Numerical solutions of reaction–diffusion equations with nonlocal boundary conditions

The purpose of this paper is to present some iterative methods for numerical solutions of a class of nonlinear reaction–diffusion equations with nonlocal boundary conditions. Using the finite-difference method and the method of upper and lower solutions we present some monotone iterative schemes for both the time-dependent and the steady-state finite-difference systems. Each monotone iterative scheme gives a computational algorithm for numerical solutions and an existence-comparison theorem for the corresponding finite-difference system. The existence-comparison theorems are used to investigate the asymptotic behavior of the discrete time-dependent solution in relation to the discrete maximal and minimal solutions of the steady-state problem. Numerical results are given to a model problem where the solution of the continuous problem is explicitly known and its values at the mesh points are used to compare with the numerical solutions obtained by the monotone iterative schemes.