Partial Differential For Finding Conservation Laws

With the development of science, people have already realized that nonlin-ear phenomena exists in nature such as physics and chemistry. The notion of conservation laws plays a vital role in the study of nonlinear systems. In fact, conservation originates from symmetries. Symmetry and conservation also have a long experience which are a long developing process from separation to inte-gration. In particular, conservation laws play an important role in linearization and integrability of partial differential equations. It is useful in getting analysis solutions and an numerical solutions of partial differential equations. As far as we know, Nother theorem plays an important part in the constructing conserva-tion laws of differential equations. However, Noether theorem also has significant restriction in evolution equations. In order to solve this problem, there are some other methods, such as characteristic methods, direct construction method, ad-joint equation approach and so on. A lot of partial differential equations depend on a perturbed parameter. These equations are called approximate partial dif-ferential equations. According to the definition of conservation laws, Baikov and Ibragimov give the definition of approximate conservation laws.The specific arrangement of this paper is as follows:In chapter 1, we introduced the developing process of the conservation laws, the elementary knowledge of conservation laws and notations in the paper.In chapter 2, we introduce how to construct conservation laws and approx-imate conservation laws.In chapter 3, we construct new conservation laws for Nonlinear heat equa-tion, (2+1)Heat equation, Nonlinear wave equation, Black-Scholes equation, Maxwellian equation, Modelling equation of Maxwellian tails depending on a small parameter.At last, we summarized our results and raised a question for further study.