Applying Legendre wavelet method with Tikhonov regularization for one-dimensional time-fractional diffusion equations

2018 ◽  
Vol 37 (4) ◽  
pp. 4793-4804 ◽  
Author(s):  
Aram Azizi ◽  
Sarkout Abdi ◽  
Jamshid Saeidian
Keyword(s):  
Tikhonov Regularization ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Wavelet Method ◽  
One Dimensional ◽  
Fractional Diffusion Equations ◽  
Legendre Wavelet Method

scholarly journals Between Waves and Diffusion: Paradoxical Entropy Production in an Exceptional Regime

Entropy ◽  
2018 ◽  
Vol 20 (11) ◽  
pp. 881 ◽  
Author(s):  
Karl Hoffmann ◽  
Kathrin Kulmus ◽  
Christopher Essex ◽  
Janett Prehl
Keyword(s):  
Entropy Production ◽  
Production Rate ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Irreversible Processes ◽  
Entropy Production Rate ◽  
Continuous Space ◽  
One Dimensional ◽  
Fractional Diffusion Equations ◽  
And Diffusion

The entropy production rate is a well established measure for the extent of irreversibility in a process. For irreversible processes, one thus usually expects that the entropy production rate approaches zero in the reversible limit. Fractional diffusion equations provide a fascinating testbed for that intuition in that they build a bridge connecting the fully irreversible diffusion equation with the fully reversible wave equation by a one-parameter family of processes. The entropy production paradox describes the very non-intuitive increase of the entropy production rate as that bridge is passed from irreversible diffusion to reversible waves. This paradox has been established for time- and space-fractional diffusion equations on one-dimensional continuous space and for the Shannon, Tsallis and Renyi entropies. After a brief review of the known results, we generalize it to time-fractional diffusion on a finite chain of points described by a fractional master equation.

scholarly journals A Second-Order Uniformly Stable Explicit Asymmetric Discretization Method for One-Dimensional Fractional Diffusion Equations

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Lin Zhu
Keyword(s):  
Difference Operator ◽  
Nodal Point ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Discrete Solution ◽  
Time Step ◽  
One Dimensional ◽  
Fractional Diffusion Equations ◽  
Discretization Technique ◽  
Explicit Finite Difference

Using the asymmetric discretization technique, an explicit finite difference scheme is constructed for one-dimensional spatial fractional diffusion equations (FDEs). The spatial fractional derivative is approximated by the weighted and shifted Grünwald difference operator. The scheme can be solved explicitly by calculating unknowns in the different nodal-point sequences at the odd time-step and the even time-step. The uniform stability is proven and the error between the discrete solution and analytical solution is theoretically estimated. Numerical examples are given to verify theoretical analysis.

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