The spectral properties of boundary value problems with eigenvalue parameter in the boundary conditions

We study three types of boundary value problems with eigenvalue parameter in the boundary conditions, called (*), (**), and (***). We develop the self-adjoint operator and the Green's function associated with each of (*), (**), and (***). We also demonstrate the convergence of the eigenfunction expansion graphically and numerically for (*). Based on these results, we then solve a nonhomogeneous heat flow problem associated with (*).