investigation and solving of initial-boundary value problem including fourth order pde by contour integral and asymptotic methods

In this paper, we consider a fourth order mixed partial differential equation with some initial and boundary conditions which is unsolvable by classical methods such as fourier, fourier-bircove and laplace transformation methods. for this problem we will apply the contour integral and asymptotic methods. the convergence of the appeared integrals, existence and uniqueness of solution, satisfying the solution and holding the given initial  and boundary conditions are stablished by complex analysis theory and related contour integrals. finally, the form of analytic  and approximate solutions are given due to different cases of eigenvalues distributions in the  complex plane.