scholarly journals Operator Splitting for an Immunology Model Using Reaction-Diffusion Equations with Stochastic Source Terms

2008 ◽  
Vol 46 (6) ◽  
pp. 3113-3135 ◽  
Author(s):  
Timothy A. Lucas
Keyword(s):  
Operator Splitting ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Source Terms

scholarly journals NEW EXACT SOLUTIONS OF SPATIALLY AND TEMPORALLY VARYING REACTION-DIFFUSION EQUATIONS

2008 ◽  
Vol 06 (04) ◽  
pp. 371-381 ◽  
Author(s):  
NALINI JOSHI ◽  
TEGAN MORRISON
Keyword(s):  
Reaction Diffusion ◽  
Elliptic Functions ◽  
Point Of View ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Source Terms ◽  
Symmetry Approach ◽  
New Exact Solutions ◽  
Special Cases ◽  
New Feature

This paper considers reaction-diffusion equations from a new point of view, by including spatiotemporal dependence in the source terms. We show for the first time that solutions are given in terms of the classical Painlevé transcendents. We consider reaction-diffusion equations with cubic and quadratic source terms. A new feature of our analysis is that the coefficient functions are also solutions of differential equations, including the Painlevé equations. Special cases arise with elliptic functions as solutions. Additional solutions given in terms of equations that are not integrable are also considered. Solutions are constructed using a Lie symmetry approach.

scholarly journals Splitting spectral element method for fractional reaction-diffusion equations

2020 ◽  
Vol 14 ◽  
pp. 174830262096670
Author(s):  
Qi Li ◽  
Fangying Song
Keyword(s):  
Spectral Element Method ◽  
Operator Splitting ◽  
Reaction Diffusion ◽  
Spectral Element ◽  
Second Order ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Nonlinear Solver ◽  
Fractional Reaction ◽  
Element Method

In this paper, we propose a second-order operator splitting spectral element method for solving fractional reaction-diffusion equations. In order to achieve a fast second-order scheme in time, we decompose the original equation into linear and nonlinear sub-equations, and combine a quarter-time nonlinear solver and a half-time linear solver followed by final quarter-time nonlinear solver. The spatial discretization is eigen-decomposition based on spectral element method. Since this method gives a full diagonal representation of the fractional operator and gets an exponential convergence in space. We have an accurate and efficient approach for solving spacial fractional reaction-diffusion equations. Some numerical experiments are carried out to demonstrate the accuracy and efficiency of this method. Finally, we apply the proposed method to investigate the effect of the fractional order in the fractional reaction-diffusion equations.

scholarly journals Analysis of Operator Splitting in the Nonasymptotic Regime for Nonlinear Reaction-Diffusion Equations. Application to the Dynamics of Premixed Flames

2014 ◽  
Vol 52 (3) ◽  
pp. 1311-1334 ◽  
Author(s):  
Stéphane Descombes ◽  
Max Duarte ◽  
Thierry Dumont ◽  
Frédérique Laurent ◽  
Violaine Louvet ◽  
...  
Keyword(s):  
Operator Splitting ◽  
Premixed Flames ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Nonlinear Reaction
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