Lie Symmetry Analysis For Several Kinds Of Fractional Porous Medium Equation

In this paper,Lie group analysis is developed to investigate the time fractional Porous Medium equation,the time fractional Dual Porous Medium equation,and three kinds of space-time fractional porous medium equations.Firstly,the time fractional Porous Medium equation is discussed by Lie group analysis,according to the obtained vector field and the corresponding optimal system,we establish the group invariant solutions of this equation.On the basis of the vector field admitted by this equation,the conservation law is investigated by the new Noether theorem.Using the homogeneous balanced method,we obtain the power series solutions and the graphs of several solutions.Then,the fractional dual Porous Medium equations are studied with Lie symmetry analysis according to the different coefficients,and we obtain some group invariant solutions and graphs of some special solutions to several kinds of equations.Finally,three kinds of time-space fractional Porous Medium equations are studied by Lie symmetry analysis,and the group invariant solutions are obtained.The main contents of this paper are as follows:First of all,the background,purpose,significance and current research situation of these studying equations and the methods used in this paper are described in the first chapter,and then the definitions and lemmas needed in the research process are listed in the second chapter.In the third chapter,we use the Lie symmetry analysis to study the time fractional Porous Medium equation and obtain the group invariant solutions and graphs of some special solutions.The power series solutions of the time fractional Porous Medium equation are derived by the Homogeneous Balanced method.And based on the obtained symmetry group,we investigate the conservation law of this equation.Then,in Chapter 4,the time fractional Dual Porous Medium equation is classified according to the coefficients and these equations are discussed using the Lie symmetry analysis.In addition,the group invariant solutions of several kinds of equations are constructed.Finally,in Chapter 5,we use the Lie symmetry analysis to study three kinds of space-time fractional Porous Medium equations,and construct the group invariant solutions of these equations.