Monotone iterative methods for numerical solutions of nonlinear integro-elliptic boundary problems

The aim of this paper is to obtain various monotone iterative schemes for numerical solutions of a class of nonlinear nonlocal reaction-diffusion-convection equations under linear boundary conditions. The boundary-value problem under consideration is discretized into a system of nonlinear algebraic equations by the finite difference method, and the iterative schemes are given for the finite difference system using upper and lower solutions as the initial iterations. The construction of the monotone sequences and the definition of upper and lower solutions depend on the quasimonotone property of the reaction function, and the iterative schemes are presented for each of the three types of quasimonotone functions. An application of the monotone iterations, including some numerical results, is given to a modified logistic diffusive equation where the kernel in the integral term may be positive, negative, or changing sign in its domain.