The Symmetry Group Analysis Of Two Types Nonlinear Partial Differential Equations

Many problems of natural and social(such as physics,biology and economics)ultimately comes down to a nonlinear partial differential equation,So for solving a nonlinear partial differential equation and the research has become the important research topicsa of a scholar in every field,especially the nonlinear partial differential equation with variable coefficients which contains so many practical factors that it has the practical significance of the more general,and more attention.Here we consider two classes nonlinear partial differential equation with great significance in physics and financial,respectively study the symmetry group classification,the classical reduced and the exact solution of them.The first is a generalized nonlinear beam equation with both second-order and fourth-order wave terms,which is extended from the classical beam equation arising in the historical events of travelling wave behavior in the Golden Gate Bridge in San Francisco.We perform a complete Lie symmetry group classification by using classical Lie symmetry method the equivalence transformation group theory for the equation under consideration.Lie symmetry reductions of two nonlinear beam-like equations which are singled out from the classification results are investigated.Some classes of exact solutions,including solitary wave solutions,triangular periodic wave solutions and rational solutions of one class of nonlinear beam-like equations are constructed by means of the reductions and symbolic Computation.The second is the generalization of a transaction-cost model which describe the pricing of derivative securities for illiquid markets.We perform symmetry classification for for the equation under consideration.The optimal systems of one-dimensional subalgebras are constructed for two families five-dimensional symmetry algebras which are admitted by the classification models.Reduced equations through these subalgebras are carried out.The research results has a certain theoretical and practical significance for structure equation more accurate solution,study the law of conservation and test the numerical solution.