# Applications of General Residual Power Series Method to Differential Equations with Variable Coefficients

Authors: Bochao Chen; Li Qin; Fei Xu; Jian Zu
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

This paper is devoted to studying the analytical series solutions for the differential equations with variable coefficients. By a general residual power series method, we construct the approximate analytical series solutions for differential equations with variable coefficients, including nonhomogeneous parabolic equations, fractional heat equations in 2D, and fractional wave equations in 3D. These applications show that residual power series method is a simple, effective, and powerful method for seeking analytical series solutions of differential equations (especially for fractional differential equations) with variable coefficients.


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## Body

## 1. Introduction

In the field of science and engineering, many physical phenomena can be described by differential equations with variable coefficients. For example, some physical problems in inhomogeneous media [1–3]. In the past, many assumptions on integral order differential equations were applied artificially to describe the systems with memory properties and hereditary properties. Some significant information will be lost by such assumptions. Generally, fractional calculus provides an effective tool to describe memory properties and hereditary properties of different materials and processes without extra assumptions. Now the fractional differential equation has attracted a great deal of interest in several areas including chemistry, physics, engineering, and even finance and social sciences [4, 5]. Some recent progress in fractional calculus can be found in [6–8].The analytical series solutions of differential equations are of fundamental importance in applied science. Various numerical and analytical methods are proposed such as Adomian decomposition method [9, 10], Fractional complex transform method [11], and Laplace transform method [12]. Although lots of methods are put forward, scientists are still looking for more effective ways to solve specific problems, especially for the fractional equations with variable coefficients.The residual power series method (RPS), proposed by Abu Arqud in [13], is an efficient and easy method for constructing power series solutions of differential equations without linearization, perturbation, or discretization. Different from the classical power series method, RPS does not need to compare the coefficients of the corresponding terms. This method computes the coefficients of the power series by a chain of equations with one or more variables. One advantage is that RPS is not affected by computational round-off errors and also does not require large computer memory and extensive time. In [14], power series solutions of higher-order ordinary differential equations are obtained by RPS. Inspired by this approach, we present a general residual power series method (GRPS) for constructing power series solutions of time-space fractional differential equations with variable coefficients:(1)Dtmαux,t+PxGu=Fx,t,Dtiαux,0=aix,i=0,…,m-1,where x=(x1,x2,…,xd)∈Rd, α∈(m-1/m,1], m∈N+, and (2)Gu≔Gu,Dtαu,…,Dtm-1αu,Dx1β11u,…,Dxdβ1du,…,Dx1βl1u,…,Dxdβldu,with p-1<βpj⩽p, p=1,…,l; j=1,…,d. Here Dtiα and Dxjβpj mean the Caputo fractional derivative with respect to t of order iα and xj of order βpj, respectively. Such type of differential equation provides an exact description of some physical phenomena in fluid dynamics, electrodynamics, and elastic mechanics.RPS has been extended to many partial differential equations (PDE), especially to fractional partial differential equations (FPDE), such as time-fractional dispersive PDE [15, 16], time-fractional KdV-Burgers equations [17], homogeneous time-fractional wave equation [18], and time-space fractional Boussinesq equations [19]. In the present paper, we will apply GRPS to a series of PDE with variable coefficients, including fourth-order parabolic equations, fractional heat equation, and fractional wave equation. For other approximation and numerical techniques for FPDE, we refer to finite difference methods [20, 21], differential transform method [22, 23], wavelet method [24], Adomian’s decomposition method [25], variational iteration method [26, 27], homotopy analysis method [28], homotopy perturbation method [29], tau method [30, 31], and so on.The paper is organized as follows: some necessary definitions and theorems will be presented in Section2. In Section 3, we propose the main steps of GRPS for the general time-space fractional equations with variable coefficients. In Section 4, the applications of GRPS to some different equations with variable coefficients are given, including fourth-order parabolic equations, fractional heat equations, and fractional wave equations. Finally, conclusions are presented in Section 5.


## 2. Concepts on Fractional Calculus Theory

There are several definitions of the fractional integration with orderα≥0, and they are not necessarily equivalent to each other. The two most common ones are Riemann-Liouville’s definition and Caputo’s definition; see [32, 33].Definition 1.
The Mittag-Leffler function is defined as follows:(3)Eαx≔∑k=0∞xkΓαk+1,α>0.Definition 2.
A real functionu(x,t), x∈I, t>0 is said to be in the space Cμ(I×R+), μ∈R, if there exists a real number p>μ such that u(x,t)=tpf(x,t), where f(x,t)∈C(I×R+), and it is said to be in the space Cμn, if ∂n/∂tnu(x,t)∈Cμ, n∈N.Definition 3.
Letu(x,t)∈Cμ(I×R+), μ≥-1. The Riemann-Liouville fractional integral operator of order α≥0 of u(x,t) is defined as follows: (4)Jtαux,t≔1Γα∫0tt-τα-1ux,τdτ,α>0,x∈I,t>τ≥0,ux,t,α=0.Definition 4.
The Caputo time-fractional derivative operator of orderα of u(x,t) is defined as follows: (5)Dtαux,t≔Jtn-α∂nux,t∂tn,n-1<α<n,n∈N+,∂nux,t∂tn,α=n.Definition 5.
The Caputo space fractional derivative operator of orderβ is defined as follows: (6)Dxβux,t≔Jxn-β∂nux,t∂xn,n-1<β<n,n∈N+,∂nux,t∂xn,α=n.Definition 6.
A power series representation of the form(7)∑n=0∞cnt-t0nα≔c0+c1t-t0α+c2t-t02α+⋯is called a fractional power series (FPS) about t0, where t is a variable and cn are the coefficients of the series.Theorem 7 (see [34]).
Suppose thatf has a FPS representation at t0 of the form (8)ft=∑n=0∞cnt-t0nα,t0⩽t<t0+R,where R is the radius of convergence of the FPS. If Dtnαf(t)∈C(t0,t0+R) for n=0,1,2,⋯, then the coefficients cn will take the form of (9)cn=Dtnαftt=t0Γnα+1,where Dtnα=Dtα·Dtα·⋯Dtα (n-times).


## 3. Algorithm of GRPS

In this section, we give a general RPS to obtain fractional power series solutions for any-order time-space fractional differential equations with variable coefficients (1). The analytic function u(x,t) can be expanded as follows:(10)ux,t=∑n=0∞unx,t≔∑n=0∞Cnxtnα,x∈I⊂Rd,t<R,where R is the radius of convergence of above series. Substitute the initial conditions in (1); we have (11)aix=CixΓiα+1,i=0,1,…,m-1,which implies (12)uix,t=Cixtiα=aixΓiα+1tiα,i=0,1,…,m-1.So we have the initial guess approximation of u(x,t) in the following form:(13)uinitialx,t≔u0x,t+u1x,t+⋯+um-1x,t=a0x+a1xΓα+1tα+⋯+am-1xΓm-1α+1tm-1α.Define the approximate solution of (1) by the kth truncated series:(14)ukx,t≔uinitialx,t+∑i=mkCixtiα,k=m,m+1,m+2⋯.Before applying GRPS to solve (1), we give some notations(15)Resu,x,t≔Dtmαux,t+PxGu-Fx,t.Substituting the kth truncated approximate solutions uk(x,t) into (1), we obtain the kth residual function(16)Resku,x,t≔Dtmαukx,t+PxGku-Fx,t,where (17)Gku=Guk,Dtαuk,…,Dtm-1αuk,Dx1β11uk,…,Dxdβ1duk,…,Dx1βl1uk,…,Dxdβlduk.Then we have the following facts: (1)
limk⟶∞⁡uk(x,t)=u(x,t);(2)
Res(u,x,t)=0;(3)
limk⟶∞⁡Resk(u,x,t)=Res(u,x,t),x∈I⊆Rd,|t|<R.Assume that(18)Dtk-mαResku,x,tt=0=0.Since (19)Dtk-mαResku,x,tt=0=CkxΓkα+1+Dtk-mαPxGku-Fx,tt=0,we have(20)Ckx=-Dtk-mαPxGku-Fx,tt=0Γkα+1,k=m,m+1,m+2,⋯.In fact, this relation is a fundamental rule in GRPS. So the FPS solution of (1) is(21)ux,t=uinitialx,t+∑i=m∞Cixtiα=∑i=0m-1aixΓiα+1tiα+∑i=m∞fixΓiα+1tiα,where(22)fkx=-Dtk-mαPxGku-Fx,tt=0,k=m,m+1,m+2,⋯.


## 4. Applications of GRPS to PDEs with Variable Coefficients

### 4.1. Fourth-Order Parabolic Equation with Variable Coefficients inR1

Let us consider the fourth-order parabolic differential equation(23)∂2u∂t2+xsin⁡x-1∂4u∂x4=0,0<x<1,t>0,where x/sin⁡x-1>0 is the ratio of flexural rigidity of the beam to its mass per unit length; see [35]. In [35], the initial conditions and the boundary conditions of (23) are(24)ux,0=x-sin⁡x,∂∂tux,0=-x+sin⁡x,and(25)u0,t=0,u1,t=e-t1-sin⁡1,∂2∂x2u0,t=0,∂2∂x2u1,t=e-tsin⁡1,respectively. According to (10), u can be written in the following form: (26)ux,t=C0x+C1xt+C2xt2+C3xt3+⋯.The initial approximation is (27)uinitialx,t=x-sin⁡x-x-sin⁡xt.Now by (15), denote (28)Resx,t=∂2ux,t∂t2+xsin⁡x-1∂4ux,t∂x4,ukx,t=uinitial+∑i=2kfixti,Reskx,t=∂2ukx,t∂t2+xsin⁡x-1∂4ukx,t∂x4.By (16) and (18), we assume that(29)Dtk-2Reskx,0=0,k=2,3,4⋯.Letting k=2 in (29), it shows(30)2!C2x+xsin⁡x-1-sin⁡x=0,which implies(31)C2x=x-sin⁡x2!.So the 2nd truncated approximate solution of (23) is(32)u2x,t=x-sin⁡x-x-sin⁡xt+x-sin⁡x2!t2.Similarly, Ck(x) can be constructed as follows: (33)Ckx=-1kk!x-sin⁡x,k=2,3,4⋯.So the kth truncated approximate solution of (23) is (34)ukx,t=x-sin⁡x∑i=0k-1kk!ti.Finally, if we define(35)ux,t≔limk⟶∞⁡ukx,t=x-sin⁡xe-t,it is easy to verify that u(x,t) in (35) is the exact solution of (23) with boundary value condition (25).Numerical comparisons are studied next. Figure1 shows the exact solution u of (23) with 0≤t≤5. In Figure 2, u9, u10, u11, and u12 represent the 9th-, 10th-, 11th-, and 12th-order truncated approximate solution of u with 0≤t≤5. It shows that these GRPS approximate solutions are convergent to the exact solution u.Figure 1
The exact solutionu of (23).Figure 2
The approximate solutionsu9, u10, u11, and u12 of (23).


### 4.2. Nonhomogeneous Parabolic Equation with Source Term inR1

Let us consider the nonhomogeneous parabolic equation (see [35]):(36)∂2u∂t2+1+x∂4u∂x4=x4+x3-67!x7cos⁡t,0<x<1,t>0,with the initial conditions(37)ux,0=67!x7,∂∂tux,0=0,and the boundary conditions (38)u0,t=0,u1,t=67!cos⁡t,∂2∂x2u0,t=0,∂2∂x2u1,t=120cos⁡t.Assume that u(x,t) is an analytical function with t>0. The initial approximation is(39)uinitialx,t=67!x7.Denote (40)Reskx,t=∂2ukx,t∂t2+1+x∂4ukx,t∂x4-x4+x3-67!x7cos⁡t,where (41)ukx,t=uinitialx,t+C2xt2+⋯+Ckxtk.Let k=2 in (40); it yields(42)Res2x,t=2C2x+1+x63!x3+C24xt2-x4+x3-67!x7cos⁡t.Using the fact that(43)Dtk-2Reskx,0=0,k=2,3,4⋯,we have(44)C2x=-62!7!x7.Thus the 2nd truncated series have the following form:(45)u2x,t=67!x7-62!7!x7t2.Similarly, taking k=3,4,5,6 in (40) we obtain(46)C3x=0,C4x=64!7!x7,C5x=0,C6x=-66!7!x7.Then 6th truncated approximate solution of (36) is (47)u6x,t=67!x7-62!7!x7t2+64!7!x7t4-66!7!x7t6.By (21), we can obtain the solution of (36): (48)ux,t=limk⟶∞⁡ukx,t=67!x7cos⁡t,which is consistent with the solution obtained by Adomian decomposition method [35].Some numerical comparisons are given next. Figure3 shows the exact solution u of (36) with 0≤t≤5. In Figure 4, u19, u20, u21, and u22 represent the 19th-, 20th-, 21st-, and 22nd-order GRPS solution of u with 0≤t≤5. It shows that these approximate solutions are convergent to the exact solution u.Figure 3
The exact solutionu of (36).Figure 4
The approximate solutionsu19, u20, u21, and u22 of (36).


### 4.3. Fractional Heat Equation with Variable Coefficients inR2

Consider the two-dimensional heat equation with variable coefficients(49)Dtαux,y,t=12y2uxxx,y,t+12x2uyyx,y,t,x,y,t∈R+3,0<α≤1,with the initial conditions(50)ux,y,0=y2.Assume that u(x,y,t) is an analytical function on t>0 and the initial approximation solution has the following form: (51)uinitialx,y,t=y2.Then the kth truncated series and kth residual function will be (52)ukx,y,t=uinitialx,y,t+∑m=1kCmx,ytmα=y2+C1x,ytα+⋯+Ckx,ytkα,and (53)Reskx,y,t=Dtαukx,y,t-12y2∂2ukx,y,t∂x2-12x2∂2ukx,y,t∂y2,respectively. By (18), we have(54)Dtk-1αReskx,y,0=0,0<α<1,k=1,2,3,⋯.When k=1 in (54), we obtain (55)C1x,y=x2Γα+1.Thus, the 1st truncated approximate solution of (49)-(50) is (56)u1x,y,t=y2+x2Γα+1tα.Let k=2 in (54); it yields that (57)C2x,y=y2Γ2α+1.Therefore, the 2nd truncated approximate solution of (49)-(50) is (58)u2x,y,t=y2+x2Γα+1tα+y2Γ2α+1t2α.In the similar way, taking k=3,4,5,6 in (54), we can obtain that(59)C3x,y=x2Γ3α+1,C4x,y=y2Γ4α+1,C5x,y=x2Γ5α+1,and (60)C6x,y=y2Γ6α+1.Thus 6th truncated approximate solution of (49)-(50) can be obtained (61)u6x,y,t=y2+x2Γα+1tα+y2Γ2α+1t2α+x2Γ3α+1t3α+y2Γ4α+1t4α+x2Γ5α+1t5α+y2Γ6α+1t6α.Following the same step, we have the exact analytical solutions of (49)-(50): (62)ux,y,t=y21+1Γ2α+1t2α+1Γ4α+1t4α+1Γ6α+1t6α+⋯+x21Γα+1tα+1Γ3α+1t3α+1Γ5α+1t5α+⋯=y2cosh⁡tα,α+x2sinh⁡tα,α.Particularly, if α=1, we obtain the following form: (63)ux,y,t=y2et+e-t2+x2et-e-t2,which is the solution of the integer order heat equation with variable coefficients.Some numerical simulation are presented next. In Figures5 and 6, u9, u10, u11, and u12 represent the 9th-, 10th-, 11th-, and 12th-order GRPS solution of (49) at time t=3 with α=1/2 and α=1, respectively. In Figures 7 and 8, u9, u10, u11, and u12 represent the 9th-, 10th-, 11th-, and 12th-order GRPS solution of (49) at time t=5 with α=1/2 and α=1, respectively. It shows that the speed of convergence of truncated approximate solution of (49) with α=1 is better than the one with α=1/2.Figure 5
The approximate solutionsu9, u10, u11, and u12 of (49) at t=3 with α=1/2.Figure 6
The approximate solutionsu9, u10, u11, and u12 of (49) at t=3 with α=1.Figure 7
The approximate solutionsu9, u10, u11, and u12 of (49) at t=5 with α=1/2.Figure 8
The approximate solutionsu9, u10, u11, and u12 of (49) at t=5 with α=1.


### 4.4. Fractional Wave Equation inR3

Consider the three-dimensional wave equation with variable coefficients(64)Dt2αux,y,z,t=12x2uxxx,y,z,t+y2uyyx,y,z,t+z2uzzx,y,z,t+x2+y2+z2,x,y,z∈R+3,t>0,0<α≤1,with the initial conditions(65)ux,y,z,0=0,Dtαux,y,z,0=x2+y2-z2.First, we construct the initial approximation solution: (66)uinitialx,y,z,t=x2+y2-z2Γα+1tα.Secondly, construct the kth truncated series and kth residual function of (64)-(65) as follows: (67)ukx,y,z,t=uinitialx,y,z,t+∑m=2kCmx,y,ztmα,Reskx,y,z,t=Dt2αukx,y,z,t-x2+y2+z2-12x2uxxkx,y,z,t+y2uyykx,y,z,t+z2uzzkx,y,z,t.By (18), we have(68)Dtk-2αReskx,y,z,0=0,0<α<1,k=2,3,4,⋯.Taking k=2 in (68), it yields (69)C2x,y,z=x2+y2+z2Γ2α+1.Then the 2nd truncated approximate solution will be (70)u2x,y,z,t=1Γα+1x2+y2-z2tα+1Γ2α+1x2+y2+z2t2α.In a similar way, taking k=3,4,5,6 in (68), we have (71)C3x,y,z=1Γ3α+1x2+y2-z2,C4x,y,z=1Γ4α+1x2+y2+z2,C5x,y,z=1Γ5α+1x2+y2-z2,C6x,y,z=1Γ6α+1x2+y2+z2.Then the 6th-order truncated approximate solution of (64)-(65) can be obtained as follows: (72)u6x,y,z,t=1Γα+1x2+y2-z2tα+1Γ2α+1x2+y2+z2t2α+1Γ3α+1x2+y2-z2t3α+1Γ4α+1x2+y2+z2t4α+1Γ5α+1x2+y2-z2t5α+1Γ6α+1x2+y2+z2t6α.Finally, the exact analytical solution of (64)-(65) can be obtained: (73)ux,y,t=x2+y2-z21Γα+1tα+1Γ3α+1t3α+1Γ5α+1t5α+⋯+x2+y2+z21Γ2α+1t2α+1Γ4α+1t4α+1Γ6α+1t6α+⋯=x2+y2-z2sinh⁡tα,α+x2+y2+z2cosh⁡tα,α-1.


## 4.1. Fourth-Order Parabolic Equation with Variable Coefficients inR1

Let us consider the fourth-order parabolic differential equation(23)∂2u∂t2+xsin⁡x-1∂4u∂x4=0,0<x<1,t>0,where x/sin⁡x-1>0 is the ratio of flexural rigidity of the beam to its mass per unit length; see [35]. In [35], the initial conditions and the boundary conditions of (23) are(24)ux,0=x-sin⁡x,∂∂tux,0=-x+sin⁡x,and(25)u0,t=0,u1,t=e-t1-sin⁡1,∂2∂x2u0,t=0,∂2∂x2u1,t=e-tsin⁡1,respectively. According to (10), u can be written in the following form: (26)ux,t=C0x+C1xt+C2xt2+C3xt3+⋯.The initial approximation is (27)uinitialx,t=x-sin⁡x-x-sin⁡xt.Now by (15), denote (28)Resx,t=∂2ux,t∂t2+xsin⁡x-1∂4ux,t∂x4,ukx,t=uinitial+∑i=2kfixti,Reskx,t=∂2ukx,t∂t2+xsin⁡x-1∂4ukx,t∂x4.By (16) and (18), we assume that(29)Dtk-2Reskx,0=0,k=2,3,4⋯.Letting k=2 in (29), it shows(30)2!C2x+xsin⁡x-1-sin⁡x=0,which implies(31)C2x=x-sin⁡x2!.So the 2nd truncated approximate solution of (23) is(32)u2x,t=x-sin⁡x-x-sin⁡xt+x-sin⁡x2!t2.Similarly, Ck(x) can be constructed as follows: (33)Ckx=-1kk!x-sin⁡x,k=2,3,4⋯.So the kth truncated approximate solution of (23) is (34)ukx,t=x-sin⁡x∑i=0k-1kk!ti.Finally, if we define(35)ux,t≔limk⟶∞⁡ukx,t=x-sin⁡xe-t,it is easy to verify that u(x,t) in (35) is the exact solution of (23) with boundary value condition (25).Numerical comparisons are studied next. Figure1 shows the exact solution u of (23) with 0≤t≤5. In Figure 2, u9, u10, u11, and u12 represent the 9th-, 10th-, 11th-, and 12th-order truncated approximate solution of u with 0≤t≤5. It shows that these GRPS approximate solutions are convergent to the exact solution u.Figure 1
The exact solutionu of (23).Figure 2
The approximate solutionsu9, u10, u11, and u12 of (23).


## 4.2. Nonhomogeneous Parabolic Equation with Source Term inR1

Let us consider the nonhomogeneous parabolic equation (see [35]):(36)∂2u∂t2+1+x∂4u∂x4=x4+x3-67!x7cos⁡t,0<x<1,t>0,with the initial conditions(37)ux,0=67!x7,∂∂tux,0=0,and the boundary conditions (38)u0,t=0,u1,t=67!cos⁡t,∂2∂x2u0,t=0,∂2∂x2u1,t=120cos⁡t.Assume that u(x,t) is an analytical function with t>0. The initial approximation is(39)uinitialx,t=67!x7.Denote (40)Reskx,t=∂2ukx,t∂t2+1+x∂4ukx,t∂x4-x4+x3-67!x7cos⁡t,where (41)ukx,t=uinitialx,t+C2xt2+⋯+Ckxtk.Let k=2 in (40); it yields(42)Res2x,t=2C2x+1+x63!x3+C24xt2-x4+x3-67!x7cos⁡t.Using the fact that(43)Dtk-2Reskx,0=0,k=2,3,4⋯,we have(44)C2x=-62!7!x7.Thus the 2nd truncated series have the following form:(45)u2x,t=67!x7-62!7!x7t2.Similarly, taking k=3,4,5,6 in (40) we obtain(46)C3x=0,C4x=64!7!x7,C5x=0,C6x=-66!7!x7.Then 6th truncated approximate solution of (36) is (47)u6x,t=67!x7-62!7!x7t2+64!7!x7t4-66!7!x7t6.By (21), we can obtain the solution of (36): (48)ux,t=limk⟶∞⁡ukx,t=67!x7cos⁡t,which is consistent with the solution obtained by Adomian decomposition method [35].Some numerical comparisons are given next. Figure3 shows the exact solution u of (36) with 0≤t≤5. In Figure 4, u19, u20, u21, and u22 represent the 19th-, 20th-, 21st-, and 22nd-order GRPS solution of u with 0≤t≤5. It shows that these approximate solutions are convergent to the exact solution u.Figure 3
The exact solutionu of (36).Figure 4
The approximate solutionsu19, u20, u21, and u22 of (36).


## 4.3. Fractional Heat Equation with Variable Coefficients inR2

Consider the two-dimensional heat equation with variable coefficients(49)Dtαux,y,t=12y2uxxx,y,t+12x2uyyx,y,t,x,y,t∈R+3,0<α≤1,with the initial conditions(50)ux,y,0=y2.Assume that u(x,y,t) is an analytical function on t>0 and the initial approximation solution has the following form: (51)uinitialx,y,t=y2.Then the kth truncated series and kth residual function will be (52)ukx,y,t=uinitialx,y,t+∑m=1kCmx,ytmα=y2+C1x,ytα+⋯+Ckx,ytkα,and (53)Reskx,y,t=Dtαukx,y,t-12y2∂2ukx,y,t∂x2-12x2∂2ukx,y,t∂y2,respectively. By (18), we have(54)Dtk-1αReskx,y,0=0,0<α<1,k=1,2,3,⋯.When k=1 in (54), we obtain (55)C1x,y=x2Γα+1.Thus, the 1st truncated approximate solution of (49)-(50) is (56)u1x,y,t=y2+x2Γα+1tα.Let k=2 in (54); it yields that (57)C2x,y=y2Γ2α+1.Therefore, the 2nd truncated approximate solution of (49)-(50) is (58)u2x,y,t=y2+x2Γα+1tα+y2Γ2α+1t2α.In the similar way, taking k=3,4,5,6 in (54), we can obtain that(59)C3x,y=x2Γ3α+1,C4x,y=y2Γ4α+1,C5x,y=x2Γ5α+1,and (60)C6x,y=y2Γ6α+1.Thus 6th truncated approximate solution of (49)-(50) can be obtained (61)u6x,y,t=y2+x2Γα+1tα+y2Γ2α+1t2α+x2Γ3α+1t3α+y2Γ4α+1t4α+x2Γ5α+1t5α+y2Γ6α+1t6α.Following the same step, we have the exact analytical solutions of (49)-(50): (62)ux,y,t=y21+1Γ2α+1t2α+1Γ4α+1t4α+1Γ6α+1t6α+⋯+x21Γα+1tα+1Γ3α+1t3α+1Γ5α+1t5α+⋯=y2cosh⁡tα,α+x2sinh⁡tα,α.Particularly, if α=1, we obtain the following form: (63)ux,y,t=y2et+e-t2+x2et-e-t2,which is the solution of the integer order heat equation with variable coefficients.Some numerical simulation are presented next. In Figures5 and 6, u9, u10, u11, and u12 represent the 9th-, 10th-, 11th-, and 12th-order GRPS solution of (49) at time t=3 with α=1/2 and α=1, respectively. In Figures 7 and 8, u9, u10, u11, and u12 represent the 9th-, 10th-, 11th-, and 12th-order GRPS solution of (49) at time t=5 with α=1/2 and α=1, respectively. It shows that the speed of convergence of truncated approximate solution of (49) with α=1 is better than the one with α=1/2.Figure 5
The approximate solutionsu9, u10, u11, and u12 of (49) at t=3 with α=1/2.Figure 6
The approximate solutionsu9, u10, u11, and u12 of (49) at t=3 with α=1.Figure 7
The approximate solutionsu9, u10, u11, and u12 of (49) at t=5 with α=1/2.Figure 8
The approximate solutionsu9, u10, u11, and u12 of (49) at t=5 with α=1.


## 4.4. Fractional Wave Equation inR3

Consider the three-dimensional wave equation with variable coefficients(64)Dt2αux,y,z,t=12x2uxxx,y,z,t+y2uyyx,y,z,t+z2uzzx,y,z,t+x2+y2+z2,x,y,z∈R+3,t>0,0<α≤1,with the initial conditions(65)ux,y,z,0=0,Dtαux,y,z,0=x2+y2-z2.First, we construct the initial approximation solution: (66)uinitialx,y,z,t=x2+y2-z2Γα+1tα.Secondly, construct the kth truncated series and kth residual function of (64)-(65) as follows: (67)ukx,y,z,t=uinitialx,y,z,t+∑m=2kCmx,y,ztmα,Reskx,y,z,t=Dt2αukx,y,z,t-x2+y2+z2-12x2uxxkx,y,z,t+y2uyykx,y,z,t+z2uzzkx,y,z,t.By (18), we have(68)Dtk-2αReskx,y,z,0=0,0<α<1,k=2,3,4,⋯.Taking k=2 in (68), it yields (69)C2x,y,z=x2+y2+z2Γ2α+1.Then the 2nd truncated approximate solution will be (70)u2x,y,z,t=1Γα+1x2+y2-z2tα+1Γ2α+1x2+y2+z2t2α.In a similar way, taking k=3,4,5,6 in (68), we have (71)C3x,y,z=1Γ3α+1x2+y2-z2,C4x,y,z=1Γ4α+1x2+y2+z2,C5x,y,z=1Γ5α+1x2+y2-z2,C6x,y,z=1Γ6α+1x2+y2+z2.Then the 6th-order truncated approximate solution of (64)-(65) can be obtained as follows: (72)u6x,y,z,t=1Γα+1x2+y2-z2tα+1Γ2α+1x2+y2+z2t2α+1Γ3α+1x2+y2-z2t3α+1Γ4α+1x2+y2+z2t4α+1Γ5α+1x2+y2-z2t5α+1Γ6α+1x2+y2+z2t6α.Finally, the exact analytical solution of (64)-(65) can be obtained: (73)ux,y,t=x2+y2-z21Γα+1tα+1Γ3α+1t3α+1Γ5α+1t5α+⋯+x2+y2+z21Γ2α+1t2α+1Γ4α+1t4α+1Γ6α+1t6α+⋯=x2+y2-z2sinh⁡tα,α+x2+y2+z2cosh⁡tα,α-1.


## 5. Conclusions

This paper concerns the analytical series solutions of the differential equations with variable coefficients (integer order or fractional order). By a general residual power residual series method, we construct the analytical approximate solutions and the analytical exact solutions of the differential equations with variable coefficients, for example, nonhomogeneous parabolic equations, fractional heat equations in 2D, and fractional wave equations in 3D. It shows that GRPS is a direct, simple, and efficient method which could be widely applied to many other PDEs with variable coefficients.



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*Source: 2394735-2018-07-09.xml*