Some Applications Of Laplace's Cascade Method

Laplace's cascade method is an effective way to construct general solutions of Euler-Poisson-Darboux(EPD)equations.In this paper,Laplace's cascade method is applied to construct a class of third-order EPD equations(?)and Tricomi equations(?)where ?,?,p,m are constants.By reduction of order and transformations of variables,these two class of equations can be transformed to second-order EPD equations,respectively.The corresponding Laplace invariants are given to constructed their solutions.Some examples of solving initial value problem of them are presented.