Finite difference reaction diffusion equations with nonlinear boundary conditions

Abstract

This article is concerned with numerical solutions of finite difference systems of reaction diffusion equations with nonlinear internal and boundary reaction functions. The nonlinear reaction functions are of general form and the finite difference systems are for both time-dependent and steady-state problems. For each problem a unified system of nonlinear equations is treated by the method of upper and lower solutions and its associated monotone iterations. This method leads to a monotone iterative scheme for the computation of numerical solutions as well as an existencecomparison theorem for the corresponding finite difference system. Special attention is given to the dynamical property of the time-dependent solution in relation to the steady-state solutions. Application is given to a heat-conduction problem where a nonlinear radiation boundary condition obeying the Boltzmann law of cooling is considered. This application demonstrates a bifurcation property of two steady-state solutions, and determines the dynamic behavior of the time-dependent solution. Numerical results for the heat-conduction problem, including a test problem with known analytical solution, are presented to illustrate the various theoretical conclusions. 0

Numerical methods for fourth‐order nonlinear elliptic boundary value problems

“…In particular, B[w i ] = w i if α = 0, β = 1 (see [16][17][18] for some detailed derivations). To treat problem (2.2) by the monotone method, we find it more convenient to express it in vector form.…”

“…The components of G (j) are mostly zero except at those mesh points where they are related to the boundary points of ∂Ω n . Because our main concern is the structure of the finite difference system, detailed derivation of (2.4) is not given here (see [17,18] for some discussions).…”

Accelerated monotone iterative methods for finite difference equations of reaction-diffusion

“…The function g * (u i,n ) is associated with the boundary function g(u i,n ) and is zero when x i is an interior mesh point of Ω (see [13,15] for a more detailed derivation). The above approximation and the connectness assumption of Λ p ensures that A n possesses the following property which is our basic hypothesis for the system (2.1):…”

“…[3,4,[6][7][8][9][12][13][14][15][16][17]). This method and its associated monotone iteration lead not only to the existence and uniqueness of a solution but the process of iterations gives also a computational algorithm for numerical solutions of both parabolic and elliptic boundary-value problems.…”

“…To increase the rate of convergence while maintaining the monotone property of the iterations we propose some accelerated iterative schemes which have often been used for the treatment of elliptic boundary-value problems (cf. [4,8,9,11,15]). An advantage of this method is that it leads to a sequence which converges either quadratically or nearly quadratically with only the usual differentiability requirement on the functions f (·, u) and g (·, u).…”