Research On Symmetry And Conservation Laws Of Two Types Of Fractional Order Partial Differential Equations (groups

The fractional derivative is a generalization of the integer order case,which has the characteristics of memory dependence and description of abnormal behavior,and has received extensive attention in the scientific community.Symmetry group theory,as one of the important methods to study integer order partial differential equations,plays an increasingly important role in the problems related to fractional order partial differential equations.This paper mainly studies the symmetry and conservation law of time-fractional partial differential equations.The content mainly includes the following two parts:In the first part,the Lie symmetry and conservation law of the(2+1)dimensional time-fractional Hirota-Satsuma-Ito equations in the sense of Riemann-Liouville are studied.Firstly,a new formula of infinitesimal generator in the case of fractional and integer mixed derivatives is given,and the Lie symmetry allowed by the time-fractional Hirota-Satsuma-Ito equations is obtained.Secondly,the original(2+1)dimensional partial differential equations are reduced to a(1+1)dimensional partial differential equations by using an one-dimensional optimal Lie subalgebra system.In particular,an explicit fractional power series solution is constructed,and Mathematica software is used to draw the image of the truncated power series solution to describe the dynamic behavior of the solution.Finally,using the nonlinear self-adjoint idea in the fractional case,the conservation law formula of the equation is given,and several new nontrivial conservation laws are obtained.In the second part,the approximate symmetry method of integer order is extended to the fractional partial differential equations with a small parameter in the sense of Riemann-Liouville derivative.The approximate symmetry method of fractional partial differential equations is proposed,and the time-fractional Korteweg-de Vries equation with small parameters is taken as an example to illustrate the effectiveness of the proposed approximate symmetry method.In addition,we reduce the coupled system corresponding to the approximate symmetry to an ordinary differential equations,and construct an explicit power series solution of the equation.Finally,the dynamical behavior of the solution is explained by the image of the truncated power series solution.