In this paper, the authors investigate an initial-boundary value problem for a partial differential equation with a complex constant coefficient and non-local-boundary conditions whose related spectral problem is a non-self-adjoint-boundary value problem. For this problem, we first derive the related spectral problem. Then the formal and analytical solutions were formulated by using Eigen functions of the related spectral problem. Finally, the existence, uniqueness and convergence of the series solution will be proved.