The Investigation And Application Of Several Solutions For Nonlinear Partial Differential Equations

In this paper, we introduce four kinds of effective methods to solve nonlinear partial differential equation(NLPDE), which including Lie symmetry, the extended simplest equation method, the extended tanh method and the homotopy perturbation method, at the same time,we put our focus on the research of combining Lie symmetry with the other three kinds of constructive methods to cope with solving NLPDE.Firstly, with the help of the computer algebraic system Mathematica, we successfully obtain several kinds of symmetry of NLPDE based on Wu-differential characteristic set algorithm; Secondly, we achieve all kinds of reduction to these NLPDE with the aid of symmetry;Finally, we respectively choose a sort of constant coefficient equation from the reduced equations in each groups and obtain a series of the exact solutions and the approximate analytic solutions based on the above-mentioned three kinds of constructive methods to amplify the result of the past document. According to examples, we have proved the availability of the three kinds of constructive methods and explained the feasibility of combining Lie symmetry with other methods to solve NLPDE.In the first chapter, we introduce the research and development status of NLPDE, many research methods of solving NLPDE and put forward the development status of Lie symmetry, the methods of determining symmetry(Wu-differential characteristic set algorithm),which make the new way of developing solving method to NLPDE become more theoretical and practical.In the second chapter, We investigate the whole idea of constructing the exact solutions to NLPDE based on the extended simplest equation and present some applied examples, thus we obtain three new kinds of the exact traveling wave solutions. According to the analysis of those results, we can prove that this approach is quite direct, effective and stable, it is also suitable for high-dimensional NLPDE, and can be applied to more study of nonlinear problem in mathematical physics.The third chapter mainly investigates the new applications of symmetry methods to solve NLPDE, that is based on symmetry, combining with the extended simplest equation method and the extended tanh method, respectively, and then solving several kinds of NLPDE. According to these two kinds of combinations, we both obtain three different kinds of solutions, which are expressed by the hyperbolic functions, the trigonometric functions and the rational functions, respectively, where the solutions of applications of combining Lie symmetry method with the extended tanh method contain 27 different kinds of cases. According to the analysis of results, not only does it reflect the idea of combining Lie symmetry with constructive methods is available and feasible, but also has become a new research of solving NLPDE.In the fourth chapter, based on the idea of combining Lie symmetry with constructive methods, further, we combine it with the homotopy perturbation method, then we obtain several sorts of approximate analytic solutions of NLPDE and prove the accuracy and availability of the homotopy perturbation method by examples, further.The fifth chapter summarizes all the content of this paper, and puts forward some new advices to the investigation of methods to solve NLPDE.