Symmetry Reductions And Exact Solutions Of Partial Differential Equations

In this paper,Lie symmetry group method is applied to a fourth-order partial differential equation and the new coupled Konno-Oono equations.The similarity reductions and group-invariant solutions of the equations are obtained.Furthermore,Riccati equation expansion method of two nonlinear integer order partial differential equations are also performed.Exact solutions and their image orientation are given.Finally,consistent Riccati expansion method for three nonlinear fractional partial differential equations and new exact solutions are performed.The main results are as follows:1.Classical Lie group analysis of a fourth-order partial differential equation is performed.Its Lie symmetries are derived.The basis similarity reductions for the equation are performed.Reduced equations and invariant solutions associ-atde to the symmetries are obtained.Finally,based on the power series method,a kind of explicit power series solutions for the reduced equation is well con-structed with a detailed derivation,accordingly the exact solutions associated to the fourth-order partial differential equation are also provided.2.Lie symmetry group method is applied to study the new coupled Konno-Oono equations Its Lie symmetry groups are determined.Firstly,3-dimensional subalgebra of the infinite Lie algebra is found and the one dimension optimal system of the Lie symmetries admitted by the equation in consideration is constructed.In ad-dition,the reduced equations or all exact solutions corresponding to the optimal system are presented.Finally,based on our results of the new conservation the-orem,we also get conservation laws of the new coupled Konno-Oono equations.3.Using the Riccati equation expansion method,we study the ill-posed Boussinesq equation and the fourth-order Burgers equation The new exact solutions of two classes of equations are obtained.These solutions contain triangular periodic solutions,hyperbolic function solutions,rational so-lutions,and so on.4.A consistent Riccati expansion method is proposed for solving nonlinear fractional partial differential equations involving Jumarie's modified Riemann-Liouville derivative with the help of a Riccati equation.This approach is suc-cessfully applied to fractional KdV equation fractional Sharma-Tasso-Olvever equation fractional generalized Hirota-Satsuma coupled KdV equation As a result,a variety of exact solutions to these equations under study are obtained.And the validity of this approach is demonstrated.