The fractional partial differential equations (FPDEs) can describe many nonlinear phenomena in physics, chemistry, biology, engineering and other areas science. So the FPDEs have been widely concerned in recent years. To seek the solutions of FPDEs is an important aspect of scientific research. At present some effective methods have been put forward, such as the variational iteration method,^[1–3] differential transform method,^[4–5] Adomian decomposition method,^[6] the sub-equation method,^[7–8] invariant subspaces method^[9–11] and so on. Lie group analysis method is also one of the effective methods.

It is well known that Lie group analysis method plays a significant role in the analysis of differential equations.^[12–15] And Lie group analysis method has been extended to fractional differential equations by Gazizov et al. in Ref. [16], who were the earliest. Then some scholars apply Lie group analysis method to study the fractional differential equations and some results^[17–27] have been gotten.

In this paper, we study nonlinear time-fractional diffusion-convection equations with the help of Lie group analysis method. The diffusion-convection equations are widely used in science and engineering as mathematical models for computational simulations, such as in oil reservoir simulations, transport of mass and energy and global weather production, etc. Recently, Sahadevan and Prakash^[9] have used invariant subspaces method to derive the solutions of two special forms of nonlinear time-fractional diffusion-convection equation:

The two special forms^[9] of Eq. (1) are f(u) = βu, g(u) = (γ/2)u² and f(u) = βu² + γu, g(u) = K = constant, respectively. The corresponding equations are as follows

When α = 1, Eq. (1) can be reduced to the general nonlinear diffusion-convection equation, which have been studied in Refs. [28–30] et al.

In this paper, we adopt Riemann–Liouville fractional partial derivative, which is defined by

Some useful formulas and properties of Riemann–Liouville fractional derivative, which will be used in the following sections, are as follows. (i)

The generalized Leibniz rule:

If f = 1, g = ψ*(x*, t), we get^[26]

The paper is organized as follows. In Sec. 2, we study the (1 + N)-dimensional time-fractional partial differential equations by Lie group theory and the expression of is given. In Sec. 3, two special forms of nonlinear time-fractional diffusion-convection equations are analyzed using Lie group analysis method. Under the corresponding group transformation the equations are reduced into fractional ordinary differential equations and invariant solutions are obtained. Some conclusions are presented at the end of the paper.

In this section, we first consider the following (1 + N)-dimensional time-fractional partial differential equations:

According to Lie group theory, Eq. (4) admits a group of one-parameter ε continuous transformations:

The infinitesimal generator of the one-parameter Lie group (5) can be expressed in the following vector form

The prolongation of infinitesimal generator should be considered in the form:

Then we focus on the expression of .

Since Eqs. (2) and (3) have two independents t, x, one dependent u and have at most second-order derivatives, the second prolongation of infinitesimal generator should be considered in the form:

3.2 Group Analysis and Invariant Solutions of Eq. (3)

(i) Group Analysis of Eq. (3)

Similarly, applying Eq. (8) to Eq. (3), we obtain Eq. (13), which is acquired by Eq. (12)

Substituting expressions of , η^x, η^xx into Eq. (13), a polynomial in terms of dependent variable u and derivatives of dependent variables is obtained. Thus we get the following over-determined system of equations

We find the three-parameter Lie group of transformation which can be shown as Eq. (14)

where a₁, a₂ are arbitrary constants. We can obtain the corresponding two-dimensional Lie algebras

whose communication relations is given by [V₁, V₂] = V₁.

Here we consider two cases: β ≠ 0, γ = 0 and β = 0, γ ≠ 0.

(ii) Invariant Solutions

Subsequently, we will find invariant solution of Eq. (3) corresponding to Lie group of transformation. (i)

For Eq. (3):

For the generator V₂ = x(∂/∂x) + (2/α)t(∂/∂t), according to the invariant surface condition

the invariant solution has the form u = ϕ(z) with z = xt^−α/2, where ϕ(z) satisfies the equation

where the prime denotes differentiation with respect to z.

(b)

For Eq. (15):

For the generator V₂ = t(∂/∂t) − (α/2)u(∂/∂u), we have invariant solution u = ϕ(z)t^−α/2, where z = x. Then Eq. (15) is reduced to

where the prime denotes differentiation with respect to z.

For the generator V₃ = x(∂/∂x) + u(∂/∂u), we obtain u = xϕ(z), z = t.

Substituting invariant solution u = xϕ(z), z = t into Eq. (15), it leads to the reduced ODE

which gives

where E_{α, α}(z) is the Mittag–Leffler function.^[31–32]

(c)

For Eq. (16):

For the generator V₂ = x(∂/∂x) + (2/α)t(∂/∂t), we obtain invariant solution u = ϕ(z), where z = xt^−α/2. Then Eq. (16) is reduced to

In this paper, on the basis of Lie group theory the (1 + N)-dimensional time-fractional partial differential equations are analyzed and the expression of is given. As applications, two special forms of nonlinear time-fractional convection-diffusion equation are studied. Using the obtained group transformation, the equations are reduced into fractional ordinary differential equations and invariant solutions, some exact solutions are obtained. The exact solution and the conservation laws of some more complex nonlinear fractional partial differential equations should be studied.