The Application Of Lie Symmetry In Boundary Problems Of Some Nonlinear Partial Differential Equations

Many problems in natural science and engineering are essentially differential equations,and the principal part of the differential equation of research is the partial differential equation(s)(PDEs),especially the nonlinear partial differential equation(NLPDEs),so the research of solving NLPDEs has important significance.Due to the complexity of the nonlinear equations,the solving of them is difficult.In order to solve the PDEs,many methods have been proposed.However,there is no unified and systematic method,which is applicable to all kinds of problems and these methods have their respective applicable scope.Therefore,the method of solving PDEs is still the basic research in mathematics,physics and mechanics discipline.In particular,it is necessary to improve the existing methods,summarize,deepen the understanding,accept the advantages and discard the defects,which is the prerequisite for the discovery of new methods.Lie symmetry method is the most universal method,with many traditional methods as its special cases.The study of PDEs symmetry theory and method has important theoretical and practical significance in modern mathematics,physics and mechanics,and it has been used widely.In this paper,we will study the application of Lie symmetry method and symmetric classification method in NLPDEs boundary value problems based on differential characteristic set algorithm.Concrete contents are as follows:The first chapter,the development of symmetric method and its importance in the research of PDEs are reviewed.In addition,the differential characteristic set algorithm,Runge Kutta method and homotopy perturbation method are introduced.The second chapter,combining the symmetry method and numerical method(Runge Kutta method)to get the numerical solution of hydrodynamics NLPDEs boundary value problem.The third chapter,the application of symmetric classification in NLPDEs boundary value problem is studied,and the symmetric classifications of two hydrodynamics NLPDEs boundary value problem are computed and solved.Steps are as follows:(1)Based on the differential characteristic set algorithm,the symmetry classification of the NLPDEs boundary value problem with parameters is determined.According to the parameter values of the equation,the main symmetry and extended symmetry can be determined.(2)The NLPDEs boundary value problem is reduced to ODEs initial value problem by using the defined extended symmetry.(3)The numerical solution of ODEs initial value problem is obtained by means of Mathematical symbol system.The fourth chapter,the two NLPDEs boundary value problems are solved by combining the symmetry method and approximate analytic solution method(such as: the homotopy perturbation method).At first,NLPDE boundary value problem is reduced to ODEs initial value problem using the symmetric method.Then,the approximate solutions can be obtained by using the homotopy perturbation method.Finally,the numerical solutions can be obtained by using the numerical methods.By comparing the numerical solutions with the approximate solutions,we verified that the approximate solutions converge to the numerical solutions.Finally,we summarize all the contents of this paper,and look forward to the relevant research on the next step.