Conservation Laws And Its Application Of The Partial Differential Equation

Conservation regards the nature moves in cycles and circulation. The study onconservation laws is one of the important problems in the field of mathematicalphysics. How to construct conservation laws has been a highlight of the study. In thereal world, many physical phenomena can be described by partial differentialequations, so it is important to construct the conservation laws of partial differentialequations. Up to now, there have many methods to construct conservation laws andscholars have tried to find out easier methods to construct conservation laws.Ibragimov introduced a method to acquire the conservation laws by the symmetriesand adjoint equations in2007. Then he provided a method of conservation laws tosolve partial differential equations in2010.This paper is organized as follows:In chapter1, we introduce the development and significance of conservationlaws, and show the way to construct conservation laws by using the adjoint equationsas well as how to solve partial differential equations by using conservation laws.In chapter2, we discuss the Lie point symmetries and conservation laws of theChen-Lee-Liu equation.In chapter3, we obtain the Lie point symmetries, Lie-B cklund symmetries andconservation laws of the double Sine-Gordon equation. We find that the conservationlaws provided by the Lie point symmetries is a special case of that provided by theLie-B cklund symmetries.From this viewpoint, the Lie-B cklund symmetries providemore conservation laws.In chapter4, we use the method of conservation laws to solve the nonconservati-ve Fokker-Planck equation. The new exact solutions we obtained that expand itsgroup invariant solutions.The result which we obtained can give more useful information to the correspon-ding equations. In the end, we sum up the conclusions and the study in the future.