Invariant subspaces and exact solutions for some types of scalar and coupled time-space fractional diffusion equations

https://doi.org/10.1007/s12043-020-01964-3

Invariant subspaces and exact solutions for some types of scalar and coupled time-space fractional diffusion equations

P PRAKASH

Department of Mathematics, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Coimbatore 641 112, India

E-mail: vishnuindia89@gmail.com; p_prakash@cb.amrita.edu

MS received 5 July 2019; revised 10 February 2020; accepted 9 April 2020

Abstract. We explain how the invariant subspace method can be extended to a scalar and coupled system of time-space fractional partial differential equations. The effectiveness and applicability of the method have been illustrated using time-space (i) fractional diffusion-convection equation, (ii) fractional reaction-diffusion equation, (iii) fractional diffusion equation with source term, (iv) two-coupled system of fractional diffusion equation, (v) two-coupled system of fractional stationary transonic plane-parallel gas flow equation and (vi) three-coupled system of fractional Hirota–Satsuma KdV equation. Also, we explicitly showed how to derive more than one exact solution of the equations as mentioned above using the invariant subspace method.

Keywords. Time-space fractional partial differential equations; invariant subspace method; Laplace transformation technique; Mittag–Leffler function.

PACS Nos 02.30.Jr; 02.70.–c; 02.90.+p; 02.30.Uu

1. Introduction

The subject of fractional calculus is one of the most rapidly developing areas of mathematical analysis. The study of fractional differential equations (FDEs) has considerable popularity and importance for the past few decades, mainly due to their widespread applica- tions in various fields of science and engineering such as fluid flow, viscoelasticity, aerodynamics, electro- magnetic theory, rheology, signal processing, electrical networks and so on [1–8]. In the last few decades, several analytical and numerical techniques have been developed to construct exact and numerical solutions of nonlinear differential equations. However, the deriva- tion of the exact solution of FDEs is not an easy task, because some properties of fractional derivatives are harder than that of the classical derivative.

For this reason, in recent years, both mathematicians and physicists have paid much attention to study the exact and numerical solutions of nonlinear fractional partial differential equations (FPDEs) using various ad- hoc methods, such as Lie group analysis method [9–

15], Adomian decomposition method [16–18], homo- topy decomposition method [19], differential transform method [20], function-expansion method [21–23] and

so on. However, recent investigations have shown that a new analytic method based on the invariant sub- space approach provides an effective tool to derive the exact solution of scalar and coupled system of time- space FPDEs. This method was originally developed by Galaktionov and Svirshchevskii [24] (see also [25–

33]) for PDEs and was further extended by Gazizov and Kasatkin [34] (see also [35–46]) for time FPDEs.

The main objective of this article is to demonstrate how the invariant subspace method provides an effective tool to derive exact solution of the following time-space FPDEs: (i) time-space fractional diffusion-convection equation, (ii) time-space fractional reaction-diffusion equation and (iii) time-space fractional diffusion equa- tion with source term.

Here we would like to point out that only a lim- ited number of applications for the coupled system of time-space FPDEs have been investigated through the invariant subspace method. The applicability and effec- tiveness of the method are illustrated through time-space fractional (i) two-coupled system of diffusion equation, (ii) two-coupled system of stationary transonic plane- parallel gas flow equation [11,24] and (iii) three-coupled system of Hirota–Satsuma KdV equation [12], and their exact solutions are derived.

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The layout of this paper is as follows: In §2, some basic concepts of fractional calculus and a brief descrip- tion of the invariant subspace method for scalar and m-component coupled system of nonlinear time-space FPDEs in the sense of Riemann–Liouville/Caputo frac- tional derivative are presented. In §3, the effectiveness of the method is illustrated by solving the above-mentioned scalar and coupled system of time-space FPDEs. Finally, a summary of our results is given in §4.

2. Preliminaries

In this section, we would like to present some basic definitions and results related to the fractional calculus.

Also, we present brief details of the invariant subspace method for scalar and coupled system of time-space FPDEs.

DEFINITION 1 [1,2]

The Riemann–Liouville (R–L) fractional derivative of order α > 0 of the function g ∈ L¹([a,b],R₊) is defined by

RLd^αg(t) dt^α =

⎧⎨

⎩ 1 (n−α)

dⁿ dtⁿ

_t

0

g(s) (t−s)^α−n+1ds

,n−1< α <n;

g⁽ⁿ⁾(t), α =n, n ∈N,

where L¹([a,b])denotes the set of all absolutely inte- grable functions on[a,b].

Note1 [1,2]. The R–L fractional derivative ofg(t)=t^μ is as follows:

RLd^αt^μ

dt^α = (μ+1)

(μ−α+1)t^μ−α,

α >0, μ >−1, t >0. (1)

DEFINITION 2 [1,2]

The Caputo fractional derivative of as order α > 0 of the functiong∈Cⁿ([a,b])is defined as

d^αg(t) dt^α

=

⎧⎨

⎩ 1 (n−α)

_t

0

g⁽ⁿ⁾(s)

(t−s)^α−n+1ds,n−1< α <n; g⁽ⁿ⁾(t), α =n, n ∈N, whereCⁿ([a,b])denotes the set of all continuouslyn- times differentiable functions on[a,b].

Note2 [1,2]. The Caputo fractional derivative ofg(t)=

t^μis as follows:

d^αt^μ dt^α =

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

0, ifμ∈{0,1, . . . ,n−1}, andn=[α]+1; (μ+1)

(μ−α+1)t^μ−α, ifμ∈Nandμ≥nor μ /∈Nandμ>n−1. For simplicity, we denote the R–L and Caputo frac- tional derivative operators respectively as^RL_dt^d_α^α and _dt^dα_α. Note3 [1,2]. The Laplace transformation of Caputo fractional derivative of orderα ∈ (n−1,n],n ∈ N, is

L d^αg(t) dt^α

=s^αg(s)−

n−1

k=0

s^α−k−1g^(k)(0), Re(s) >0.

DEFINITION 3 [1,2]

Two-parametric Mittag–Leffler function is defined as

E_β,γ(z)= ∞ r=0

z^r

(βr +γ ), β, γ,z ∈C, Re(β) >0, Re(γ ) >0.

Some properties of the Mittag–Leffler function are as follows:

E1,1(z)=e^z, E1,k(z)= 1

z^k−1

e^z−

k−2

r=0

z^r r!

, k ∈N.

Note thatE_β,1(z)≡E_β(z).

Note4 [1,2]. The Laplace transformation of t^γ−1E_β,γ(±kt^β)is

L{t^γ−1E_β,γ(±kt^β)} = s^β−γ

(s^β ∓k), Re(s) >|k|^1/β. Caputo fractional derivative of the Mittag–Leffler functions are

d^α

dt^α[t^γ−1E_β,γ(kt^β)] =t^γ−α−1E_β,γ−α(kt^β), d^α

dt^α

E_α(kt^α)

=k E_α(kt^α), α, β, γ >0, k ∈R.

Theorem 1. If L{ϕ(t)} = ¯ϕ(s)and L{φ(t)} = ¯φ(s), then

ϕ(t) φ(t)= _t

0

ϕ(t−τ)φ(τ)dτ =L⁻¹

¯

ϕ(s)φ(¯ s) , where ϕ(t) φ(t) is called a convolution ofϕ(t) and φ(t).

2.1 Invariant subspace method for scalar and coupled FPDEs

2.1.1 Scalar time-space FPDE. We consider the fol- lowing generalised scalar time-space FPDE:

m i=0

λi∂^α+iu(x,t)

∂t^α+i = ˆG[u(x,t)]

=G

x,u,∂^βu

∂x^β, . . . , ∂^β

∂x^β ∂^βu

∂x^β

,∂^rβu

∂x^rβ,∂^β+ku

∂x^β+k

, (2) where α, β > 0, k,r ∈ N, λi ∈ R, and Gˆ[u] is a linear or nonlinear fractional differential operator.

Here,(∂^β/∂x^β)(·)and(∂^α/∂t^α)(·)are space- and time- fractional derivatives in the R–L or Caputo sense and Gˆ(·) is a sufficiently given smooth function. First, we define the linear space

Wn =L{ϕ1(x), . . . , ϕn(x)}

=

⎧⎨

⎩ n

j=1

ajϕj(x)|aj ∈R, j =1,2, . . . ,n

⎫⎬

⎭, whereLdenotes the linear span and the functionsϕ1(x), . . . , ϕn(x) are linearly independent. The linear space Wn is said to be invariant with respect to the fractional differential operator Gˆ[u] if Gˆ : Wn → Wn, that is, Gˆ[Wn] ⊆ Wn or Gˆ[u] ∈ Wn, for all u ∈ Wn. This means that there existn-functions1,2,. . .,n such that

Gˆ

⎡

⎣ⁿ

j=1

a_jϕj(x)

⎤

⎦= n

j=1

j(a₁,a₂, . . . ,a_n) ϕj(x), foraj ∈R.

Theorem 2. LetWn be ann-dimensional linear space overR. IfWnis invariant under the fractional differen- tial operatorGˆ[u],then the time-space FPDE(2)admits the following exact solution:

u(x,t)= A1(t)ϕ1(x)+A2(t)ϕ2(x)+· · ·+An(t)ϕn(x), (3) where the coefficients Aj(t) (j = 1,2, . . . ,n)satisfy the following system of fractional ordinary differential equations(FODEs):

m i=0

λi

d^α+iAj(t)

dt^α+i =j(A₁(t),A₂(t), . . . ,A_n(t)),

j =1, . . . ,n. (4)

Proof. Using the linearity of the fractional derivative with eq. (3), we obtain

m i=0

λi∂^α+iu(x,t)

∂t^α+i = n

j=1

_m

i=0

λi

d^α+iAj(t) dt^α+i

ϕj(x).

(5) Let Wn be an invariant subspace with respect to the fractional differential operatorGˆ[u]. Then there existn functions1, 2, . . . , nsuch that

Gˆ

⎡

⎣ⁿ

j=1

ajϕj(x)

⎤

⎦= n

j=1

j(a1,a2, . . . ,an)ϕj(x), (6) where aj ∈ Randj’s are expansion coefficients of Gˆ[u] ∈ Wn corresponding toϕj’s. From eqs (3) and (6), we have

Gˆ[u(x,t)] = ˆG

⎡

⎣ⁿ

j=1

A_j(t)ϕj(x)

⎤

⎦

= n

j=1

j(A1(t), . . . ,An(t))ϕj(x). (7)

Substituting eqs (7) and (5) in eq. (2), we have n

j=1

_m

i=0

λi

d^α+iAj

dt^α+i −j(A1(t),A2(t), . . . ,An(t))

×ϕj(x)=0. (8)

From eq. (8) and using their linear independence of {ϕj(x), j = 1,2, . . . ,n}, we yield the system of FODEs

m i=0

λi

d^α+iA_j(t)

dt^α+i =j(A1(t),A2(t), . . . ,An(t)),

j =1,2, . . . ,n. (9)

2.1.2 Two-coupled system of time-space FPDEs.

Consider the following two-coupled system of time- space FPDEs:

∂^α1u1

∂t^α1 =G1

x,u1,u2,∂^βu1

∂x^β ,∂^βu2

∂x^β , . . . ,

∂^rβu1

∂x^rβ ,∂^rβu2

∂x^rβ ,∂^β+k1u1

∂x^β+k1 ,∂^β+k1u2

∂x^β+k1

,

∂^α2u₂

∂t^α2 =G2

x,u1,u2,∂^βu₁

∂x^β ,∂^βu₂

∂x^β , . . . ,

∂^rβu1

∂x^rβ ,∂^rβu2

∂x^rβ ,∂^β+k2u1

∂x^β+k2 ,∂^β+k2u2

∂x^β+k2

, (10)

where α1, α2, β > 0, k1,k2,r ∈ N, and G1, G2 are generalised linear/nonlinear fractional differential oper- ators and it can be considered as the given sufficient smooth functions, and(∂^α/∂t^α)(·)and(∂^β/∂x^β)(·)are time- and space-fractional derivatives in R–L/Caputo sense. Hereafter, we shall use the following notations throughout the article:

Gˆp[u1,u2] =Gp

x,u1,u2,∂^βu1

∂x^β ,∂^βu2

∂x^β , . . . , ∂^β

∂x^β ∂^βu1

∂x^β

,∂^rβu1

∂x^rβ ,∂^rβu2

∂x^rβ ,∂^β+kpu1

∂x^β+kp ,∂^β+kpu2

∂x^β+kp

,

up =up(x,t), p =1,2.

Estimation of invariant subspace: Following the proce- dure same as above for scalar time-space FPDEs, we develop the following result for the two-coupled system of time-space FPDEs. First, we define the linear spaces

Wn^p_p =L{ϕ₁^p(x), . . . , ϕn^p_p(x)}

≡

⎧⎨

⎩

n_p

j=1

a^p_jϕ^p_j(x)a^p_j ∈R, j=1, . . . ,np

⎫⎬

⎭,

wherep=1,2 and the functionsϕ₁^p(x), . . . , ϕn^p_p(x)are linearly independent. The linear spacesWn^p_p, p=1,2, are called invariant under the vector fractional differ- ential operator Gˆ = (G1,G2) ifGˆ : Wn¹₁ ×Wn²₂ → Wn¹₁ ×Wn²₂, which means that Gˆp : Wn¹₁ ×Wn²₂ → Wn^pp, p = 1,2, that is, Gˆ_p[W_n¹₁ ×W_n²₂] ⊆ Wn^pp or Gˆp[u1,u2] ∈ Wn^pp, for all (u1,u2) ∈ Wn¹₁ ×Wn²₂, p=1,2. Then, we have

Gˆ_p

⎡

⎣ⁿ¹

j=1

a¹_jϕ¹_j(x),

n₂

j=1

a²_jϕ²_j(x)

⎤

⎦

=

n_p

j=1

_j^p(a₁¹, . . . ,a_n¹₁,a₁², . . . ,a_n²₂)ϕ^p_j(x), p=1,2.

Theorem 3. Let Wn^p_p be a finite-dimensional linear space over R. If Wn^pp is invariant with respect to the fractional differential operator Gˆp[u1,u2], then the two-coupled system of time-space FPDEs (10) admit the following exact solution:

u_p(x,t)=

n_p

j=1

A^p_j(t)ϕ^p_j(x), p=1,2, (11)

where the coefficientsA^p_j(t)satisfy the following system of FODEs:

d^αpAj(t) dt^αp

=^p_j(A¹₁(t),A¹₂(t), . . . ,A¹_n1(t),A²₁(t), . . . ,A²_n2(t)), j =1, . . . ,np, p=1,2.

Proof. Using the linearity of the fractional derivative with eq. (11), we obtain

∂^αpup(x,t)

∂t^αp = n p

j=1

d^αpAj(t)

dt^αp ϕ^p_j(x), p =1,2. (12) LetWn^p_p be an invariant subspace under the fractional differential operatorGˆp[u1,u2]. Then there exists the functions₁^p, ₂^p, . . . , n^p_p (p=1,2)such that Gˆp

⎡

⎣ⁿ¹

j=1

a¹_jϕ¹j(x),

n₂

j=1

a²_jϕ²j(x)

⎤

⎦

=

n_p

j=1

^p_j(a₁¹, . . . ,a_n¹₁,a²₁, . . . ,a_n²₂)ϕ_j^p(x), (13)

wherea^p_j ∈R,p=1,2,and^p_j’s are expansion coeffi- cients ofG[uˆ 1,u2] ∈Wn^p_pcorresponding toϕ^p_j’s. From eqs (11) and (13), we have

Gˆ_p[u₁(x,t),u₂(x,t)]

= ˆGp

⎡

⎣ⁿ¹

j=1

A¹_j(t)ϕ¹j(x),

n₂

j=1

A²_j(t)ϕ²j(x)

⎤

⎦

=

n_p

j=1

^p_j(A¹₁(t), . . . ,A¹_n1(t),A²₁(t), . . . ,A²_n2)ϕj(x),

p=1,2. (14)

Substituting eqs (14) and (12) into eq. (10), we have

np

j=1

d^αpA^p_j

dt^αp −^p_j(A¹₁(t),A¹₂(t), . . . ,A¹_n1(t),A²₁(t), A²₂(t), . . . ,A²_n2(t))

ϕ_j^p(x)=0, p=1,2. (15)

By their linear independence of{ϕ^p_j, j =1,2, . . . ,np, p=1,2}, we have

d^αpA^p_j(t) dt^αp

=(A¹1(t),A¹₂(t), . . . ,A¹_n(t),A²₁(t),A²₂(t), . . . ,A²_n(t)), j =1,2, . . . ,np, p=1,2. (16) 2.1.3 m-coupled system of time-space FPDEs. Con- sider the followingm-coupled system of time and space FPDEs

∂^αpU

∂t^αp = ˆG(U)≡(G1(U), . . . ,Gm(U))∈R^m, (17) where αp > 0, p = 1,2, . . . ,m, and the operators Gq(·)/(q = 1,2, . . . ,m) are generalised linear/non- linear fractional differential operators and can be con- sidered as sufficient smooth functions, and(∂^α/∂t^α)(·) and(∂^β/∂x^β)(·)are time- and space-fractional deriva- tives in R–L/Caputo sense, and U = (u1,u2, . . . , um)∈R^m,up=up(x,t),

Gˆp[U] =Gp

x,u1, . . . ,um,∂^βu1

∂x^β , . . . ,∂^βum

∂x^β ,

∂^rβu1

∂x^rβ , . . . ,∂^rβum

∂x^rβ , . . . ,∂^β+kpup

∂x^β+kp ,∂^β+kpum

∂x^β+kp

,

r,kp ∈N,β >0, p=1,2, . . . ,m.

Estimation of invariant subspace: Proceeding as above, we can develop the following result for anm-coupled system of time-space FPDEs. Here, we define the linear spaces as

Wn^p_p =L{ϕ₁^p(x), . . . , ϕn^p_p(x)}

≡

⎧⎨

⎩

np

j=1

a^p_jϕ^p_j(x)(a^p_j, . . . ,an^p_p)∈R^np

⎫⎬

⎭, where p = 1,2, . . . ,m, and the functions ϕ₁^p(x), . . . , ϕn^p_p(x) (np ≥ 1)are linearly independent. The linear spacesWn^pp, p=1,2, . . . ,m, are called invariant with respect to the vector fractional differential operatorGˆ = (G₁,G₂, . . . ,G_m)ifGˆ :W_n¹₁× · · · ×W_n^m_m →W_n¹₁×

· · ·×Wn^m_m, which means thatGˆp :Wn¹₁×· · ·×Wn^m_m → Wn^p_p, p =1,2, . . . ,m, that is,Gˆp[Wn¹₁×· · ·×Wn^m_m] ⊆ Wn^p_p orGˆp[u1, . . . ,um] ∈Wn^p_p, for all(u1, . . . ,um)∈ Wn¹₁×· · ·×Wn^m_m,p=1, . . . ,m. Then there exists^p_j,

j=1,2, . . . ,np, p =1,2, . . . ,m, such that Gˆp

⎡

⎣ⁿ¹

j=1

a¹_jϕ¹j(x), . . . ,

n_m

j=1

a^m_j ϕ^mj (x)

⎤

⎦

=

np

j=1

^p_j(a¹₁, . . . ,a¹_n1, . . . ,a₁^m, . . . ,a_n^m_m)ϕ_j^p(x), for all(a₁^p, . . . ,an^p_p)∈R^np, p=1,2, . . . ,m.

Theorem 4. Let Wn^p_p be a finite-dimensional linear space overR and if Wn^pp is invariant under the frac- tional differential operatorGˆp[U], then them-coupled system(17)has a solution of the form

up(x,t)=

n_p

j=1

A^p_j(t)ϕ^p_j(x), p=1,2, . . . ,m, (18) where the coefficientsA^p_j(t)satisfy the following system of FODEs:

d^αpAj(t)

dt^αp =j(A¹₁(t),A²₂(t), . . . ,A²_n2(t)),

j =1, . . . ,np, p=1,2, . . . ,m. (19) Proof. Similar to the proof of Theorem 3. Let us assume that the invariant subspace

Wn^pp =L{ϕ₁^p, . . . , ϕn^pp}

is defined as space generated by solutions of the follow- ing linear fractional order ODEs:

Lp[yp] =y^(α)_p +c_n^p

p−1(x)y^(α−_p ¹⁾+ · · · +c₀^p(x)yp =0, whereα ∈(np−1,np], np ∈N, p =1,2, . . . ,m, and y^(α)p = d^αyp/dx^α. Thus, the invariant condition reads as

Lp[ ˆGp[U]]|[H₁]∩···∩[H_m] =0, p=1,2, . . . ,m,

where [Hp] denotes the equation Lp[up] = 0 and its differential consequences with respect tox.

3. Construction of invariant subspaces and exact solutions

3.1 Time-space fractional diffusion-convection equation

Consider the following time-space fractional diffusion- convection equation:

∂^αu

∂t^α = ˆG[u]

= ∂^βu

∂x^β 2

∂p

∂u

+p(u) ∂^β

∂x^β ∂^βu

∂x^β

−∂^βu

∂x^β ∂q

∂u

, (20)

where t > 0, α, β ∈ (0,1], and the functions p(u) and q(u) represent the phenomenon of diffusion and convection respectively. Equation (20) withα =1 and β = 1 is a combination of the diffusion and con- vection (advection) equations, and describes physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion and convection [8]. It was discussed with α = 1 and β = 1 through the invari- ant subspace method in [33,42]. We would like to point out that the operator Gˆ[u] admits no invariant sub- space for arbitrary functions p(u)andq(u). Hence, we choose p(u)=anuⁿ+an−1uⁿ⁻¹+ · · · +a1u+a0and q(u) = bn+1uⁿ⁺¹ +bnuⁿ + · · · +b1u +b0, n ∈ N, where a_n, a_n−1, . . . ,a₀,b_n+1, . . . ,b₁,b₀ are arbitrary constants.

Then, eq. (20) reduces to

∂^αu

∂t^α = ˆG[u]

= [nanuⁿ⁻¹+(n−1)an−1uⁿ⁻² + · · · +a₁]

∂^βu

∂x^β 2

+ [anuⁿ +an−1uⁿ⁻¹+ · · · +a1u+a0]

× ∂^β

∂x^β ∂^βu

∂x^β

− [(n+1)bn+1uⁿ+nbnuⁿ⁻¹+ · · · +b1]

×∂^βu

∂x^β, t >0, (21)

whereα, β ∈ (0,1]. It is easy to find that the differen- tial operator Gˆ[u] admits a one-dimensional invariant

subspace W1 = L{E_β(kx^β)}, k ∈ R, ifark = br+1, r =1,2. . .n,n∈N, because

Gˆ[A1E_β(kx^β)] =(a0k²−b1k)A1E_β(kx^β)∈W1. Thus, we can write the exact solution in the form u(x,t)= A1(t)E_β(kx^β), (22) where A1(t)is an unknown function to be determined.

Substituting (22) in (21), we get d^αA1

dt^α =(a0k²−b1k)A1(t). (23) First, we considerα =β =1. In this case, we have u(x,t)=k0e^{(a0k2−b1k)t+kx}, k0,k,a0,b1∈R. (24) Next, we consider α, β ∈ (0,1]. Applying Laplace transformation technique on both sides of eq. (23), we get

s^αA¯₁(s)−s^α−1A₁(0)=(a₀k²−b₁k)A¯₁(s) which can be written as

A¯1(s)= k0s^α−1 s^α−(a0k²−b1k),

wherek0 = A1(0). Applying inverse Laplace transfor- mation to the above equation, we get

A1(t)=k0E_α((a0k²−b1k)t^α).

Hence, we obtain an exact solution for time-space frac- tional diffusion-convection equation (21) as follows:

u(x,t)=k0E_α((a0k²−b1k)t^α)E_β(kx^β), (25) whereα, β ∈(0,1]andk0,k,a0,b1∈R.Note that, for α = β = 1, eq. (25) is exactly the same as (24). The graphical representation of solution (25) for a0 = 2, k =k0 =b1 =1,t =1, and different values ofα and βis shown in figure1.

Case1. Letp(u)=a1u+a0,q(u)= −ka1u²+b1u+b0, k,a0,a1,b1,b0 ∈Randk(=0). Then, eq. (20) can be written as follows:

∂^αu

∂t^α =a1

∂^βu

∂x^β 2

+(a1u+a0) ∂^β

∂x^β ∂^βu

∂x^β

+(2ka1u−b1)∂^βu

∂x^β. (26)

It is easy to find that eq. (26) admits the following two- dimensional invariant subspaces:

(i) W2 =L{1,E_β(−kx^β)}. (ii) W2 =L{1,x^β}.

Here we consider the invariant subspace (i). Thus, we can write the exact solution of eq. (26) as follows:

u(x,t)= A1(t)+A2(t)E_β(−kx^β), (27)

Figure 1. Graphical representation of solution (25) fora0=2,k=k0=b1=1,t =1, and different values ofαandβ.

where the coefficients A1(t)and A2(t)satisfy the fol- lowing system of FODEs:

d^αA1

dt^α =0, d^αA2

dt^α =(−k²a₁A₁+a₀k²+kb₁)A₂. (28) First, we considerα =β =1. Solving system (28), we have

u(x,t)=k₁+k₂e^{k((−kk1a1+a0k+b1)t−x)}, (29) where k,k1,k2,a0,a1,b1 ∈ R. Next, we consider α, β ∈ (0,1]. Applying Laplace transformation tech- nique to system (28), we obtain an exact solution of eq.

(26) as follows:

u(x,t)=k1+k2E_α((−k²a1k1

+a₀k²+kb₁)t^α)E_β(−kx^β), (30) where k,k1,k2,a0,a1,b1 ∈ R.Observe that, forα = β = 1, eq. (30) is exactly the same as (29). It is also noted that whenα =β =1,a1=b1,a0 =b0,k1 =d1, k2=d2,k =a1,b1 =c1andb0 =c0, the exact solution (30) is exactly the same as given in [33]. The graphical representation of the exact solution (30) fork1 = −1, a1 =2,a0 =0,k2 =k =b1 =1,t =2, and different values ofαandβis shown in figure2.

Case2. Letp(u)=a0andq(u)=b1u+b0,a0,b1,b0∈ R. Then, eq. (20) reduces to

∂^αu

∂t^α =a0 ∂^β

∂x^β ∂^βu

∂x^β

−b1∂^βu

∂x^β (31)

which admits the following distinct invariant subspaces:

(i) W2 =L{1,x^β}.

(ii) Wn =L{E_β(k₁x^β), . . . ,E_β(k_nx^β)},n ∈N, (iii) Wn+1=L{1,E_β(k1x^β), . . . ,E_β(knx^β)},n∈N, (iv) Wn+2=L{1,x^β,E_β(k1x^β), . . . ,E_β(knx^β)},n ∈

N,

wherek_i ∈R, i=1, . . . ,n.

First, we consider the invariant subspace Wn =L{E_β(k₁x^β), . . . ,E_β(k_nx^β)}, n∈N.

Thus, we can write the exact solution in the form u(x,t)= A₁(t)E_β(k₁x^β)+· · ·+A_n(t)E_β(k_nx^β), (32) where Ai(t),i =1, . . . ,nsatisfy the following system ofn-FODEs:

d^αA1

dt^α =(a0k₁²−b1k1)A1, ...

d^αAn

dt^α =(a0k²_n−b1kn)An. (33) Applying Laplace transformation technique to system (33), we obtain an exact solution of eq. (31) withα = β =1, which reads as

u(x,t)= n s=1

r_se^{((a0ks−b1)kst+ksx)}. (34) Whenα ∈(0,1]

u(x,t)= n s=1

rsE_α

(a0ks−b1)kst^α

E_β(ksx^β), (35)

Figure 2. Graphical representation of solution (30) fork1= −1,a1 =2,a0 =0,k2 =k =b1 =1,t =2, and different values ofαandβ.

wherers,ks,a0,b1 ∈R(s =1,2, . . . ,n).We observe that forα =β =1, eq. (35) is exactly the same as (34).

Proceeding the same way as before, we can derive another more general exact solution associated with the more general invariant subspaceWn+2 =L{1,x^β, E_β(k1x^β), . . . ,E_β(knx^β)}, n ∈ N. For this case, we obtain the more general exact solution of (31) as follows:

u(x,t)=c1−c2b1(β+1)

(α+1)t^α+c2x^β +

n s=1

r_sE_α

(a₀k_s−b₁)k_st^α

E_β(k_sx^β), (36) whereα, β ∈(0,1]andc1,c2,rs,ks(s =1,2, . . . ,n), a0, b1 are arbitrary constants. Observe that, for c1 = c2 = 0, eq. (36) is exactly the same as (35). Simi- larly, we can derive different types of exact solutions for time-space fractional diffusion-convection equation (31) using the afore-mentioned invariant subspaces.

Case3. Let p(u) =a₁uandq(u) =(b₂/2)u²,b₂ and a1 are constants. Then, eq. (20) reduces to

∂^αu

∂t^α =a₁ ∂^βu

∂x^β 2

+a₁u ∂^β

∂x^β ∂^βu

∂x^β

−b₂u∂^βu

∂x^β. (37) It is easy to find that eq. (37) admits the following two- dimensional invariant subspaces:

(i) W2=L{1,E_β(kx^β)}ifa1=b2/2k.

(ii) W2=L{1,x^β}.

Here we consider the invariant subspace (i). Thus, we obtain an exact solution of (37) withα =β =1, which

reads as

u(x,t)=k₀+k₁e^{k(−b22k0t+x)}, (38) and whenα ∈(0,1],

u(x,t)=k₀+k₁E_α

−b2

2 kk₀t^α

E_β(kx^β), (39) wherek0, k1, b2, k ∈ R.Note that, for α = β = 1, eq. (39) is exactly the same as (38). We would like to mention that whenβ = 1, solution (39) is exactly the same as given in [42].

Case4. Letp(u)=uandq(u)=b0,b0 ∈R. Then, eq.

(20) describes only the time-space fractional diffusion equation as follows:

∂^αu

∂t^α = ∂^βu

∂x^β 2

+u ∂^β

∂x^β ∂^βu

∂x^β

(40) which admits an invariant subspaceW2 =L{1,x^β}. In this case, we have

u(x,t)=k0+k₁²((β+1))²

(α+1) t^α+k1x^β, (41) wherek₀,k₁ ∈ Randα, β ∈ (0,1].We would like to point out that whenβ =1, solution (41) is exactly the same as given in [42].

Case5. We note that if p(u) = 1 and q(u) = −¹₂u², then the diffusion-convection equation (20) reduces to the time-space fractional Burgers equation

∂^αu

∂t^α = ∂^β

∂x^β ∂^βu

∂x^β

+u∂^βu

∂x^β, (42)