scholarly journals On alternative to partial differential equations for the modelling of reaction-diffusion systems

2013 ◽  
pp. 35-47
Author(s):  
M.N. Nazarov ◽  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Reaction Diffusion ◽  
Partial Differential ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems

scholarly journals CAS WAVELETS METHOD TO SOLVE REACTION-DIFFUSION SYSTEM AND COMPARE IT WITH (G.F.E) METHOD

2020 ◽  
Vol 17 (35) ◽  
pp. 1110-1123
Author(s):  
Badran Jasim SALIM ◽  
Oday Ahmed JASIM
Keyword(s):  
Steady State ◽  
Finite Elements ◽  
Reaction Diffusion ◽  
Partial Differential ◽  
Reaction Diffusion Systems ◽  
Time Frequency ◽  
Nonlinear Reaction ◽  
Diffusion Systems ◽  
Cas Wavelets ◽  
Steady State Solutions

Wavelet analysis plays a prominent role in various fields of scientific disciplines. Mainly, wavelets are very successfully used in signal analysis for waveform representation and segmentation, time-frequency analysis, and fast algorithms in the propagation equations and reaction. This research aimed to guide researchers to use Cos and Sin (CAS) to approximate the solution of the partial differential equation system. This method has been successfully applied to solve a coupled system of nonlinear Reaction-diffusion systems. It has been shown CAS wavelet method is quite capable and suited for finding exact solutions once the consistency of the method gives wider applicability where the main idea is to transform complex nonlinear partial differential equations into algebraic equation systems, which are easy to handle and find a numerical solution for them. By comparing the numerical solutions of the CAS and Galerkin finite elements methods, the answer of nonlinear Reaction-diffusion systems using the CAS wavelets for all tˆ and x values is accurate, reliable, robust, promising, and quickly arrives at the exact solution. When parameters 𝜀1 𝑎𝑛𝑑 𝜀2 are growing and with L decreasing, then the CAS method converges to steady-state solutions quickly (the less L, the more accurate the solution). It is converging towards steady-state solutions faster than and loses steps over time. Moreover, the results also show that the solution of the CAS wavelets is more reliable and faster compared to the Galerkin finite elements (G.F.E).

scholarly journals On the solution of reaction-diffusion equations with double diffusivity

1987 ◽  
Vol 10 (1) ◽  
pp. 163-172
Author(s):  
B. D. Aggarwala ◽  
C. Nasim
Keyword(s):  
Heat Conduction ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Partial Differential ◽  
Coupled Partial Differential Equations ◽  
Special Cases ◽  
Two Temperatures

In this paper, solution of a pair of Coupled Partial Differential equations is derived. These equations arise in the solution of problems of flow of homogeneous liquids in fissured rocks and heat conduction involving two temperatures. These equations have been considered by Hill and Aifantis, but the technique we use appears to be simpler and more direct, and some new results are derived. Also, discussion about the propagation of initial discontinuities is given and illustrated with graphs of some special cases.

scholarly journals Development of Methods of Asymptotic Analysis of Transition Layers in Reaction–Diffusion–Advection Equations: Theory and Applications

2021 ◽  
Vol 61 (12) ◽  
pp. 2068-2087
Author(s):  
N. N. Nefedov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Asymptotic Methods ◽  
General Scheme ◽  
Rapid Change ◽  
Reaction Diffusion ◽  
Singularly Perturbed ◽  
Parabolic Problems ◽  
Singularly Perturbed Problems ◽  
Partial Differential

Abstract This work presents a review and analysis of modern asymptotic methods for analysis of singularly perturbed problems with interior and boundary layers. The central part of the work is a review of the work of the author and his colleagues and disciples. It highlights boundary and initial-boundary value problems for nonlinear elliptic and parabolic partial differential equations, as well as periodic parabolic problems, which are widely used in applications and are called reaction–diffusion and reaction–diffusion–advection equations. These problems can be interpreted as models in chemical kinetics, synergetics, astrophysics, biology, and other fields. The solutions of these problems often have both narrow boundary regions of rapid change and inner layers of various types (contrasting structures, moving interior layers: fronts), which leads to the need to develop new asymptotic methods in order to study them both formally and rigorously. A general scheme for a rigorous study of contrast structures in singularly perturbed problems for partial differential equations, based on the use of the asymptotic method of differential inequalities, is presented and illustrated on relevant problems. The main achievements of this line of studies of partial differential equations are reflected, and the key directions of its development are indicated.

scholarly journals Comparison theorems for multicomponent diffusion systems: developments since 1961

2000 ◽  
Vol 4 (2) ◽  
pp. 151-163
Author(s):  
M. I. Nelson
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Existence And Uniqueness ◽  
Multicomponent Diffusion ◽  
Comparison Theorems ◽  
Numerical Techniques ◽  
Uniqueness Of Solutions ◽  
Partial Differential ◽  
Diffusion Systems ◽  
Component Systems

Comparison theorems may be used to prove the existence and uniqueness of solutions to certain types of partial differential equations. They provide bounds for solutions and can be used as the basis of numerical techniques for the computation of solutions. In 1961 Alex McNabb published one of the first papers extending their use to multi-component systems. Developments in the theory and applications of such results, through citations of this original paper, are reviewed.

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