The Solition Solutions And Invariance Solutions And Boundary Value Probolems' Solutions Of Partial Differential Equations

First tanh-function method is extended then used to solve BBM equation. we also used deformation mapping method to obtain solutions of BBM equation. With both methods we can obtain abundant explicit and exact traving wave solutions. Which coation Soliton solutions, Plural line soliton solutions, periodic wave solutions, Jacobi elliptic fuction solutions,Weierstrass elliptic function solutions and other exact solutions.Second we apply infinitesimal transformations to the construction of solutions of partial differential equations. As for ODE's we will show that the infinitesimal criterion for invariance of PDE's leads directly to an algorithm to determine infinitesimal generators X admitted by given PDE's . Invariant surfaces of the corresponding Lie group of point transformations lead to similarity solutions. These solutions are obtained by solving PDE's with fewer independent variables than the given PDE's. Now we obtain the set of determining equations is an overdermined system of PDE's which is composed of the arbitrary element of axisymmetric wave equation and the coefficient of infinitesimal generators, that derived by classical infinitesimal Lie method. Second we give some infinitesimal generators of axisymmetric wave equation with the help of symbols computer sorftware, after we find out the PDE'S one-parameter Lie group of transformations by first fundamental theorem of Lie. Last take the infinitesimal generators that we find out into invariant surface condition then we can get group invariant solutions of axisymmetric wave equation by use invariant form method or direct substitution.Last we discuss how one can use infinitesimal transformations to solve boundary value problems for PDE's .If a one-parameter Lie group of transformation admitted by a PDE leaves the domain and boundary conditions of a BVP invariant , then the solution of the BVP is an invariant solution, and hence the given BVP is reduced to a BVP with one less independent variable .we also consider the invariant of BVP's under multi-parameter Lie groups of transformations. We now apply the given method to solve the boundary value probolems'solutions of Green function.