scholarly journals The space spectral interpolation collocation method for reaction-diffusion systems

2021 ◽  
pp. 22-22
Author(s):  
Xiao-Li Zhang ◽  
Wei Zhang ◽  
Yu-Lan Wang ◽  
Ting-Ting Ban
Keyword(s):  
Collocation Method ◽  
Complex Dynamics ◽  
Solution Process ◽  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Spectral Interpolation ◽  
Fractal Calculus ◽  
Pattern Formations ◽  
Good Agreement

A space spectral interpolation collocation method is proposed to study nonlinear reaction-diffusion systems with complex dynamics characters. A detailed solution process is elucidated, and some pattern formations are given. The numerical results have a good agreement with theoretical ones. The method can be extended to fractional calculus and fractal calculus.

Different Types of Instabilities and Complex Dynamics in Fractional Reaction-Diffusion Systems

2009 ◽  
Author(s):  
Vasyl Gafiychuk ◽  
Bohdan Datsko
Keyword(s):  
Fractional Derivatives ◽  
Complex Dynamics ◽  
Reaction Diffusion ◽  
Main Attention ◽  
Van Der Pol ◽  
Reaction Diffusion Systems ◽  
Brusselator Model ◽  
Diffusion Systems ◽  
Different Types ◽  
Pattern Formations

In this article we analyze a influence of possible instabilities on pattern formations in the reaction-diffusion systems with fractional derivatives. The results of qualitative analysis are confirmed by numerical simulations. The main attention is paid to two models: a fractional order reaction diffusion system with Bonhoeffer-van der Pol kinetics and to Brusselator model.

Different Types of Instabilities and Complex Dynamics in Reaction-Diffusion Systems With Fractional Derivatives

2012 ◽  
Vol 7 (3) ◽  
Author(s):  
Vasyl Gafiychuk ◽  
Bohdan Datsko
Keyword(s):  
Complex Dynamics ◽  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Different Types ◽  
Linearized System ◽  
The Stability ◽  
Nonlinear Solutions ◽  
Activator Inhibitor ◽  
Fractional Reaction

In this article we analyze conditions for different types of instabilities and complex dynamics that occur in nonlinear two-component fractional reaction-diffusion systems. It is shown that the stability of steady state solutions and their evolution are mainly determined by the eigenvalue spectrum of a linearized system and the fractional derivative order. The results of the linear stability analysis are confirmed by computer simulations of the FitzHugh-Nahumo-like model. On the basis of this model, it is demonstrated that the conditions of instability and the pattern formation dynamics in fractional activator- inhibitor systems are different from the standard ones. As a result, a richer and a more complicated spatiotemporal dynamics takes place in fractional reaction-diffusion systems. A common picture of nonlinear solutions in time-fractional reaction-diffusion systems and illustrative examples are presented. The results obtained in the article for homogeneous perturbation have also been of interest for dynamical systems described by fractional ordinary differential equations.

Pattern Formation in Reaction Diffusion Systems: A Moving Boundary Model

2004 ◽  
Vol 97-98 ◽  
pp. 125-132 ◽  
Author(s):  
George Varghese ◽  
Jacob George
Keyword(s):  
Pattern Formation ◽  
Boundary Problem ◽  
Moving Boundary ◽  
Reaction Diffusion ◽  
Moving Boundary Problem ◽  
Precipitation Pattern ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Boundary Model ◽  
Good Agreement

Periodic precipitation pattern formation in reaction diffusion systems is interpreted as a moving boundary problem. All the existing laws are reexamined on the basis of the moving boundary assumption. Experimental observations were found to be in good agreement with the new equations suggested.

Complex dynamics in initially separated reaction-diffusion systems

1995 ◽  
Vol 221 (1-3) ◽  
pp. 1-14 ◽  
Author(s):  
S. Havlin ◽  
M. Araujo ◽  
Y. Lereah ◽  
H. Larralde ◽  
A. Shehter ◽  
...  
Keyword(s):  
Complex Dynamics ◽  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems

OSCILLATING LOCALIZED STRUCTURES IN REACTION–DIFFUSION SYSTEMS

2004 ◽  
Vol 14 (12) ◽  
pp. 4097-4104 ◽  
Author(s):  
ORAZIO DESCALZI ◽  
YUMINO HAYASE ◽  
HELMUT R. BRAND
Keyword(s):  
Reaction Diffusion ◽  
Point Of View ◽  
Reaction Diffusion Systems ◽  
Localized Structures ◽  
Diffusion Systems ◽  
Reaction Diffusion Model ◽  
Analytical Point ◽  
Analytical Expressions ◽  
Particle Solution ◽  
Good Agreement

Oscillating localized structures are studied for a simple reaction–diffusion model from an analytical point of view. The result is a particle solution which acts as a source of traveling waves. The analytical expressions obtained are in good agreement with direct numerical simulations.

scholarly journals Numerical Solution of a Class of Predator-Prey Systems with Complex Dynamics Characters Based on a Sinc Function Interpolation Collocation Method

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-34
Author(s):  
Mingjing Du ◽  
Pengfei Ning ◽  
Yulan Wang
Keyword(s):  
Collocation Method ◽  
Complex Dynamics ◽  
Sinc Function ◽  
New Approach ◽  
Predator Prey ◽  
Prey System ◽  
Simulation Results ◽  
Pattern Formations ◽  
Theoretical Results ◽  
Good Agreement

Although many kinds of numerical methods have been announced for the predator-prey system, simple and efficient methods have always been the direction that scholars strive to pursue. Based on this problem, in this paper, a new interpolation collocation method is proposed for a class of predator-prey systems with complex dynamics characters. Some complex dynamics characters and pattern formations are shown by using this new approach, and the results have a good agreement with theoretical results. Simulation results show the effectiveness of the method.

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