A Kind Of Variable Coefficient Bernoulli Auxiliary Equation Method And Exact Solutions

The solution of the nonlinear evolution equation,especially the exact solution,is an important subject in modern mathematics and physics.In recent years,function expansion method which based on homogeneous balance principle and computer algebra,developed by using the known the accurate solution of ordinary differential equation,is a effective method to seek exact solutions of nonlinear evolution equation.By using this method we can easily get solitary wave solutions and periodic solutionsFirstly,in this paper,on the basis of Riccati equation methods we propose a new expansion method with variable coefficient auxiliary equation,which is called variable coefficient Bernoulli equation method,int this method the auxiliary equation Bernoulli equation is a variable coefficient like thisφ(ξ)=p(ξ)φ(ξ)+φ~2(ξ),we put this method in the Burgers equation and Drinfel-Sokolov equations,and obtain rich exact solutions.Secondly,when solving nonlinear partial differential equation,we generally use wave transform convert the partial differential equation into ordinary differential equation,so many different partial differential equations can be converted into the same type of ordinary differential equations.So in the third chapter,we convert MKdV equations,Klein-Gordon equation and Schr(o|¨)dinger equation into the elliptic-like sub-ODE equation,and use projective Riccati equation method to obtained its rich exact solutions.There are many partial differential equations,which can be converted by wavelet transform into ellipse-like sub-ODE equation,then obtain their exact solutions,and this method is very effective and convenient.