A spline collocation approach for a generalized parabolic problem subject to non-classical conditions

A numerical method based on finite differences and spline collocation is proposed for solving a generalized parabolic problem subject to non-classical conditions. It is shown numerically that this scheme has second-order convergence. The stability conditions relating the time step size to mesh sizes were examined through numerical examples. It turns out that the stability of the scheme in certain cases is subjected to a stability condition and depends on the nature of the problem. A number of specific test problems are solved to assess the efficiency and performance of the method. The numerical results obtained indicate that the approach is viable and yields accurate solutions when compared with the existing closed-form solutions and generates results consistent with recent numerical approaches.