Bifurcation, stability diagrams, and varying diffusion coefficients in reaction-diffusion equations

1982 ◽  
Vol 44 (1) ◽  
pp. 87-102 ◽  
Author(s):  
K. J. Brown ◽  
J. C. Eilbeck
1997 ◽  
Vol 1 (2) ◽  
pp. 91-102 ◽  
Author(s):  
Arun V. Holden

The propagation of electrical activity in cardiac tissue can be modelled by reaction diffusion equations, where a tensor of diffusion coefficients represents anisotropy due to fiber orintation, and excitation is represented by high-order, stiff differential systems. The effects of external electrical stimulation, as in artifical pacemakers, or in defibrillators, requires bidomain models, in wich intra- and extracellular currents are treated separately.simplified approaches are taken to this problem to illustrate two methods of defibrillation: by a sinhle large pulse, that eliminates all propagating activity, and by a series of smaller amplitude perturbations, that drive out re-entrant sources of excitation.


2019 ◽  
Vol 29 (11) ◽  
pp. 4093-4129 ◽  
Author(s):  
J.I. Ramos

Purpose The purpose of this paper is to develop a new transversal method of lines for one-dimensional reaction–diffusion equations that is conservative and provides piecewise–analytical solutions in space, analyze its truncation errors and linear stability, compare it with other finite-difference discretizations and assess the effects of the nonlinear diffusion coefficients, reaction rate terms and initial conditions on wave propagation and merging. Design/methodology/approach A conservative, transversal method of lines based on the discretization of time and piecewise analytical integration of the resulting two-point boundary-value problems subject to the continuity of the dependent variables and their fluxes at the control-volume boundaries, is presented. The method provides three-point finite difference expressions for the nodal values and continuous solutions in space, and its accuracy has been determined first analytically and then assessed in numerical experiments of reaction-diffusion problems, which exhibit interior and/or boundary layers. Findings The transversal method of lines presented here results in three-point finite difference equations for the nodal values, treats the diffusion terms implicitly and is unconditionally stable if the reaction terms are treated implicitly. The method is very accurate for problems with the interior and/or boundary layers. For a system of two nonlinearly-coupled, one-dimensional reaction–diffusion equations, the formation, propagation and merging of reactive fronts have been found to be strong function of the diffusion coefficients and reaction rates. For asymmetric ignition, it has been found that, after front merging, the temperature and concentration profiles are almost independent of the ignition conditions. Originality/value A new, conservative, transversal method of lines that treats the diffusion terms implicitly and provides piecewise exponential solutions in space without the need for interpolation is presented and applied to someone.


2020 ◽  
Vol 18 (1) ◽  
pp. 1552-1564
Author(s):  
Huimin Tian ◽  
Lingling Zhang

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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