Lie Symmetry Analysis Of Several Fractional Partial Differential Equations

In recent years,with the developing of the nonlinear models,people begin to pay attention to the study of fractional differential equation while studying the integer differential equation.Fractional-order nonlinear models are mainly originated from two aspects.One is physics,signal processing,biology and other practical application fields,on the other hand is to manually transform the original integer equation into fractional order.As one of the important methods to study fractional partial differential equations,Lie symmetry group has made great progress in theory and application.In this paper,the symmetric classification and exact solution construction of three kinds of fractional partial differential equations are studied by using Lie symmetry method.The paper is arranged as follows:The first chapter introduces the development background and the current research progress of fractional differentiation and Lie symmetry.Then,providing the definitions of fractional derivative and Lie symmetry that will be used in this paper.In the second chapter,firstly,the Lie symmetry of the biological population model is analyzed and seven sets of finite dimensional infinitesimal generators are obtained.Moreover,the model is reduced to(1+1)-dimensional partial differential equation and three kinds of exact solutions of biological population model are obtained by F-expansion method.In the end,a power series solution of the model is constructed by power series method.In the third chapter,the symmetry group of a class of fractional order generalized diffusion equation is analyzed.In the process of analysis,the equation is divided into two cases according to the parameters of the equation,and the symmetry classification of the two cases is obtained.Then,the characteristic equation method is used to reduce the equation to ordinary differential equation.Finally,the power series solution of the equation is obtained by using the method of traveling wave reduction and power series.In the fourth chapter,at first,we introduce fractional sub-equation method.Then,a class of fractional order generalized Kd V equation is studied,and the exact solutions of eight kinds of hyperbolic and trigonometric functions are obtained.Moreover,we show the graphs of these exact solutions.