Investigation of solvability conditions of certain boundary value problem by perturbation method

In this paper we study the solvability conditions under which certain partial differential equation with homogenous boundary conditions containing small parameter has a solution by using perturbation technique. Here we deal with the following partial differential equation 0 2 = + + φ φ φ w yy xx with boundary conditions 0 ) , ( ) , ( = − b x a x y y φ φ and Investigation of solvability conditions of certain boundary value problem ... ٥٥ 0 ) , ( ) , ( sin sin = − + = + = x w k b y x w k a y y x y x ε ε φ φ where w kw , are constants and ε is small parameter. According to this method, the solution of the problem is represented by the first few terms of a perturbation expansion.