Lie Symmetry Analysis And Exact Solutions Of Fractional Differential Equations

Fractional differential equations are generalizations of integer-order differential equations,which are studied by using fractional calculus theory.Fractional differential equations are recognized as basic equations not only in pure mathematics,but also in engineering,statistical mechanics and other fields because of their various applications.So finding its exact solutions become the main purpose of studying fract.ional differential equations.In order to study the exact solutions of a class of fractional differential equations,many literatures have proposed auxiliary equation method,homogeneous balance method and other methods,this paper focuses on the Lie symmetry method and the invariant subspace method.The structure of the article is as follows:In the first chapter:the background of the research on fractional differential equation and its solution and some preliminary knowledge related to this paper are briefly introduced.In t.he second chapter:the Lie symmetry of a class of fractional differential equations is obtained by using the classical Lie symmetry method.By means of similarity transformation and similarity variables,the equation is reduced to a nonlinear ordinary differential equation with Erdelyi-Kober fractional operator.Furthermore,on the basis of Lie algebra,a new conservation theorem with regular Lagrangian function is used to discuss the conservation laws and the detailed derivation is given.In the third chapter:taking Whitham-Broer-Kaup equation as an example,the fractional differential equation is constructed by using variational method and semi inverse method based on the variational principle.The Whitham-Broer-Kaup equation is constructed as a mixed fractional Whitham-Broer-Kaup equation,and Lie symmetry is applied to the constructed equationIn the fourth chapter:the invariant subspace method is mainly introduced,the invariant subspace method is extended to time fractional coupled nonlinear partial differential equations,the given discrete fractional differential equations are classified,the sufficient conditions for the existence of invariant subspaces allowed by the system of equations are given,and the exact solutions under different conditions are also discussed.In the fifth chapter:by summarizing this paper,the direction that we should explore in the future is put forward.