A Numerical Study of Fibrous Tissue Formation of Type I Collagen under Mechanical Stimuli by Reaction - Diffusion Equations with Motion of Fibroblasts~!2008-03-31~!2008-05-29~!2008-06-12~!

We investigate the Brusselator reaction–diffusion equations with periodic boundary conditions. We consider the range of values of the parameters used by Kuramoto in his study of chaotic concentration waves. We determine numerically the bifurcation diagram of the long-time travelling and standing wave solutions using a highly accurate Fourier pseudo-spectral method. For moderate values of the bifurcation parameter, we have found a sequence of instabilities leading either to periodic and quasiperiodic standing waves, or to chaotic regimes. However, for large values of the control parameter, we have found only uniform time-periodic solutions or time-periodic travelling wave solutions. Our numerical study motivates a new asymptotic analysis of the Brusselator equations for large values of the control parameter and small diffusion coefficients. This analysis explains the numerical predictions. The chaotic regime is limited to moderate values of the control parameter and periodic solutions are the only solutions for large values of the control parameter. We identify the stabilizing mechanism as the relaxation oscillations which appear when the control parameter is large. Our asymptotic result on the stability of periodic solutions is then generalized to a class of two-variable reaction-diffusion equations.

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A linearized spectral collocation method for Riesz space fractional nonlinear reaction‐diffusion equations

Computational and Mathematical Methods ◽

10.1002/cmm4.1177 ◽

2021 ◽

Author(s):

Mustafa Almushaira

Keyword(s):

Collocation Method ◽

Reaction Diffusion ◽

Riesz Space ◽

Reaction Diffusion Equations ◽

Spectral Collocation Method ◽

Diffusion Equations ◽

Spectral Collocation ◽

Nonlinear Reaction

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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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