A Study On Exact Solutions And Lie Symmetries Of Differential Equation With Symbolic Computation

The nonlinear differential equation (NDE) based on physics is one of the important aspects in the contemporary study of nonlinear science. Exploring and developing new method to solve the NDE is one of the forefront topics in the studies of nonlinear physics. Now there are many methods for finding the exact solutions of NDE. In this paper, not only some methods are studied, especially Lie symmetry method, but also the related theory and algorithms are improved. With computer symbolic system Maple, some packages to implement the important algorithms are presented. The theory, algorithms and implementations are very instructive for constructing the exact solution of NDE.We present and extend some methods such as the tanh-function method, the Jacobi elliptic function method, the ansatz method and so on. Based on the above methods and the theory of mathematics mechanization proposed by famous mathematician Wu Wentsiin, some type of particular exact solutions to nonlinear equations are obtained, which are helpful in clarifying the movement of matter under the nonlinear interactivities and play an important role in scientifically explaining of the corresponding physical phenomenon.However these methods have some common shortcomings: dispersive, unsystematically. It is well known that Lie groups techniques brought diverse integration methods for solving special classes of differential equations under a common concept. Indeed, Lie's infinitesimal transformation method provides a widely applicable technique to find closed form solutions of ordinary differential equations (ODEs). Applied to partial differential equations (PDEs), Lie's method can lead to symmetries. Exploiting the symmetries of PDEs, new solutions can be derived.Nowadays, the concept of symmetry plays a key role in the study and development of mathematics and physics. But the application of Lie group method to concrete physical systems involves tedious computations. So it is necessary to design some symbolic packages for it.We discuss methods and algorithms used in the computation of Lie nonclassical symmetries as well as classical symmetries, and provide the symbolic packages GDS and NGDS, which are used to generate the determining system of differential equation's classical symmetry and nonclassical symmetry respectively. For GDS, some bugs are found in the package liesymm on Maple.Because the determining systems are a linear or nonlinear overdetermined PDEs, it is very hard to solve them completely. After introducing the concept of involutive division, we can complete them to involutive forms which include all integrable conditions and maybe solve them more easily.For the case of classical symmetry, the minimal involutive base algorithm and Janet base algorithm are analyzed and described, and the associated implementations which are named MinilB and Janet are presented and some examples are tested. With GDS and Janet, we study thepotential symmetry of a generalized Burgers equation, for which an infinite parameter potential symmetry and a new exact solution are obtained.For the case of nonclassical symmetry, an involutive characteristic set algorithm(ICS) which reduces a nonlinear algebraic partial differential equation system to passive involution is described and improved. This algorithm converts all the existed methods using the multiplier variable approach such as Ritt's algorithm based on Janet division and Wu's algorithm based on Thomas division. Recently some new involutive divisions and algorithms are proposed, which can significantly reduce the computational steps in Wu-Ritt's characteristic set method for nonlinear algebraic PDEs. Based on the algorithm ICS, a package ICS is designed in Maple for computing involutive characteristic set. By testing many examples, we analysis the dependency of the algorithm ICS for the various orderings and involutive divisions. Some experimental results are obtained which may show a hint for computing involutive characteristic set of arbitrary algebraic partial differential equation systems there...