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Amplitude equation for the stochastic reaction‐diffusion equations with random Neumann boundary conditions
Author(s) -
Mohammed Wael W.
Publication year - 2015
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.3402
Subject(s) - mathematics , neumann boundary condition , mathematical analysis , stochastic partial differential equation , heat equation , partial differential equation , boundary value problem , stochastic differential equation , degenerate energy levels , amplitude , nonlinear system , boundary (topology) , physics , quantum mechanics
In this paper, we consider a quite general class of reaction‐diffusion equations with cubic nonlinearities and with random Neumann boundary conditions. We derive rigorously amplitude equations, using the natural separation of time‐scales near a change of stability and investigate whether additive degenerate noise and random boundary conditions can lead to stabilization of the solution of the stochastic partial differential equation or not. The nonlinear heat equation (Ginzburg–Landau equation) is used to illustrate our result. Copyright © 2015 John Wiley & Sons, Ltd.