scholarly journals Comparing Numerical Methods for Solving Time-Fractional Reaction-Diffusion Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-28 ◽  
Author(s):  
Veyis Turut ◽  
Nuran Güzel
Keyword(s):  
Fractional Derivatives ◽  
Padé Approximation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Pade Approximation ◽  
Transform Method ◽  
Differential Transform ◽  
Generalized Differential ◽  
Fractional Reaction

Multivariate Padé approximation (MPA) is applied to numerically approximate the solutions of time-fractional reaction-diffusion equations, and the numerical results are compared with solutions obtained by the generalized differential transform method (GDTM). The fractional derivatives are described in the Caputo sense. Two illustrative examples are given to demonstrate the effectiveness of the multivariate Padé approximation (MPA). The results reveal that the multivariate Padé approximation (MPA) is very effective and convenient for solving time-fractional reaction-diffusion equations.

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 Fractional Reaction-Diffusion Equations for Modelling Complex Biological Patterns

2014 ◽  
Vol 8 (3) ◽  
Author(s):  
Ku Azlina Ku Akil ◽  
Sithi V Muniandy ◽  
Einly Lim
Keyword(s):  
Fractional Derivatives ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Difference Approximation ◽  
Standard Reaction ◽  
Biological Patterns ◽  
Derivatives Of ◽  
Fractional Reaction

We consider a system of nonlinear time-fractional reaction-diffusion equations (TFRDE) on a finite spatial domain x ∈ [0, L], and time t ∈ [0, T]. The system of standard reaction-diffusion equations with Neumann boundary conditions are generalized by replacing the first-order time derivatives with Caputo time-fractional derivatives of order α ∈ (0, 1). We solve the TFRDE numerically using Grünwald-Letnikov derivative approximation for time-fractional derivative and centred difference approximation for spatial derivative. We discuss the numerical results and propose the applications of TFRDE for the modelling of complex patterns in biological systems.

scholarly journals The GDTM-Padé Technique for the Nonlinear Lattice Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
Junfeng Lu
Keyword(s):  
Exact Solutions ◽  
Padé Approximation ◽  
High Accuracy ◽  
Numerical Results ◽  
Differential Transform Method ◽  
Pade Approximation ◽  
Transform Method ◽  
Differential Transform ◽  
Nonlinear Lattice ◽  
Generalized Differential

The GDTM-Padé technique is a combination of the generalized differential transform method and the Padé approximation. We apply this technique to solve the two nonlinear lattice equations, which results in the high accuracy of the GDTM-Padé solutions. Numerical results are presented to show its efficiency by comparing the GDTM-Padé solutions, the solutions obtained by the generalized differential transform method, and the exact solutions.

Optimal control of fractional reaction-diffusion equations with Poisson jumps

2018 ◽  
Vol 27 (2) ◽  
pp. 605-621 ◽  
Author(s):  
N. Durga ◽  
P. Muthukumar
Keyword(s):  
Optimal Control ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Poisson Jumps ◽  
Fractional Reaction

Fast implicit integration factor method for nonlinear space Riesz fractional reaction–diffusion equations

2020 ◽  
Vol 378 ◽  
pp. 112935 ◽  
Author(s):  
Huan-Yan Jian ◽  
Ting-Zhu Huang ◽  
Xian-Ming Gu ◽  
Xi-Le Zhao ◽  
Yong-Liang Zhao
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Implicit Integration ◽  
Factor Method ◽  
Fractional Reaction

scholarly journals Solutions of the time fractional reaction–diffusion equations with residual power series method

2016 ◽  
Vol 8 (10) ◽  
pp. 168781401667086 ◽  
Author(s):  
Fairouz Tchier ◽  
Mustafa Inc ◽  
Zeliha S Korpinar ◽  
Dumitru Baleanu
Keyword(s):  
Power Series ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Series Method ◽  
Power Series Method ◽  
Residual Power Series Method ◽  
Residual Power Series ◽  
Residual Power ◽  
Fractional Reaction
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