https://doi.org/10.1007/s12043-019-1829-9
An efficient technique for a fractional-order system of equations describing the unsteady flow of a polytropic gas
P VEERESHA1 , D G PRAKASHA2 ,∗and HACI MEHMET BASKONUS3
1Department of Mathematics, Karnatak University, Dharwad 580 003, India
2Department of Mathematics, Faculty of Science, Davangere University, Shivagangothri, Davangere 577 007, India
3Department of Mathematics and Science Education, Faculty of Education, Harran University, ¸Sanliurfa, Turkey
∗Corresponding author. E-mail: prakashadg@gmail.com
MS received 2 February 2019; revised 26 April 2019; accepted 22 May 2019
Abstract. In the present investigation, theq-homotopy analysis transform method (q-HATM) is applied to find approximated analytical solution for the system of fractional differential equations describing the unsteady flow of a polytropic gas. Numerical simulation has been conducted to prove that the proposed technique is reliable and accurate, and the outcomes are revealed using plots and tables. The comparison between the obtained solutions and the exact solutions shows that the proposed method is efficient and effective in solving nonlinear complex problems.
Moreover, the proposed algorithm controls and manipulates the obtained series solution in a huge acceptable region in an extreme manner and it provides us a simple procedure to control and adjust the convergence region of the series solution.
Keywords. q-Homotopy analysis transform method; polytropic gas; Laplace transform.
PACS Nos 02.60.–Cb; 02.60.–x; 05.45.Df
1. Introduction
The concept of fractional calculus (FC) was debated at the end of the 17th century. Systems with an arbitrary order have been lately attracting significant attention and gaining more acceptance as generalisation to the classi- cal order system. The fundamental foundation of FC was laid nearly 324 years ago and since then it has proved deeply rooted mathematical concepts. Every real-world problem can be effectively described by the system of integer- and fractional-order differential equation [1–5]. In practice we can encounter such systems in con- trol theory, medicine, biology, thermodynamics, signal processing, electronics etc. [6–10].
The basic and essential results related to the solutions of fractional differential equations are found in [11–14].
The integer-order derivatives are local in nature, whereas the fractional derivatives are non-local. By using the integer-order derivative, we can analyse the variations in the neighbourhood of a point but by employing the fractional derivative, we can study the changes in the complete interval. The problems relating to applica- tions of FC are present in various connected branches
of science and engineering such as fluid and contin- uum mechanics [15], electrodynamics [16], chaos [17], optics [18], cosmology [19] and many other branches [20–23].
In the present investigation, we consider the fract- ional-order system of gas dynamic equations describing the evolution of the two-dimensional unsteady flow of a perfect gas. The polytropic gas in astrophysics is given by [24]
P = Kρ1+(1/n),
where ρ = U/V is the energy density, U is the total energy of the gas, V is the container volume, K is a constant andnis the polytropic index. Degenerate elec- tron gas and adiabatic gas are two examples of such gases. The study of polytropic gases plays a vital role in cosmology and astrophysics [25] and these gases can behave like dark energy [26]. Now consider the system of gas dynamic equations describing the evolution of unsteady flow of a perfect gas having an arbitrary order [27,28]:
Dtμu+uxu+vuy+ px
ρ =0, 0123456789().: V,-vol
Dtμv+uvx+vvy+ py
ρ =0, Dtμρ+uρx+vρy+ρux+ρvy =0,
Dtμp+upx+vpy+τpux+τpvy =0, (1) under initial conditions
u(x,y,0)=a(x,y), v(x,y,0)=b(x,y),
ρ(x,y,0)=c(x,y)andp(x,y,0)=d(x, y), (2) whereu(x,y,t)andv(x,y,t)are the velocity compo- nents,ρ(x,y,t)is the density,p(x,y,0)is the pressure andτ is the ratio of the specific heat and it represents the adiabatic index.
The analytical and numerical solutions for the non- linear fractional differential equations have fundamental importance as most of the complex phenomena are mod- elled mathematically by nonlinear fractional differential equations. The last three decades have seen the dis- covery of a number of new techniques to elucidate a nonlinear differential system having a fractional order, in parallel with the development of new symbolic pro- gramming and computational algorithms. In connection with this, in 1992, Liao [29,30] introduced the homo- topy analysis method (HAM), and it has been effectively employed to find solution to problems arising in science and technology. It does not require any discretisation, linearisation and perturbation. But, it requires more computation and computer memory to solve nonlinear problems that arise in complex phenomena. Hence, it necessitates a mixture of transformation algorithm to overcome these limitations.
In the present framework, we considerq-homotopy analysis transform method (q-HATM) to find analyti- cally approximated solution for the system of equations describing a polytropic gas with an unsteady flow. The proposed technique is a modified technique which is an elegant blend of q-HAM with Laplace transform (LT) [31,32]. Asq-HATM is a modified technique of HAM, it does not require linearisation, discretisation or perturbation, and additionally, it will decrease huge mathematical computations, needs less computer mem- ory and is free from difficult polynomials, integrations and physical parameters. It provides us exceptional freedom to pick the equation type of linear subprob- lems, physical parameters and initial assumption. Due to this, complicated nonlinear differential equations can often be solved in a simple way. The novelty of the proposed technique is that it offers a simple solution pro- cedure, large convergence region and non-local effect in the obtained solution. The future scheme controls and manipulates the series solution, which quickly con- verges to the analytical solution in a small acceptable region. Recently, many researchers like Srivastavaet al
[33] studied the model of vibration equation of arbitrary order, Singhet al[34] found the solution for the frac- tional Drinfeld–Sokolov–Wilson equation, Bulut et al [35] analysed the fractional model of HIV infection of CD4+T cells, Kumaret al[36] analysed the model of Lienard’s equation and many others [37–39] with the help ofq-HATM.
On the other hand, the solution for the considered system of equations was analysed using distinct tech- niques, such as the Adomian decomposition technique [40], variational iteration technique [27], fractional nat- ural decomposition scheme [41] and others [28,42]. But in the above cited papers, the researchers do not present numerical simulation and behaviour of system (1) with fractional order. Hence, the authors in the present inves- tigation present the behaviour of the obtained solution and the numerical simulation of the future problem.
2. Preliminaries
We recall the definitions and notations of FC and LT, which shall be employed in the present framework:
DEFINITION 1
The fractional integral of a function f(t)∈Cδ(δ≥ −1) and of order μ > 0, initially defined by Riemann–
Liouville, is presented as Jμf(t)= 1
(μ) t
0
(t−ϑ)μ−1f(ϑ)dϑ,
J0f(t)= f(t). (3)
DEFINITION 2
The fractional derivative of f ∈ C−n1 in the Caputo sense is defined as [8]
Dtμf(x,y,t)
= ∂μf(x,y,t)
∂tμ
=
⎧⎪
⎪⎨
⎪⎪
⎩
dn f(x,y,t)
dtn μ=n ∈N, In−μ
∂nf(x,y,t)
∂tn
n−1< μ <n, n ∈N.(4) DEFINITION 3
The LT of a Caputo fractional derivative Dtμf(t) is represented as
L[Dtμf(t)] =sμF(s)
−
n−1
r=0
sμ−r−1f(r)(0+)(n−1< μ≤n), (5) where F(s)symbolises the LT of the function f(t).
3. Fundamental idea ofq-HATM
In this section, we present the fundamental solution pro- cedure of the proposed method, the fractional partial differential equation of the form
Dtμu(x,y,t)+Ru(x,y,t)+N u(x,y,t)
= f(x,y,t), n−1< μ≤n, (6) where Dtμv(x,y,t) denotes the Caputo’s fractional derivative of the function u(x,y,t), R is the bounded linear differential operator inx,yandt(i.e. for a num- ber ε > 0 we have Ru ≤ εu), N specifies the nonlinear differential operator and Lipschitz continu- ous withσ >0 satisfying|N u−Nv| ≤σ|u−v|, and
f(x,y,t)represents the source term.
Now, by employing the LT on eq. (6), we get sμL[u(x,y,t)]
−
n−1
k=0
sμ−k−1u(k)(x,y,0)
+L[Ru(x,y,t)]+L[N u(x,y,t)]
=L[f(x,y,t)]. (7) On simplifying eq. (7), we have
L[u(x,y,t)]− 1 sμ
n−1
k=0
sμ−k−1uk(x,y,0)
+ 1
sμ{L[Ru(x,y,t)]
+L[N u(x,y,t)]−L[f(x,y,t)]} =0. (8) According to the HAM, the nonlinear operator is defined as
N[ϕ(x,y,t;q)]
=L[ϕ(x,y,t;q)]
− 1 sμ
n−1
k=0
sμ−k−1ϕ(k)(x,y,t;q)(0+) + 1
sμ{L[Rϕ(x,y,t;q)]
+L[Nϕ(x,y,t;q)]−L[f(x,y,t)]}, (9)
whereq ∈ 0,n1
andϕ(x,y,t;q)is the real function ofx,y,t andq. For a non-zero auxiliary function, we construct a homotopy as follows:
(1−nq)L[ϕ(x,y,t;q)−u0(x,y,t)]
= ¯hq N[ϕ(x,y,t;q)], (10) where L is a symbol of the LT, h¯ = 0 is an auxil- iary parameter,q ∈ [0,1/n](n≥1)is the embedding parameter,u0(x,y,t)is an initial condition ofu(x,y,t) andϕ(x,y,t;q)is an unknown function. The following results hold, respectively, forq =0 and 1/n:
ϕ(x,y,t;0)=u0(x,y,t), ϕ
x,y,t; 1 n
=u(x,y,t). (11) Thus, by amplifying q from 0 to 1/n, the solution ϕ(x,y,t;q)converges fromu0(x,y,t)to the solution u(x,y,t). On expanding the function ϕ(x,y,t;q) in series form by employing Taylor’s theorem nearq, one can get
ϕ(x,y,t;q)=u0(x,y,t)+
∞ m=1
um(x,y,t)qm, (12) where
um(x,y,t)= 1 m!
∂mϕ(x,y,t;q)
∂qm
q=0. (13)
On choosing the auxiliary linear operator,u0(x,y,t),n and h, series (12) converges at¯ q = 1/n and then it yields one of the solutions for eq. (6):
u(x,y,t)=u0(x,y,t)+
∞ m=1
um(x,y,t) 1
n m
. (14) Now, differentiating the zeroth-order deformation equation. (10)mtimes with respect toqand then divid- ing bym!and finally takingq =0, gives
L
um(x,y,t)−kmum−1(x,y,t)
= ¯hRm( um−1), (15) where
um={u0(x,y,t),u1(x,y,t), ...,um(x,y,t)}. (16) Employing the inverse LT on eq. (15), it yields um(x,y,t)=kmum−1(x,y,t)+ ¯h L−1
Rm(um−1) , (17) where
Rm( um−1)
=L
um−1(x,y,t)
−
1−km
n
n−1 k=0
sμ−k−1u(k)(x,y,0) +1
sμL[f(x,y,t)]
+ 1 sμL
Rum−1 +Hm−1
(18) and
km =
0, m≤1,
n, m>1. (19)
In eq. (18),Hmdenotes the homotopy polynomial and is defined as
Hm = 1 m!
∂mϕ(x,y,t;q)
∂qm
q=0
and
ϕ(x,y,t;q)=ϕ0+qϕ1+q2ϕ2+ · · ·. (20) By eqs (17) and (18), we have
um(x,y,t)
=(km+ ¯h)um−1(x,y,t)
−
1−km
n
L−1 n−1
k=0
sμ−k−1u(k)(x,y,0)
+1
sμL[f(x,y,t)]
+ ¯h L−1 1
sμL
Rum−1+Hm−1
. (21)
Finally, on solving eq. (21), we get the iterative terms of um(x,y,t). Theq-HATM series solution is presented as u(x,y,t)=
∞ m=0
um(x,y,t). (22)
4. Solution for a fractional system of nonlinear equations of unsteady flow of a polytropic gas To validate the applicability and the accuracy of the pro- posed technique, in this section, we consider a system of equations which describes the unsteady flow of a poly- tropic gas of fractional order.
Example1. Consider the fractional system of equations of the form [27,40]:
Dtμu+uxu+vuy+ px
ρ =0, Dtμv+uvx+vvy+ py
ρ =0,
Dtμρ+uρx+vρy+ρux+ρvy =0,
Dtμp+upx+vpy+τpux+τpvy=0, (23) under initial conditions:
u(x,y,0)=ex+y, v(x,y,0)= −1−ex+y, ρ(x,y,0)=ex+y,
p(x,y,0)=η, (24)
whereηis the real constant.
By applying LT on system (23) and then employing the condition given in system (24), we have
L[u(x,y,t)]−1 s{ex+y} + 1
sμL
u∂u
∂x +v∂u
∂y + 1 ρ
∂p
∂x
=0, L[v(x,y,t)]+1
s{1+ex+y} + 1
sμL
u∂v
∂x +v∂v
∂y + 1 ρ
∂p
∂y
=0, L[ρ(x,y,t)]−1
s{ex+y} + 1
sμL
u∂ρ
∂x +v∂ρ
∂y +ρ∂u
∂x +ρ∂v
∂y
=0, L[p(x,y,t)]−1
s{η}
+ 1 sμL
u∂p
∂x +v∂p
∂y +τp∂u
∂x +pτ∂v
∂y
=0. (25) By using the proposed algorithm, the nonlinear oper- atorN is defined as
N1[ϕ1(x,y,t;q), ϕ2(x,y,t;q), ϕ3(x,y,t;q), ϕ4(x,y,t;q)]
=L[ϕ1(x,y,t;q)]− 1 s{ex+y} + 1
sμL
ϕ1(x,y,t;q)∂ϕ1(x,y,t;q)
∂x +ϕ2(x,y,t;q)∂ϕ1(x,y,t;q)
∂y
+ 1
ϕ3(x,y,t;q)
∂ϕ4(x,y,t;q)
∂x
, N2[ϕ1(x,y,t;q), ϕ2(x,y,t;q), ϕ3(x,y,t;q),
ϕ4(x,y,t;q)]
=L[ϕ2(x,y,t;q)]+ 1
s{1+ex+y} + 1
sμL
ϕ1(x,y,t;q)∂ϕ2(x,y,t;q)
∂x
+ϕ2(x,y,t;q)∂ϕ2(x,y,t;q)
∂y
+ 1
ϕ3(x,y,t;q)
∂ϕ4(x,y,t;q)
∂y
, N3[ϕ1(x,y,t;q), ϕ2(x,y,t;q), ϕ3(x,y,t;q),
ϕ4(x,y,t;q)]
=L[ϕ3(x,y,t;q)]−1 s{ex+y} + 1
sμL
ϕ1(x,y,t;q)∂ϕ3(x,y,t;q)
∂x +ϕ2(x,y,t;q)∂ϕ3(x,y,t;q)
∂y +ϕ3(x,y,t;q)∂ϕ1(x,y,t;q)
∂x +ϕ3(x,y,t;q)∂ϕ2(x,y,t;q)
∂y
,
N4[ϕ1(x,y,t;q), ϕ2(x,y,t;q), ϕ3(x,y,t;q), ϕ4(x,y,t;q)]
=L[ϕ4(x,y,t;q)]−1 s{η}
+ 1 sμL
ϕ1(x,y,t;q)∂ϕ4(x,y,t;q)
∂x +ϕ2(x,y,t;q)∂ϕ4(x,y,t;q)
∂y +τϕ4(x,y,t;q)∂ϕ1(x,y,t;q)
∂x +τϕ4(x,y,t;q)∂ϕ2(x,y,t;q)
∂y
. (26)
By adopting the foregoing procedure ofq-HATM, the deformation equation ofmth order atH(x,y,t)=1, is given as
L[um(x,y,t)−kmum−1(x,y,t)]
= ¯hR1,m[ um−1,vm−1,ρm−1,pm−1], L[vm(x,y,t)−kmvm−1(x,y,t)]
= ¯hR2,m[ um−1,vm−1,ρm−1, pm−1], L[ρm(x,y,t)−kmρm−1(x,y,t)]
= ¯hR3,m[ um−1,vm−1,ρm−1,pm−1], L[pm(x,y,t)−kmpm−1(x,y,t)]
= ¯hR4,m[ um−1,vm−1,ρm−1, pm−1], (27)
Figure 1. Surfaces of (a)q-HATM solution, (b) exact solu- tion and (c) absolute error = |uExac.−uq-HATM| when
¯
h= −1,t =0.1,n=1 andμ=1.
where R1,m
um−1,vm−1,ρm−1,pm−1
=L
um−1(x,y,t)
−
1−km
n 1
s
ex+y
+ 1 sμL
m−1
i=0
ui∂um−1−i
∂x
+
m−1
i=0
vi∂um−1−i
∂y +χm−1
ρ,px
, R2,m
um−1,vm−1,ρm−1,pm−1
Figure 2. Nature of (a)q-HATM solution, (b) exact solu- tion and (c) absolute error = |vExac.−vq-HATM| when t =0.1,h¯ = −1,n=1 andμ=1.
=L
vm−1(x,y,t) +
1− km
n 1
s (1+ex+y) + 1
sμL m−1
i=0
ui∂vm−1−i
∂x
+
m−1
i=0
vi∂vm−1−i
∂y +ψm−1(ρ,py)
, R3,m
um−1,vm−1,ρm−1,pm−1
=L
ρm−1(x,y,t)
−
1− km
n 1
s (ex+y)
Figure 3. Behaviour of (a) q-HATM solution, (b) exact solution and (c) absolute error= |ρExac.−ρq-HATM| when
¯
h= −1,n =1,t=0.1 andμ=1.
+ 1 sμL
m−1
i=0
ui∂ρm−1−i
∂x +
m−1
i=0
vi∂ρm−1−i
∂y
+
m−1
i=0
ρi∂um−1−i
∂x +
m−1
i=0
ρi∂vm−1−i
∂y
,
R4,m
um−1,vm−1,ρm−1,pm−1
=L
pm−1(x,y,t)
−
1−km
n 1
s(η) + 1
sμL m−1
i=0
ui∂pm−1−i
∂x +
m−1
i=0
vi∂pm−1−i
∂y
Figure 4. Nature of (a)q-HATM solution, (b) exact solu- tion and (c) absolute error = |pExac.−pq-HATM| when t =0.1,h¯ = −1,n=1 andμ=1.
+τ
m−1
i=0
pi∂um−1−i
∂x +τ
m−1
i=0
pi∂vm−1−i
∂y
, (28)
where
χm−1(ρ,px)= 1 ρ0
∂p0
∂x + 1 (ρ0)2
ρ0∂p1
∂x −ρ1∂p0
∂x
+ · · · and
ψm−1(ρ,py)
= 1 ρ0
∂p0
∂y + 1 (ρ0)2
ρ0∂p1
∂y −ρ1∂p0
∂y
+ · · · .
Figure 5. Surface of q-HATM solution for the system of equations describing the unsteady flow of a polytropic gas whenh¯ = −1,t =0.1,n=1 andμ=1.
By applying inverse LT on both sides of system (27), we get
um(x,y,t)=kmum−1(x,y,t) + ¯h L−1
R1,m
um−1,vm−1,ρm−1,pm−1
, vm(x,y,t)=kmvm−1(x,y,t)
+ ¯h L−1 R2,m
um−1,vm−1,ρm−1,pm−1
, ρm(x,y,t)=kmρm−1(x,y,t)
+ ¯h L−1 R3,m
um−1,vm−1,ρm−1,pm−1
,
pm(x,y,t)=kmpm−1(x,y,t) + ¯h L−1
R3,m
um−1,vm−1,ρm−1,pm−1
. (29) On solving the forgoing system of equations system- atically, we obtain
u0(x,y,t)=ex+y, v0(x,y,t)= −1−ex+y, ρ0(x,y,t)=ex+y,
p0(x,y,t)=η,
u1(x,y,t)= −¯hex+ytμ [μ+1], v1(x,y,t)= ¯hex+ytμ
[μ+1], ρ1(x,y,t)= −¯hex+ytμ [μ+1],
(a)
(b)
(c)
(d)
Figure 6. Plot of the q-HATM solution: (a)u(x,y,t), (b) v(x,y,t), (c) ρ(x,y,t) and (d)p(x,y,t)with respect to t when h¯ = −1,n = 1,x = 0.1 and y = 0.1 with diverseμ.
(a)
(b)
(c)
(d)
Figure 7. The h-curves¯ drawn for (a)u(x,y,t), (b) v(x,y,t), (c) ρ(x,y,t) and (d)p(x,y,t) with diverse μwhenn=1,t=0.0 1,x=0.1 andy=0.1.
(a)
(b)
(c)
(d)
Figure 8. Behaviour of the obtained solution for (a)u(x,y,t), (b) v(x,y,t), (c) ρ(x,y,t)and(d)p(x,y,t) with diverseh¯ whenn=1, μ=1,x=0.1 andy=0.1.
p1(x,y,t)=0,
u2(x,y,t)= −(n+ ¯h)he¯ x+ytμ
[μ+1] + ¯h2ex+yt2μ [2μ+1], v2(x,y,t)= (n+ ¯h)¯hex+ytμ
[μ+1]
¯
h2ex+yt2μ [2μ+1], ρ2(x,y,t)= −(n+ ¯h)he¯ x+ytμ
[μ+1] + ¯h2ex+yt2μ [2μ+1], p2(x,y,t)=0,
u3(x,y,t)= −(n+ ¯h)2he¯ x+ytμ [μ+1]
+(n+ ¯h)h¯2ex+yt2μ
[2μ+1] − ¯h3ex+yt3μ [3μ+1], v3(x,y,t)= (n+ ¯h)2he¯ x+ytμ
[μ+1] −(n+ ¯h)h¯2ex+yt2μ [2μ+1]
+ ¯h3ex+yt3μ [3μ+1], ρ3(x,y,t)= −(n+ ¯h)2he¯ x+ytμ
[μ+1]
+(n+ ¯h)¯h2ex+yt2μ
[2μ+1] − ¯h3ex+yt3μ [3μ+1], p3(x,y,t)=0,
u4(x,y,t)= −(n+ ¯h)3he¯ x+ytμ [μ+1]
+(n+ ¯h)2h¯2ex+yt2μ [2μ+1]
−(n+ ¯h)¯h3ex+yt3μ
[3μ+1] + ¯h4ex+yt4μ [4μ+1], v4(x,y,t)=(n+ ¯h)3he¯ x+ytμ
[μ+1] −(n+ ¯h)2h¯2ex+yt2μ [2μ+1]
+(n+ ¯h)h¯3ex+yt3μ
[3μ+1] − ¯h4ex+yt4μ [4μ+1], ρ4(x,y,t)= −(n+ ¯h)3he¯ x+ytμ
[μ+1]
+(n+ ¯h)2h¯2ex+yt2μ [2μ+1]
−(n+ ¯h)¯h3ex+yt3μ
[3μ+1] + ¯h4ex+yt4μ [4μ+1],
Table 1. Error analysis of the obtained solution for the proposed problem with distinctxandtwhen
¯
h = −1,n=1,y=1 andμ=1.
x t |uExac.−uq-HATM| |vExac.−vq-HATM| |ρExac.−ρq-HATM| |pExac.−pq-HATM| 00.2 0.2 8.64762×10−9 8.64762×10−9 8.64762×10−9 0
0.4 1.05622×10−8 1.05622×10−8 1.05622×10−8 0.6 1.29007×10−8 1.29007×10−8 1.29007×10−8 0.8 1.57570×10−8 1.57570×10−8 1.57570×10−8 1 1.92456×10−8 1.92456×10−8 1.92456×10−8 0.4 0.2 1.13576×10−6 1.13576×10−6 1.13576×10−6 0.4 1.38723×10−6 1.38723×10−6 1.38723×10−6 0.6 1.69436×10−6 1.69436×10−6 1.69436×10−6 0.8 2.06950×10−6 2.06950×10−6 2.06950×10−6 1 2.52769×10−6 2.52769×10−6 2.52769×10−6 0.6 0.2 1.99220×10−5 1.99220×10−5 1.99220×10−5 0.4 2.43328×10−5 2.43328×10−5 2.43328×10−5 0.6 2.97201×10−5 2.97201×10−5 2.97201×10−5 0.8 3.63002×10−5 3.63002×10−5 3.63002×10−5 1 4.43372×10−5 4.43372×10−5 4.43372×10−5 0.8 0.2 1.53301×10−4 1.53301×10−4 1.53301×10−4 0.4 1.87240×10−4 1.87240×10−4 1.87240×10−4 0.6 2.28696×10−4 2.28696×10−4 2.28696×10−4 0.8 2.79330×10−4 2.79330×10−4 2.79330×10−4 1 3.41174×10−4 3.41174×10−4 3.41174×10−4 1 0.2 7.51252×10−4 7.51252×10−4 7.51252×10−4 0.4 9.17582×10−4 9.17582×10−4 9.17582×10−4 0.6 1.12074×10−3 1.12074×10−3 1.12074×10−3 0.8 1.36887×10−3 1.36887×10−3 1.36887×10−3 1 1.67194×10−3 1.67194×10−3 1.67194×10−3
p4(x,y,t)=0,
u5(x,y,t)= −(n+ ¯h)4h¯ex+ytμ [μ+1]
+(n+ ¯h)3h¯2ex+yt2μ [2μ+1]
−(n+ ¯h)2h¯3ex+yt3μ [3μ+1]
+(n+ ¯h)h¯4ex+yt4μ
[4μ+1] − ¯h5ex+yt5μ [5μ+1], v5(x,y,t)= (n+ ¯h)4he¯ x+ytμ
[μ+1]
−(n+ ¯h)3h¯2ex+yt2μ [2μ+1]
+(n+ ¯h)2h¯3ex+yt3μ [3μ+1]
−(n+ ¯h)h¯4ex+yt4μ
[4μ+1] + ¯h5ex+yt5μ [5μ+1],
ρ5(x,y,t)= −(n+ ¯h)4he¯ x+ytμ [μ+1]
+(n+ ¯h)3h¯2ex+yt2μ [2μ+1]
−(n+ ¯h)2h¯3ex+yt3μ [3μ+1]
+(n+ ¯h)¯h4ex+yt4μ
[4μ+1] − ¯h5ex+yt5μ [5μ+1], p5(x,y,t)=0,
...
Similarly, we can get the rest of the term. Then, the q-HATM series solution of system (23) is given by u(x,y,t)=u0(x,y,t)
+
∞ m=1
um(x,y,t) 1
n m
, v(x,y,t)=v0(x,y,t)