Numerical solutions for some coupled systems of nonlinear boundary value problems (Q1095620)

The aim of the paper is to discuss the following coupled system of nonlinear boundary value problems: \(-\nabla^ 2u=f^{(1)}(x,u,v),\quad -\nabla^ 2v=f^{(2)}(x,u,v),\quad \quad (x\in \Omega),\) \(B^{(1)}[u]=g^{(1)}(x)\), \(B^{(2)}[v]=g^{(2)}(x)\), (x\(\in \Omega)\), where \(f^{(1)}\) and \(f^{(2)}\) are any two continuous functions on \(\Omega \times {\mathbb{R}}^+\times {\mathbb{R}}^+\). The system is motivated by applications in physical and biochemical reaction-diffusion systems, especially the Volterra-Lotka model. The problem can be replaced by a finite difference system. An iterative scheme for the finite difference equations is presented and the convergence of the iterations is shown. The author gives an application to the Volterra-Lotka model. Some numerical results for the boundary value problem with a different type of boundary conditions are given.