Energy-Norm A Posteriori Error Estimates for Singularly Perturbed Reaction-Diffusion Problems on Anisotropic Meshes: Neumann Boundary Conditions

This paper developed the anti-derivative wavelet bases to handle the more general types of boundary conditions: Dirichlet, mixed and Neumann boundary conditions. The boundary value problem can be formulated by the variational approach, resulting in a system involving unknown wavelet coefficients. The wavelet bases are constructed to solve the unknown solutions corresponding to the types of solution spaces. The augmentation method is presented to reduce the dimension of the original system, while the convergence rate is in the same order as the multiresolution method. Some numerical examples have been shown to confirm the rate of convergence. The examples of the singularly perturbed problem with Neumann boundary conditions are also demonstrated, including highly oscillating cases.

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Periodic Solutions with a Boundary Layer of Reaction–Diffusion Equations with Singularly Perturbed Neumann Boundary Conditions

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127414400197 ◽

2014 ◽

Vol 24 (08) ◽

pp. 1440019 ◽

Cited By ~ 7

Author(s):

Valentin F. Butuzov ◽

Nikolay N. Nefedov ◽

Lutz Recke ◽

Klaus R. Schneider

Keyword(s):

Boundary Layer ◽

Boundary Conditions ◽

Boundary Layers ◽

Neumann Boundary ◽

Reaction Diffusion ◽

Neumann Boundary Conditions ◽

Reaction Diffusion Equations ◽

Singularly Perturbed ◽

Diffusion Equations ◽

Time Periodic

We consider singularly perturbed reaction–diffusion equations with singularly perturbed Neumann boundary conditions. We establish the existence of a time-periodic solution u(x, t, ε) with boundary layers and derive conditions for their asymptotic stability. The boundary layer part of u(x, t, ε) is of order one, which distinguishes our case from the case of regularly perturbed Neumann boundary conditions, where the boundary layer is of order ε. Another peculiarity of our problem is that — in contrast to the case of Dirichlet boundary conditions — it may have several asymptotically stable time-periodic solutions, where these solutions differ only in the description of the boundary layers. Our approach is based on the construction of sufficiently precise lower and upper solutions.

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Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions

Open Mathematics ◽

10.1515/math-2020-0088 ◽

2020 ◽

Vol 18 (1) ◽

pp. 1552-1564

Author(s):

Huimin Tian ◽

Lingling Zhang

Keyword(s):

Boundary Conditions ◽

Blow Up ◽

Sufficient Conditions ◽

Neumann Boundary ◽

Reaction Diffusion ◽

Neumann Boundary Conditions ◽

Reaction Diffusion Equations ◽

Time Dependent ◽

Diffusion Equations ◽

Nonlocal Reaction

Abstract In this paper, the blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients are investigated under Neumann boundary conditions. By constructing some suitable auxiliary functions and using differential inequality techniques, we show some sufficient conditions to ensure that the solution u ( x , t ) u(x,t) blows up at a finite time under appropriate measure sense. Furthermore, an upper and a lower bound on blow-up time are derived under some appropriate assumptions. At last, two examples are presented to illustrate the application of our main results.

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