https://doi.org/10.1007/s12043-019-1791-6
Symmetry analysis of some nonlinear generalised systems of space–time fractional partial differential equations with time-dependent variable coefficients
SACHIN KUMAR and BALJINDER KOUR ∗
Department of Mathematics and Statistics, Central University of Punjab, Bathinda 151 001, India
∗Corresponding author. E-mail: baljindervirk172@gmail.com
MS received 15 August 2018; revised 15 January 2019; accepted 12 April 2019; published online 21 May 2019 Abstract. In this paper, the Lie group analysis method is applied to carry out the Lie point symmetries of some space–time fractional systems including coupled Burgers equations, Ito’s system, coupled Korteweg–de-Vries (KdV) equations, Hirota–Satsuma coupled KdV equations and coupled nonlinear Hirota equations with time- dependent variable coefficients with the Riemann–Liouville derivative. Symmetry reductions are constructed using Lie symmetries of the systems. To the best of our knowledge, nobody has so far derived the invariants of space–time nonlinear fractional partial differential equations with time-dependent coefficients.
Keywords. Fractional differential equations with time-dependent variable coefficients; Lie symmetry analysis;
Erdèlyi–Kober operators; Riemann–Liouville fractional derivative.
PACS Nos 02.30.Jr; 02.20.Hj
1. Introduction
Nonlinear phenomena consistently appear in the study of applied mathematics, engineering and many related scientific fields [1–3]. Solving nonlinear systems is a great task in mathematical analysis and applications.
In daily life, we come across many real-life phenom- ena which can be described by mathematical structure of nonlinear partial differential equations (NLPDEs) of integer or non-integer order. In order to interpret the nonlinear phenomena better, in place of NLPDEs of integer order, nonlinear fractional partial differen- tial equations (NLFPDEs) can be used. There are many methods for solving NLFPDEs [4–9]. Numerical meth- ods give approximate solutions [10–13]. But the study of exact solution gives proper understanding of NLFPDEs.
Exact solution of NLFPDEs facilitates the authentica- tion of numerical solvers and also supports the stability analysis of the solution. Various methods have been used by many researchers for solving NLPDEs in the last two or three decades. Some of the very important methods are: the exp-function method [14,15], the fractional sub- equation method [16], the homotopy method [17,18], the inverse scattering method, the adomian decompo- sition method [19], the Laplace decomposition method
[20], the variational iteration method [21], the Lie sym- metry method [22–26], etc.
The application of the Lie symmetry method to NLF- PDEs is novel. This method is basically introduced by the Norwegian mathematician Sophus Lie at the end of the 19th century. The Lie symmetry analysis is one of the very effective methods for obtaining exact solutions of NLFPDEs. This method has been introduced, firstly, for solving ordinary differential equations. After that, it has been demonstrated that by the Lie symmetry method, systems of differential equations can be converted to other equivalent systems of differential equations in reduced form. After using the one-parameter Lie group of the infinitesimal transformations with compliance of the invariant conditions, the solution space of the dif- ferential equation remains invariant into another space.
There have been many approaches for solving NLF- PDEs [3,27–35], yet not much work has been performed on the Lie symmetry analysis of space–time NLFPDEs.
The main aim of this paper is to extend the appli- cation of the Lie symmetry approach from systems of space–time NLFPDEs with constant coefficients to the systems of NLFPDEs with time-dependent variable coefficients. The invariance of space–time NLFPDEs with constant coefficients under the Lie group of scaling
transformations has been studied by only a few researchers [36,37]. The principal aim of this paper is to deliberate the space–time fractional systems with vari- able coefficients. In this study, we propose appropriate prolongation operators and used them to analyse some space–time fractional nonlinear systems with time- dependent variable coefficients. The considered systems of NLFPDEs are reduced to systems of nonlinear fractional ordinary differential equations (NLFODEs) containing the left- and right-hand side Erdelyi–Kober´ fractional differential operators [37,38].
This paper is divided into four sections. The sec- tions are arranged as follows: Section 1 is introductory.
In §2, the Lie symmetry method is introduced to deal with systems of space–time fractional PDEs. The Lie symmetry analysis of five systems of NLFPDEs with time-dependent variable coefficients with space–
time derivatives of fractional order is presented in §3.
Section 4 contains conclusion.
2. Lie symmetry analysis for a system of fractional partial differential equations
In this section, the Lie symmetry method has been intro- duced [39,40] for a system of space–time NLFPDEs with two independentx=(x,t)andp-dependent vari- ables v= (v1, v2, . . . , vp). Some basic definitions and formulas are given below.
2.1 Basic definitions
2.1.1 Riemann–Liouville fractional derivative. The formal definition of Riemann–Liouville fractional deri- vative [41–43] is given as follows.
Let f: [a,b] ⊆ R → R such that ∂mf/∂xm is continuous and integrable ∀m ∈ N0 = N∪ {0} and m ≤ [γ] +1. Then, the Riemann–Liouville fractional derivative of orderγ >0 is defined as
∂γ f(x,t)
∂tγ
=
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
1 (1+[γ]−γ )
∂1+[γ]
∂t1+[γ] t
0
(t−s)[γ]−γ f(x,s)ds, t >0, [γ]< γ <[γ] +1,
∂mf(t)
∂tm , γ = [γ] =m∈N,
(1)
where(γ )is the Euler’s gamma function.
2.1.2 Erdèlyi–Kober operator. The left-hand side Erdèlyi–Kober fractional differential operator(Tϑ,α)is defined as
(Tϑ,αg)(y)
=
r−1
k=0
ϑ+k−1 y d
dy (Mϑ+α,r−αg)(y)), y>0, >0, α >0,
r =
[α] +1 if α /∈N,
α if α ∈N, (2)
where (Mϑ,α g)(y)
=
⎧⎪
⎨
⎪⎩ 1 (α)
∞
1
(ρ−1)α−1ρ−(ϑ+α)g(yρ1/)dρ if α >0,
g(y) if α=0
(3)
is the left-hand side Erdèlyi–Kober fractional integral operator.
The right-hand side Erdèlyi–Kober fractional differ- ential operator(Rϑ,β )is defined as
(Rϑ,β f)(y) :=
r k=1
ϑ+k+ 1 y d
dy (Iϑ+β, r−βf)(y)), y>0, >0, β >0,
r =
[β] +1 if β /∈N,
β if β ∈N, (4)
where
(Iϑ,β f)(y)=
⎧⎪
⎪⎨
⎪⎪
⎩ 1 (β)
1
0
(1−ρ)β−1ρϑf(yρ1/)dρ if β >0,
f(y) if β =0
(5)
is the right-hand side Erdèlyi–Kober fractional integral operator.
2.2 Symmetry analysis
Consider a system of space–time fractional PDEs having the following form:
Fh
x,v,∂γv
∂tγ ,∂βv
∂xβ, ∂v
∂x, ∂2v
∂x2, . . . ,∂nv
∂xn =0,
h=1,2, . . . , (6)
where x = (x,t) is the independent variable, v = (v1, v2, . . . , vp) are the dependent variables, ∂v/∂x and ∂2v/∂x2 denote the integer-order derivatives of orders 1 and 2 and∂γv/∂tγ,∂βv/∂xβare the Riemann–
Liouville fractional derivatives of ordersγ, β >0. Let
us consider the one-parameter Lie group of transforma- tions
x∗=x+εξ(x,t,v)+O(ε2), t∗=t+ετ(x,t,v)+O(ε2), vr∗=vr +εη(r)(x,t, vr)+O(ε2),
∂γvr∗
∂tγ = ∂γvr
∂tγ +εη(r)γ,t +O(ε2),
∂βvr∗
∂xβ = ∂βvr
∂xβ +εη(r)β,x+O(ε2),
∂vr∗
∂x = ∂vr
∂x +εη(r)x+O(ε2),
∂2vr∗
∂x2 = ∂2vr
∂x2 +εη(r)x x+O(ε2),
... (7)
where r = 1,2, . . . ,p, ε is the group parameter, ξ, τ, η(r) are the infinitesimals and η(r)γ,t are the extended infinitesimals of order γ. η(r)β,x are the extended infinitesimals of order β. η(r)x and η(r)x x are the integer-order extended infinitesimals, which leave system (6) invariant. The infinitesimal transfor- mations (7) are such that if vr, r = 1,2, . . . ,p are solutions of (6), then so are vr∗,r = 1,2, . . . ,p. The associated symmetry generator is given by the following form:
V =ξ(x,t,v) ∂
∂x+τ(x,t,v)∂
∂t+ p r=1
η(r)(x,t,v) ∂
∂vr. (8) The corresponding prolonged symmetry generator is given by
pr(γ,β,n)V =V + p r=1
η(r)γ,t∂∂tγvr + p r=1
η(r)β,x∂∂β xvr
+ p r=1
η(r)x∂vrx + p r=1
η(r)x x∂vrx x + · · ·, (9) where n is the order of system (6) and ∂tγvr =
∂γvr/∂tγ, ∂xβvr = ∂βvr/∂xβ, vrx = ∂vr/∂x, vrx x =
∂2vr/∂x2forr =1,2, . . . ,p.
Also, the lower limit of the Riemann–Liouville frac- tional derivative (1) is fixed, and so the invariance condition yields
ξ(x,t,v)|x=0 =0, τ(x,t,v)|t=0 =0. (10)
Theγth-order extended infinitesimal η(k)γ,t related to the Riemann–Liouville fractional derivative is given by
η(k)γ,t = Dtγ(η(k))+ξDγt (vkx)−Dγt (ξvkx)
+τDtγ(vtk)−Dtγ(τvtk). (11) The generalised Leibnitz rule [41,42] forγ ∈Ris Dzγ(f(z)g(z))=
∞ n=0
γ
n Dnz f(z)Dzγ−ng(z), (12) where
Dznf = dnf dzn, γ
n = (γ +1)
(n+1)(γ +1−n), n∈N.
Using the Leibnitz rule (12), eq. (11) can be expanded as
η(k)γ,t =Dγt (η(k))−γDtτ∂γvk
∂tγ
− ∞ n=1
γ
n Dtn(ξ)Dtγ−n(vkx)
− ∞ n=1
γ
n+1 Dtn+1(τ)Dtγ−n(vk), (13) where Dt represents the total derivative operator given as
Dt =∂t +vkt∂vk +vttk∂vkt + · · ·. (14) Using the generalised chain rule [42,44] and generalised Leibnitz rule (12), the first term in (13) can be written as
Dtγ(η(k))= ∂γη(k)
∂tγ + p r=1
∞ n=1
γ n
∂nη(vkr)
∂tn Dtγ−n(vr) +
p r=1
μη(k),γ,r
+ p r=1
ηv(kr)∂γvr
∂tγ −vr∂γηv(kr)
∂tγ
, (15)
where μη(k),γ,r =
∞ n=2
n m=2
m j=2
j−1
q=2
γ n
n m
j q
1 j!
× tn−γ
(n−γ +1)(−v(r))q
× ∂m
∂tm((v(r))j−q) ∂n−m+jη
∂tn−m∂(v(r))j,
r =1,2, . . . ,p. (16) Here,η(vkr)= ∂η(k)/∂vr.
Therefore, theγth-order extended infinitesimalη(k)γ,t for the system of fractional partial differential equations (6) is
η(k)γ,t
= ∂γη(k)
∂tγ +(ηv(k(k)) −γDt(τ))∂γvk
∂tγ −vk∂γηvk
∂tγ +
p r=1
μη(k),γ,r+ p r=k,r=1
ηvr∂γvr
∂tγ −vr∂γηvr
∂tγ
− ∞ n=1
γ
n Dtn(ξ)Dγt −n(vk,x) +
∞ n=1
γ n
∂nη(vkk)
∂tn − γ
n+1 Dtn+1(τ)
×Dtγ−n(vk) +
p r=k,r=1
∞ n=1
γ n
∂nη(vkr)
∂tn Dtγ−n(vr), (17) whereμη(k),γ,r is given by (16).
Similarly, theβth-order extended infinitesimalη(k)β,x forβ >0 is proposed by
η(k)β,x
= ∂βη(k)
∂xβ +(η(vkk)−βDx(τ))∂βvk
∂xβ −vk∂βηvk
∂xβ +
p r=1
μη(k),β,r+ p r=k,r=1
ηvr∂βvr
∂xβ −vr∂βηvr
∂xβ
− ∞ n=1
β
n Dnx(ξ)Dxβ−n(vtk) +
∞ n=1
β n
∂nηv(kk)
∂xn − β
n+1 Dn+1x (τ)
×Dβ−x n(vk) +
p r=k,r=1
∞ n=1
β n
∂nη(vkr)
∂xn Dβ−nx (vr), (18) where
μη(k),β,r =∞
n=2
n m=2
m j=2
j−1
q=2
β n
n m
j q
1 j!
× xn−β
(n−β+1)(−vr)q ∂m
∂xm((v(k))j−q)
× ∂n−m+jφ
∂xn−m∂(v(k))j, r =1,2, . . . ,p. (19)
3. Application to some system of NLFPDEs with time-dependent variable coefficients
In this section, the proposed symmetry approach is applied to investigate the Lie point symmetries and reductions for some well-known nonlinear system of fractional partial differential equations with time-depen- dent variable coefficients.
3.1 Space–time fractional coupled Burgers system with time-dependent variable coefficients
In this subsection, we consider the space–time fractional coupled Burgers system [28,33] with time-dependent variable coefficients:
∂γu
∂tγ = A1(t)u∂βu
∂xβ +A2(t)u∂βv
∂xβ +A3(t)v∂βu
∂xβ +A4(t)∂2u
∂x2,
∂γv
∂tγ =B1(t)v∂βv
∂xβ +B2(t)u∂βv
∂xβ +B3(t)v∂βu
∂xβ +B4(t)∂2v
∂x2. (20)
The invariance criteria for system (20) under one- parameter Lie group of transformation are obtained as ηγ,t =A4(t)ηx x+(τA1(t)u+A1(t)η+τA3(t)v
+A3(t)φ)∂xβu+A2(t)uφβ,x+τA4(t)ux x +(A1(t)u+A3(t))ηβ,x
+(τA2(t)u+A2(t)η)∂xβv,
φγ,t =B4(t)φx x+τB4(t)vx x+(τB1(t)v+B1(t)φ +τB2(t)u+B2(t)η)∂xβv
+(B1(t)v+B2(t)u)φβ,x
+b3(t)vηβ,x+(τB3(t)v+B3(t)φ)∂xβu, (21)
whereηx x,φx xare the extended infinitesimals of order 2, ηγ,t, φγ,t are the extended infinitesimals of order γ and ηβ,x, φβ,x are the extended infinitesimals of orderβ.
Now substituting the values of the prolongations and equating the coefficient of various linearly independent variables to zero, for 0 < γ, β ≤ 1, the determining equations are obtained as
ξt =ξu =ξv =0, τx =τu =τv =0,
ηvv=ηuu =φvv =φuu =0, A4(t)τ +A4(t)(γ τt −2ξx)=0, B4(t)τ+B4(t)(γ τt −2ξx)=0, τB3(t)v+B3(t)φ+B3(t)v(γ τt −βξx)
+(B1(t)v+B2(t)u)φu −(A1(t)u+ A3(t)v)φu=0, τB1(t)v+B1(t)φ+τB2(t)u+B3(t)vηv+(B1(t)v
+B2(t)u)(γ τt −βξx)+B2(t)η−A2(t)uφu=0, τA2(t)u+ A2(t)η+(A1(t)u+ A2(t)u(γ τt −βξx)
+A3(t)ηv =0,
τA1(t)u+A1(t)η+τA3(t)v+A3(t)φ+(A1(t)u +A3(t)v)(γ τt −βξx)+ A2(t)uφu =0,
A4(t)ηx x+(A1(t)u+A3(t)v)(∂xβη−u∂xβηu
−v∂xβηv)+u∂tγηu+v∂tγηv
+A2(t)u(∂xβφ−v∂xβφv−u∂xβφu)−∂tγη=0, B4(t)φx x+(B1(t)+B2(t)u)(∂xβφ−u∂xβφu −v∂xβφv)
+u∂tγφu +v∂tγφv+B3(t)v(∂xβη−u∂xβηu
−v∂xβηv)−∂tγφ =0, γ
n ∂tnηu − γ
n+1 Dnt+1τ =0, n ∈N, γ
n ∂tnφv− γ
n+1 Dtn+1τ =0, n∈N, β
n ∂xnηu− β
n+1 Dnx+1ξ =0, n ∈N, β
n ∂xnφv− β
n+1 Dnx+1ξ =0, n ∈N. (22) In a particular case, take A4(t)=atmandB4(t)=bts, where a, b are arbitrary constants and m, s are real numbers.
Solving these PDEs and FPDEs (22) together, and also using (10), we obtain the infinitesimals given as
ξ = c1
2x, τ = c1
γ +mt, η= (γ −1)c1
2(γ +m)u,
φ = (γ −1)c1
2(γ +m)v, (23)
where c1 is an arbitrary constant and variable coeffi- cients are governed by the following conditions:
A1(t)=M1t1−23γ+β(γ2+m), A2(t)=K1A1(t), A3(t)=K2A1(t), B1(t)=K3A1(t),
B2(t)=K4A1(t), B3(t)=K5A1(t), (24) where M1,K1,K2,K3,K4 and K5 are arbitrary constants.
For the symmetry generator, given by X = x
2
∂
∂x + t γ +m
∂
∂t + (γ −1)u 2(γ +m)
∂
∂u +(γ −1)v
2(γ +m)
∂
∂v, (25)
the auxiliary equations are written as dx
x 2
= dt
γ+tm
= du
γ−1
2(γ+m)u = dv
γ−1
2(γ+m)v. (26) Solving (26), the similarity variable of the considered system (20) is obtained as
y =xt−γ+m2 (27)
and the corresponding similarity transformations are u(x,t)=tγ−21g(y),
v(x,t)=tγ−12 h(y) (28) and the admissible coefficients given by (24), with (27) and (28), reduce the system of NLFPDEs (20) for γ, β >0 to the system of NLFODEs.
Let us assumen−1< γ <n;n ∈Nthen by the def- inition of Riemann–Liouville fractional differentiation (1), we have
∂γu
∂tγ = ∂n
∂tn
1 (n−γ )
t
0
(t−ρ)n−γ−1ργ−21
×g
xρ−γ+2m
ds . (29)
Letρ = t/s and using the similarity variable (27), we get the following equation:
∂γu
∂tγ = ∂n
∂tn
tn−γ+12 (n−γ )
× ∞
1
(s−1)n−γ−1s−(n−γ+γ+21) g(ysγ+2m)ds .
By using the left-hand side Erdèlyi–Kober fractional integral operator (3), we have
∂γu
∂tγ = ∂n
∂tn
tn−γ+21
Mγ+221,n−γ
γ+m g (y) . (30)
Now for more simplification, letμ(y) be the continu- ously differentiable function fory =xt−((γ+m)/2)from (27). Then
t ∂
∂tμ(y)=t x
−γ +m
2 t−(γ+2m)−1μ(y)
=
−γ +m
2 y d
dyμ(y).
Therefore, (30) becomes
∂γu
∂tγ = ∂n
∂tn
tn−γ+21
Mγ+221,n−γ
γ+m g (y)
= ∂n−1
∂tn−1 ∂
∂ttn−γ+21
Mγ+122 ,n−γ γ+m
g (y)
= ∂n−1
∂tn−1
tn−γ+12 −1
n−γ +1
2 − γ +m 2 y d
dy
×
Mγ+122 ,n−γ γ+m
g (y) . Continuing in this way, we get
∂γu
∂tγ = ∂n
∂tn
tn−γ+21
Mγ+122 ,n−γ γ+m
g (y)
=t−γ+12
n−1 j=0
1− γ +1 2 + j−
γ +m
2 y d
dy
×
Mγ+122 ,n−γ γ+m
g (y). (31)
Using the left-hand side Erdèlyi–Kober fractional dif- ferential operator (2) in (31), we have
∂γu
∂tγ =t−γ+21
T1−γ22 ,γ
γ+m g (y). (32)
Analogous to the result obtained above, it can also be concluded that
∂γv
∂tγ =t−γ+21
T1−γ22 ,γ γ+m
h (y). (33)
Similarly, the partial fractional derivatives of order β >0,∂βu/∂xβand∂βv/∂xβ are obtained by
∂βu
∂xβ =tγ−12 x−β(R−β,β1 g)(y), (34)
∂βv
∂xβ =tγ−12 x−β(R−β,β1 h)(y), (35) where (R−β,β1 ) is the right-hand side Erdèlyi–Kober fractional differential operator by (4).
Using (32)–(35) and the similarity variable (27), the reduced NLFODEs of the coupled Burger’s system (20) is
T1−γ22 ,γ γ+m g (y)
= y−β(M1g(y)+M3h(y))(R−β,β1 g)(y)) +M2y−βg(y)(R−β,β1 h)(y)+ag(y),
T1−γ22 ,γ γ+m h (y)
= y−β(M4h(y)+M5g(y))(R−β,β1 h)(y)
+M6y−βh(y)(R−β,β1 g)(y)+bh(y), (36) where (T1−γ22 ,γ
γ+m )and(R−β,β1 ) are given by (2) and (4), respectively, and arbitrary constants are given by M2 = K2M1, M3 = K1M1, M4 = K3M1, M5 = K4M1, M6= K5M1.
Remark1. In the obtained symmetries of the considered system (20), if we take values A1(t)=2, A2(t)= −1, A3(t) = −1, A4(t) = 1, B1(t) = 2, B2(t) = −1, B3(t)= −1 andB4(t)=1, we can find the symmetries of the system considered in [33]. Also, if we takeβ =1 along with these values, then we can find symmetries of the system considered in [36].
3.2 Space–time fractional Ito’s system with time-dependent variable coefficients
Consider the space–time fractional Ito’s system with time-dependent variable coefficients [33] in the follow- ing form:
∂γu
∂tγ = A1(t)u∂βu
∂xβ +A2(t)v∂βu
∂xβ +A3(t)∂3u
∂x3,
∂γv
∂tγ =B1(t)u∂βv
∂xβ +B2(t)v∂βu
∂xβ. (37)
The invariance criteria for system (37) under the one- parameter Lie group of transformation is obtained as ηγ,t =A3(t)ηx x x+τA3(t)ux x x+(τA1(t)u+A1(t)η
+τA2(t)v+A2(t)φ)∂xβu +(A1(t)u+A2(t)v)ηβ,x,
φγ,t =B1(t)uφβ,x+(τB1(t)u+B1(t)η)∂xβv
+b2(t)vηβ,x+(τB2(t)v+B2(t)φ)∂xβu, (38) whereηx x xis the extended infinitesimal of order 3,ηγ,t, φγ,t are the extended infinitesimals of orderγ andηβ,x, φβ,x are the extended infinitesimals of orderβ.
System (37) for A3(t) = atm (a is an arbitrary constant and mis a real number) admits the group of
transformations (7). Then the obtained Lie symmetries are
ξ = c1x
3 , τ = c1t γ +m, η= γ −1
2(γ +m)c1u, φ = γ −1
2(γ +m)c1v, (39) where c1 is an arbitrary constant and variable coeffi- cients are governed by the following conditions:
A1(t)= M1t1−23γ+β(γ+3m), A2(t)= K1A1(t),
B1(t)= K2A1(t), B2(t)= K3A1(t), (40) where M1, K1, K2andK3are arbitrary constants.
The Lie symmetry generator of system (37) is X = x
3
∂
∂x + t γ +m
∂
∂t +(γ −1)u
2(γ +m)
∂
∂u + (γ −1)v 2(γ +m)
∂
∂v. (41)
The solution of the auxiliary equations of the infinitesi- mal generator (41) gives the symmetry variable
y =xt−γ+m3 (42)
AND the corresponding symmetry transformations are obtained as
u(x,t)=tγ−21g(y),
v(x,t)=tγ−21h(y). (43) The admissible coefficients given by (40), with (42) and (43), reduce the system of NLFPDEs (37) forγ, β >0 to the system of NLFODEs, given by
T1−γ32 ,γ
γ+m
g (y)=y−β(M1g(y)+M2h(y))
×(R1−β,βg)(y))+ag(y),
T1−γ32 ,γ
γ+m h (y)=M3y−βg(y)(R−β,β1 h)(y)
+M4y−βh(y)(R−β,β1 g)(y), (44) where (T1−γ32 ,γ
γ+m
) is the left-hand side Erdèlyi–Kober fractional differential operator given by (2) and(R−β,β1 ) is the right-hand side Erdèlyi–Kober fractional dif- ferential operator given by (4) and M2 = K1M1, M3 = K3M1, M4 = K2M1 are arbitrary constants.
Remark2. In the obtained symmetries of the considered system (45), if we putA1(t)=3,A2(t)=1,A3(t)=1, B1(t)= 1 andB2(t) =1, we can find the symmetries
of the system considered in [33]. Also, if we takeβ =1 along with these values, then we can find symmetries of the system considered in [36].
3.3 Space–time fractional coupled Korteweg–de Vries (KdV) equations with time-dependent variable
coefficients
Various methods have been applied to study the time fractional coupled KdV equations [45] such as homo- topy decomposition method [46]. Also, the space–time fractional coupled KdV equations have been studied by the Lie symmetry method [33] with constant variables.
We consider the space–time fractional coupled KdV equations with time-dependent variable coefficients
∂γu
∂tγ = A1(t)u∂βu
∂xβ +A2(t)v∂βv
∂xβ +A3(t)∂3u
∂x3,
∂γv
∂tγ =B1(t)u∂βv
∂xβ +B2(t)∂3v
∂x3. (45)
The invariance criteria for system (37) under the one- parameter Lie group of transformation is obtained as ηγ,t = A3(t)ηx x x+τA3(t)ux x x
+(τA1(t)u+ A1(t)η)∂xβu
+A2(t)vφβ,x+(τA2(t)v+A2(t)φ)∂xβv +A1(t)uηβ,x,
φγ,t =B2(t)ηx x x+τB2(t)vx x x
+B1(t)uφβ,x+(τB1(t)u
+B1(t)η)∂xβv, (46) where ηx x x, φx x x are the extended infinitesimals of order 3, ηγ,t, φγ,t are the extended infinitesimals of orderγ andηβ,x, φβ,x are the extended infinitesimals of orderβ.
System (45) for A3(t)=atm andB2(t) =bts (a, b are arbitrary constants andm,nare real numbers) admits the group of transformations (7). Then the obtained Lie symmetries are
ξ = c1x
3 , τ = c1t γ +m, η= γ −1
2(γ +m)c1u, φ = γ −1
2(γ +m)c1v, (47) where c1 is an arbitrary constant and variable coeffi- cients are governed by the following conditions:
A1(t)=M1t1−3γ2 +β(γ3+m), A2(t)=K1A1(t),
B1(t)=K2A1(t), (48)