Spectral problem with spectral parameter in the boundary condition in the theory of the radial heat equation (Q946104)

At application of the Fourier method of separation of variables to the radial heat equation for the cylinder \(u_t=a^2(u_{rr}+\frac{1}{r}u_r)\) with initial condition \(u(r,0)=g(r),0\leq r\leq R,\) and the boundary condition \(-ku_r| _{r-R}=cu_t| _{r=R}\) the authors get the equation \[ U''(r)+\tfrac{1}{r}U'(r)+\lambda U(r)=0 \tag{1} \] with spectral parameter in boundary condition \[ U'(r)| _{r=R}=d\lambda U(r)| _{r=R},\; d=\frac{Ca^2}{k}.\tag{2} \] The authors determine the system of eigenfunctions for (1), (2) which is a Riesz basis in the space \(L_2(0,R)\)