Numerical algorithms for a sideways parabolic problem with variable coefficients

We investigate in this paper an inverse problem (IP) of reconstructing inaccessible boundary values for parabolic equation with variable coefficients. The implicit finite difference (FD) for the IP is firstly introduced, which indicates different choices of the mesh-ratio compared with the same FD scheme for the direct problem. By means of the discrete Fourier transform, the FD scheme has a regularizing effect which prevents the solution from blowing up. Apart from the FD, a novel forward collocation (FC) method is formulated, which is based on the formulation of the IP into a sequence of well-posed direct problems and an ill-posed system of algebraic equations. The continuous dependence of a quasi-solution for the unknown boundary profile is firstly proven from which the existence of the quasi-solution in a compact set of admissible boundary profile is deduced. The corresponding dual problem is introduced with given proofs on the existence and error estimate of its quasi-solution. For numerical illustration, we apply the implicit FD and FC with Chebyshev nodes to solve the IP and its corresponding quasi-solution problem, respectively, in one dimension in which the inaccessible boundary values are reconstructed from the given right boundary values. Convergence rates for both numerical methods are derived and the numerical results validate the feasibility and effectiveness of the proposed numerical algorithms.