International Journal of Differential Equations

Volume 2016 (2016), Article ID 4270724, 8 pages

http://dx.doi.org/10.1155/2016/4270724

## Symmetry Classification and Exact Solutions of a Variable Coefficient Space-Time Fractional Potential Burgers’ Equation

Department of Mathematics, Jaypee University of Information Technology, Waknaghat, Solan 173234, India

Received 29 April 2016; Revised 13 July 2016; Accepted 9 August 2016

Academic Editor: Wen-Xiu Ma

Copyright © 2016 Manoj Gaur and K. Singh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We investigate the symmetry properties of a variable coefficient space-time fractional potential Burgers’ equation. Fractional Lie symmetries and corresponding infinitesimal generators are obtained. With the help of the infinitesimal generators, some group invariant solutions are deduced. Further, some exact solutions of fractional potential Burgers’ equation are generated by the invariant subspace method.

#### 1. Introduction

Partial differential equations (PDEs) have a wide range of applications in many fields, such as physics, engineering, and chemistry, which are fundamental for the mathematical formulation of continuum models [1, 2]. Burgers’ equation is a one-dimensional nonlinear partial differential equation, which is a simple form of the one-dimensional Navier-Stokes equation. It was presented for the first time in a paper in 1940s by Burgers. Later, Burgers’ equation was studied by Cole [3] who gave a theoretical solution, based on Fourier series analysis, using the appropriate initial and boundary conditions. Burgers’ equation has a large variety of applications in the modelling of water in unsaturated soil, dynamics of soil water, statistics of flow problems, mixing and turbulent diffusion, cosmology, and seismology [3, 4].

Recently, fractional differential equations have found extensive applications in many fields. Many important phenomena in viscoelasticity, electromagnetics, material science, acoustics, and electrochemistry are elegantly described with the help of fractional-order differential equations [4–9]. It has been revealed that the nonconservative forces can be described by fractional differential equations. Therefore, as most of the processes in the real physical world are nonconservative, the fractional calculus can be used to describe them. Fractional integrals and derivatives also appear in the theory of control of dynamical systems, when the controlled system and/or the controller is described by a fractional differential equation [10, 11]. In the last few decades, the subject of the fractional calculus has caught the consideration of many researchers who contributed to its development. Recently, some analytical and numerical methods have been introduced to solve a fractional-order differential equation [7, 8, 11–24]. However, all the methods have insufficient development as they allow one to find solutions only in case of linear equations and for some isolated examples of nonlinear equations [4, 7, 8, 11, 23, 24].

It is very well known that the Lie group method is the most effective technique in the field of applied mathematics to find exact solutions of ordinary and partial differential equations [25, 26]. However, this approach is not yet applied much to investigate symmetry properties of fractional differential equations (FDEs). To the best of our knowledge, there are a few papers (e.g., [17, 26–32]) in which Lie symmetries and similarity solutions of some fractional differential equations have been discussed by some researchers. More recently, Jumarie [33] proposed the modified Riemann-Liouville derivative and Jumarie-Lagrange method [34], after which a generalized fractional characteristic method and a fractional Lie group method have been introduced by Wu [32, 35] in order to solve a fractional-order partial differential equation.

In this paper, we intend to apply Lie group method to solve the space-time fractional potential Burgers’ equation of the formwhere and are the modified Riemann-Liouville derivatives with respect to time and space variables, respectively, and and are arbitrary functions of . Equation (1) is connected to the fractional Burgers’ equation by the well-known Hopf-Cole transformations and is a generalization of the time-fractional Burgers’ equation with constant coefficients examined by Wu [32]. This work is based on some basic elements of fractional calculus, with special emphasis on the modified Riemann-Liouville type derivative. The paper is organized as follows. In Section 2, we briefly describe some definitions and properties of fractional calculus. In Section 3, we obtain the symmetries for Burgers’ equation having six-dimensional Lie algebra. In Section 4, we analyze the reduced systems and find some invariant solutions of (1). Section 5 contains application of invariant subspace method on fractional Burgers’ equation (1). Finally, a conclusion is given in Section 6.

#### 2. Some Concepts from Fractional Calculus

In this paper, the modified Riemann-Liouville derivative proposed by Jumarie [33] has been adopted. Some definitions are given which have been used in this work.

##### 2.1. Fractional Riemann-Liouville Integral

The fractional Riemann-Liouville integral of a continuous (but not necessarily differentiable) real valued function with respect to is defined as [7, 33, 36]The fractional integral with respect to was introduced by Jumarie [36] in order to study the fractional derivative of nondifferentiable functions in modified Riemann-Liouville sense. Here we are fully in Leibniz framework; that is to say denote finite increment in fractional sense. As a result, we shall be able to duplicate, in a straightforward manner, most of the known standard formulae by merely making the substitution .

##### 2.2. Riemann-Liouville Fractional Derivative

The fractional Riemann-Liouville derivative of is defined as [7]

##### 2.3. Modified Riemann-Liouville Derivative

Through the fractional Riemann-Liouville integral, Jumarie [33] proposed the modified Riemann-Liouville derivative of as

##### 2.4. Some Useful Formulae

Here, some properties of modified Riemann-Liouville derivative are given which have been used in this paper:(i).(ii).(iii), given that exists.(iv), , and .(v).(vi). The above formulae and details thereof along with the scope of applications and limitations can be found in [33, 37].

##### 2.5. Characteristic Method for Fractional-Order Differential Equations

Applying the fractional chain rule proposed by Jumarie [33], we haveWu [32, 35] extended the characteristic method of first-order linear partial differential equation to a linear fractional differential equation of the form and introduced the fractional characteristic equations [32] as given herein:On solving the characteristic equations (7) for various infinitesimal generators of (1), one can obtain the similarity variables and reduction of (1) to an ODE.

#### 3. Symmetry Classification of (1)

Herein, we investigate the symmetries and reductions of space-time fractional potential Burgers’ equation (1). A fractional Lie symmetry of (1) is a continuous group of point transformations of independent and dependent variables which leaves (1) invariant.

Let us assume that (1) admits the Lie symmetries of the formwhere is the group parameter and , , and are the infinitesimals of the transformations for the independent and dependent variables, respectively. A group invariant solution of space-time fractional potential Burgers’ equation (1) is a solution which can be mapped into another solution of (1) under the point transformations (8). The associated Lie algebra of infinitesimal symmetries of (1) is then the fractional vector field of the formThe fractional second-order prolongation of (9) is given byNow, for the invariance of (1) according to [26] under (8), we must havewhere or equivalently whereThe generalized fractional prolongation vector fields , , and are given by Now, using the above generalized fractional prolongation vector fields in (12) and equating the coefficient of various derivative terms to zero, we get the following simplified set of determining equations:On solving (20) by using fractional Lie group method, we obtain a particular solution as where and , , and are arbitrary constants. Using this value of in (17) and (18), we get which brings forth the following possibilities:(i).(ii).

*Case (i)*. In this case, from the determining equations, we getwhere , , and are six arbitrary constants. Using (23) in (17), we also get , where is an arbitrary constant. This covers case (ii); as for on solving the determining equations, we get . Further, for and , the infinitesimals can be reduced to those reported in [32] by setting the coefficients , , , , , and .

Hence, the fractional point symmetry generators admitted by (1) are given by

These infinitesimal generators can be used to determine a six-parameter fractional Lie group of point transformation acting on -space. It can be verified easily that the set forms a six-dimensional Lie algebra under the Lie bracket , which reduces to the well-known generalized Galilea algebra [26] for . The commutator table is as given in Table 1.