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Numerical solutions for some coupled systems of nonlinear boundary value problems. (English) Zbl 0632.65111

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.
Reviewer: M.Nagel

MSC:

65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
35K57 Reaction-diffusion equations
65H10 Numerical computation of solutions to systems of equations
92Cxx Physiological, cellular and medical topics
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References:

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