ux,tDxτVx,t.The inverse Sumudu transform of (67) implies that(68)Ux,t=a0x+S-1uαSaDx3βUx,t+6aUx,tDxγUx,t+2bVx,tDxδVx,t,Vx,t=b0x-S-1vαSDx3λVx,t+3Ux,tDxτVx,t.Analogously, using homotopy perturbation method gives (69)∑n=0∞pnUnx,t=a0x+p×S-1uαSaDx3β∑n=0∞pnUnx,t+∑n=0∞pnHnUx,t,∑n=0∞pnVnx,t=b0x+p×S-1vαS-Dx3λ∑n=0∞pnVnx,t+∑n=0∞pnHnVx,t,where HnUx,t and HnVx,t are homotopy polynomials coefficients of the nonlinear term, which are given by (70)∑n=0∞pnHnUx,t=6aUx,tDxγUx,t+2bVx,tDxδVx,t,∑n=0∞pnHnVx,t=-3Ux,tDxτVx,t.Setting (71)Unx,t≔U~nxtnαΓnα+1,Vnx,t≔V~nxtnαΓnα+1,then (72)HnUx,t=6a∑i=0nU~ixDxγU~n-ixtnαΓiα+1Γn-iα+1+2b∑i=0ntnαΓiα+1Γn-iα+1V~ixDxδV~n-ix,HnVx,t=-3∑i=0nU~ixDxτV~n-ixtnαΓiα+1Γn-iα+1.Comparing the coefficients of p, (73)p0:U0x,t=U~0x=a0x,V0x,t=V~0x=b0x,p1:U1x,t=S-1uαSaDx3βU0x,t+H0Ux,t=S-1uαSaDx3βU~0x+6aU~0xDxγU~0x+2bV~0xDxδV~0x=tαΓα+1aDx3βU~0x+6aU~0xDxγU~0x+2bV~0xDxδV~0x=tαΓα+1U~1x,V1x,t=S-1vαS-Dx3λV0x,t+H0Vx,t=S-1vαS-Dx3λV~0x-3U~0DxτV~0x=tαΓα+1-Dx3λV~0x-3U~0DxτV~0x=tαΓα+1V~1x.Generally, we get (74)pk:Ukx,t=tkαΓkα+1aDx3βU~k-1x+∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+16aU~ixDxγU~k-1-ix+2bV~ixDxδV~k-1-ix=tkαΓkα+1U~kx,Vkx,t=tkαΓkα+1-Dx3λV~k-1x-∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+13U~ixDxτV~k-1-ix=tkαΓkα+1V~kx,where(75)U~kx=aDx3βU~k-1x+∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+16aU~ixDxγU~k-1-ix+2bV~ixDxδV~k-1-ix,V~kx=-Dx3λV~k-1x-∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+13U~ixDxτV~k-1-ix.Thus, the series solution of (65) is (76)Ux,t=∑k=0∞Ukx,t=∑k=0∞U~kxtkαΓkα+1,Vx,t=∑k=0∞Vkx,t=∑k=0∞V~kxtkαΓkα+1.


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

Under the Boussinesq approximation, Whitham [41], Broer [42], and Kaup [43] obtained the following nonlinear WBK system. In this subsection, we construct the approximate solution by the HPST method to the time-space fractional coupled WBK system.Consider the time-space fractional coupled WBK system(77)DtαU+UDxβU+DxγV+aDx2δU=0,DtαV+DxλUV-aDx2τV+bDx3θU=0with respect to the initial conditions (78)Ux,0=a0x,Vx,0=b0x,where 0<α,β,γ,λ≤1, 1/2<δ,τ≤1, 2/3<θ≤1, x,t∈R×0,∞, a, b∈R denote different dispersive power, U=Ux,t is the field of horizontal velocity, and V=Vx,t is the height deviating equilibrium position of liquid.Applying the Sumudu transform on both sides of (77) with the initial conditions, we obtain(79)SUx,t=a0x+uαS-Ux,tDxβUx,t-DxγVx,t-aDx2δUx,t,SVx,t=b0x+vαS-DxλUx,tVx,t+aDx2τVx,t-bDx3θUx,t.The inverse Sumudu transform of (79) implies that(80)Ux,t=a0x+S-1uαS-Ux,tDxβUx,t-DxγVx,t-aDx2δUx,t,Vx,t=b0x+S-1vαS-DxλUx,tVx,t+aDx2τVx,t-bDx3θUx,t.Using homotopy perturbation method, it leads to(81)∑n=0∞pnUnx,t=a0x+p×S-1uαS-Dxγ∑n=0∞pnVnx,t-aDx2δ∑n=0∞pnUnx,t-∑n=0∞pnHnUx,t,∑n=0∞pnVnx,t=b0x+p×S-1vαSaDx2τ∑n=0∞pnVnx,t-bDx3θ∑n=0∞pnUnx,t-∑n=0∞pnHnVx,t,where HnUx,t and HnVx,t are polynomials of the nonlinear term and are given by (82)∑n=0∞pnHnUx,t=Ux,tDxβUx,t,∑n=0∞pnHnVx,t=DxλVx,tUx,t.Setting (83)Unx,t≔U~nxtnαΓnα+1,Vnx,t≔V~nxtnαΓnα+1,then (84)HnUx,t=∑i=0nU~ixDxβU~n-ixtnαΓiα+1Γn-iα+1,HnVx,t=Dxλ∑i=0nU~ixV~n-ixtnαΓiα+1Γn-iα+1.Comparing the coefficients of p, this gives(85)p0:U0x,t=U~0x=a0x,V0x,t=V~0x=b0x,p1:U1x,t=S-1uαS-DxγV0x,t-aDx2δU0x,t-H0Ux,t=S-1uαS-DxγV~0x-aDx2δU~0x-U~0xDxβU~0x=tαΓα+1-DxγV~0x-aDx2δU~0x-U~0xDxβU~0x=tαΓα+1U~1x,V1x,t=S-1vαSaDx2τV0x,t-bDx3θU0x,t-H0Vx,t=S-1vαSaDx2τV~0x-bDx3θU~0x-DxλU~0xV~0x=tαΓα+1aDx2τV~0x-bDx3θU~0x-DxλU~0xV~0x=tαΓα+1V~1x.Generally, we have (86)pk:Ukx,t=tkαΓkα+1-DxγV~k-1x-aDx2δU~k-1x-∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+1U~ixDxβU~k-1-ix=tkαΓkα+1U~kx,Vkx,t=tkαΓkα+1aDx2τV~k-1x-bDx3θU~k-1x-∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+1DxλU~ixV~k-1-ix=tkαΓkα+1V~kx,where (87)U~kx=-DxγV~k-1x-aDx2δU~k-1x-∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+1U~ixDxβU~k-1-ix,V~kx=aDx2τV~k-1x-bDx3θU~k-1x-∑i=0k-1Γk-1α+1Γiα+1Γk-i-1α+1DxλU~ixV~k-1-ix.Hence, the series solution of (77) is (88)Ux,t=∑k=0∞Ukx,t=∑k=0∞U~kxtkαΓkα+1,Vx,t=∑k=0∞Vkx,t=∑k=0∞V~kxtkαΓkα+1.Remark 10.
Whenalpha=τ=δ=λ=θ=1, β≠0, and γ=0, (77) reduces to the classical long-wave system that describes the shallow water wave with diffusion.Remark 11.
Whenα=τ=δ=λ=θ=1, β=0, and γ=1, (77) reduces to the variant Boussinesq system.


## 5. Concluding Remarks

In this paper, we apply the HPSTM to the nonlinear time-space fractional coupled equations. Applying the HPSTM, we can obtain analytic and approximate solutions to different coupled systems, for example, the coupled Burgers system, the coupled KdV system, the generalized Hirota-Satsuma coupled KdV system, the coupled WBK system, and the coupled shallow water system. The advantage of the HPSTM is its capability of combining two powerful methods for obtaining exact and approximate analytical solutions for nonlinear system. It provides the solutions in terms of convergent series with easily computable components in a direct way without using linearization, perturbation, or restrictive assumptions. The numerical results indicate that this method is effective and simple in constructing analytic or approximate solutions to fractional coupled systems.



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*Source: 3540364-2017-12-25.xml*