Numerical methods for nonlinear integro-parabolic equations of fredholm type

This paper is concerned with iterative methods for numerical solutions of a class of nonlocal reaction-diffusion-convection equations under either linear or nonlinear boundary conditions. The discrete approximation of the problem is based on the finite-difference method, and the computation of the finite-difference solution is by the method of upper and lower solutions. Three types of quasi-monotone reaction functions are considered and for each type, a monotone iterative scheme is obtained. Each of these iterative schemes yields two sequences which converge monotonically from above and below, respectively, to a unique solution of the finite-difference system. This monotone convergence leads to an existence-uniqueness theorem as well as a computational algorithm for the computation of the solution. An error estimate between the computed approximations and the true finite-difference solution is obtained for each iterative scheme. These error estimates are given in terms of the strength of the reaction function and the effect of diffusion-convection, and are independent of the true solution. Applications are given to three model problems to illustrate some basic techniques for the construction of upper and lower solutions and the implementation of the computational algorithm.