scholarly journals An Exact Solution of the Reaction-Diffusion Equation for the Speed of the Interface Propagation in Superconductors

2021 ◽  
Vol 41 (1) ◽  
pp. 85-95
Author(s):  
Neelufar Panna
Keyword(s):  
Exact Solution ◽  
Diffusion Equation ◽  
Reaction Diffusion ◽  
Front Propagation ◽  
Reaction Diffusion Equation ◽  
Analytical Treatment ◽  
Ginzburg Landau ◽  
Interface Propagation ◽  
Tdgl Equation ◽  
Good Agreement

The speed of interface propagation in superconductors for the scalar reaction-diffusion equation ut  =   ∇2u+ F(u)   is studied in detail. Here the non linear reaction term F (u) is the time-dependent Ginzburg-Landau or TDGL equation F(u)=u-u3 which describes the dynamics of the order-disorder transition. In contrast to what has been done in previous work [1] here an improved exact solution has derived by using TDGL equation to determine the speed of the front propagation. The analytical treatment of this study has been found in good agreement with the numerical simulation of V. Mendez et al. [2] and Di Bartolo and Dorsey [3]. The Chittagong Univ. J. Sci. 40(1) : 85-95, 2019

scholarly journals Amplitude Equations and Chemical Reaction–Diffusion Systems

1997 ◽  
Vol 07 (07) ◽  
pp. 1539-1554 ◽  
Author(s):  
M. Ipsen ◽  
F. Hynne ◽  
P. G. Sørensen
Keyword(s):  
Diffusion Equation ◽  
Chemical Reaction ◽  
Bifurcation Point ◽  
Landau Equation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Amplitude Equations ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation ◽  
Spatio Temporal

The paper discusses the use of amplitude equations to describe the spatio-temporal dynamics of a chemical reaction–diffusion system based on an Oregonator model of the Belousov–Zhabotinsky reaction. Sufficiently close to a supercritical Hopf bifurcation the reaction–diffusion equation can be approximated by a complex Ginzburg–Landau equation with parameters determined by the original equation at the point of operation considered. We illustrate the validity of this reduction by comparing numerical spiral wave solutions to the Oregonator reaction–diffusion equation with the corresponding solutions to the complex Ginzburg–Landau equation at finite distances from the bifurcation point. We also compare the solutions at a bifurcation point where the systems develop spatio-temporal chaos. We show that the complex Ginzburg–Landau equation represents the dynamical behavior of the reaction–diffusion equation remarkably well, sufficiently far from the bifurcation point for experimental applications to be feasible.

On the Hybrid of Fourier Transform and Adomian Decomposition Method for the Solution of Nonlinear Cauchy Problems of the Reaction-Diffusion Equation

2012 ◽  
Vol 67 (6-7) ◽  
pp. 355-362 ◽  
Author(s):  
Salman Nourazar ◽  
Akbar Nazari-Golshan ◽  
Ahmet Yıldırım ◽  
Maryam Nourazar
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Fourier Transform ◽  
Diffusion Equation ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Adomian Decomposition ◽  
The Cauchy Problem

The physical science importance of the Cauchy problem of the reaction-diffusion equation appears in the modelling of a wide variety of nonlinear systems in physics, chemistry, ecology, biology, and engineering. A hybrid of Fourier transform and Adomian decomposition method (FTADM) is developed for solving the nonlinear non-homogeneous partial differential equations of the Cauchy problem of reaction-diffusion. The results of the FTADM and the ADM are compared with the exact solution. The comparison reveals that for the same components of the recursive sequences, the errors associated with the FTADM are much lesser than those of the ADM. We show that as time increases the results of the FTADM approaches 1 with only six recursive terms. This is in agreement with the physical property of the density-dependent nonlinear diffusion of the Cauchy problem which is also in agreement with the exact solution. The monotonic and very rapid convergence of the results of the FTADM towards the exact solution is shown to be much faster than that of the ADM

scholarly journals Numerical Analysis WSGD Scheme for One- and Two-Dimensional Distributed Order Fractional Reaction–Diffusion Equation with Collocation Method via Fractional B-Spline

2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Mohammad Ramezani
Keyword(s):  
Diffusion Equation ◽  
Collocation Method ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
B Spline ◽  
Computational Work ◽  
Distributed Order ◽  
Fractional Reaction

AbstractThe main propose of this paper is presenting an efficient numerical scheme to solve WSGD scheme for one- and two-dimensional distributed order fractional reaction–diffusion equation. The proposed method is based on fractional B-spline basics in collocation method which involve Caputo-type fractional derivatives for $$0 < \alpha < 1$$ 0 < α < 1 . The most significant privilege of proposed method is efficient and quite accurate and it requires relatively less computational work. The solution of consideration problem is transmute to the solution of the linear system of algebraic equations which can be solved by a suitable numerical method. The finally, several numerical WSGD Scheme for one- and two-dimensional distributed order fractional reaction–diffusion equation.

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