Related Studies On Some Differential Equations And Their Solutions

In this paper,we study the properties of some differential equations and their solutions in four aspects: the symmetry property,conservation law,exact solution and differential inequality.The paper is mainly divided into three parts to elaborate: in the first part,we are intend to explore the symmetry properties and conservation laws of the time fractional Sawada-Kotera differential equation.First of all,a parameter transformation group acts on Sawada-Kotera differential equation,we obtain symmetries,similar variable and similar transformation.making use of similar variable and similar transformation,the time fractional Sawada-Kotera partial differential equation is reduced to a fractional ordinary differential equation.Furtherly,we make Sawada-Kotera equation transformed into the form of the components of the conservation vector.Based on the Ibragimov theorem,the conservation laws of the equation are obtained.In the second part,we mainly study the exact solutions of Sawada-Kotera equation and generalized symmetric regularized long wave equation by a new method.According to the definition of Jumarie's modified Riemann-Liouville derivative and its correlation property,we take advantage of traveling wave solution,the fractional equations are reduced to an integer order ordinary differential equation respectly.Making balance between the nonlinear term and derivative term of the highest order,we get the relationship between N and M.Assuming the values of N,M,the expression of exact solution is determined.On the basis of the transformations of Q?,thus obtain some new solutions: the forms of the trigonometric function and the Weierstrass elliptic function.Since the differential inequality is always an effective method to study the properties of the differential equations and their solutions,in the last part of this paper,we mainly make an exploration on a differential inequality and its application in the study of Camassa-Holm shallow water equation.Finally,considering the initial condition for the blow-up of the solution,the blow-up rat and the blow-up time of the solution are obtained.