A Revisit of the Semi-Adaptive Method for Singular Degenerate Reaction-Diffusion Equations

2012 ◽  
Vol 2 (3) ◽  
pp. 185-203 ◽  
Author(s):  
Qin Sheng ◽  
A. Q. M. Khaliq
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Numerical Stability ◽  
Numerical Experiments ◽  
Reaction Diffusion ◽  
Adaptive Method ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Key Characteristics

AbstractThis article discusses key characteristics of a semi-adaptive finite difference method for solving singular degenerate reaction-diffusion equations. Numerical stability, monotonicity, and convergence are investigated. Numerical experiments illustrate the discussion. The study reconfirms and improves several of our earlier results.

scholarly journals A New Approach and Solution Technique to Solve Time Fractional Nonlinear Reaction-Diffusion Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Inci Cilingir Sungu ◽  
Huseyin Demir
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Time Stepping ◽  
Nonlinear Reaction ◽  
Differential Transform ◽  
Generalized Differential

A new application of the hybrid generalized differential transform and finite difference method is proposed by solving time fractional nonlinear reaction-diffusion equations. This method is a combination of the multi-time-stepping temporal generalized differential transform and the spatial finite difference methods. The procedure first converts the time-evolutionary equations into Poisson equations which are then solved using the central difference method. The temporal differential transform method as used in the paper takes care of stability and the finite difference method on the resulting equation results in a system of diagonally dominant linear algebraic equations. The Gauss-Seidel iterative procedure then used to solve the linear system thus has assured convergence. To have optimized convergence rate, numerical experiments were done by using a combination of factors involving multi-time-stepping, spatial step size, and degree of the polynomial fit in time. It is shown that the hybrid technique is reliable, accurate, and easy to apply.

scholarly journals Stable finite difference method for fractional reaction–diffusion equations by compact implicit integration factor methods

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Rongpei Zhang ◽  
Mingjun Li ◽  
Bo Chen ◽  
Liwei Zhang
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Difference Method ◽  
Reaction Diffusion ◽  
Second Order ◽  
Reaction Diffusion Equations ◽  
Implicit Integration ◽  
Fractional Reaction

AbstractIn this paper we propose a stable finite difference method to solve the fractional reaction–diffusion systems in a two-dimensional domain. The space discretization is implemented by the weighted shifted Grünwald difference (WSGD) which results in a stiff system of nonlinear ordinary differential equations (ODEs). This system of ordinary differential equations is solved by an efficient compact implicit integration factor (cIIF) method. The stability of the second order cIIF scheme is proved in the discrete $L^{2}$ L 2 -norm. We also prove the second-order convergence of the proposed scheme. Numerical examples are given to demonstrate the accuracy, efficiency, and robustness of the method.

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