Eigenfunction expansions for second-order boundary value problems with separated boundary conditions

چکیده مقاله

‎In this paper‎, ‎we investigate some properties of eigenvalues and eigenfunctions of boundary value problems‎ ‎with separated boundary conditions‎. ‎Also‎, ‎we obtain formal series solutions for some partial differential equations associated with the second order differential equation‎, ‎and study necessary and sufficient conditions for the negative and positive eigenvalues of the boundary value problem‎. ‎Finally‎, ‎by the sequence of orthogonal eigenfunctions‎, ‎we provide the eigenfunction expansions for twice continuously differentiable functions‎