scholarly journals Semi-fractional diffusion equations

2019 ◽  
Vol 22 (2) ◽  
pp. 326-357 ◽  
Author(s):  
Peter Kern ◽  
Svenja Lage ◽  
Mark M. Meerschaert
Keyword(s):  
Fourier Series ◽  
Lévy Processes ◽  
Fractional Derivatives ◽  
Levy Processes ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Periodic Part ◽  
Fractional Diffusion Equations ◽  
Type Formula ◽  
Semigroup Approach

Abstract It is well known that certain fractional diffusion equations can be solved by the densities of stable Lévy motions. In this paper we use the classical semigroup approach for Lévy processes to define semi-fractional derivatives, which allows us to generalize this statement to semistable Lévy processes. A Fourier series approach for the periodic part of the corresponding Lévy exponents enables us to represent semi-fractional derivatives by a Grünwald-Letnikov type formula. We use this formula to calculate semi-fractional derivatives and solutions to semi-fractional diffusion equations numerically. In particular, by means of the Grünwald-Letnikov type formula we provide a numerical algorithm to compute semistable densities.

scholarly journals Fractional Diffusion Equations for Lattice and Continuum: Grünwald-Letnikov Differences and Derivatives Approach

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Fractional Derivatives ◽  
Three Dimensional ◽  
Lattice Models ◽  
Fractional Diffusion ◽  
Lattice Diffusion ◽  
Diffusion Equations ◽  
Difference Operators ◽  
Fractional Diffusion Equations ◽  
Nonlocal Media ◽  
Derivatives Of

Fractional diffusion equations for three-dimensional lattice models based on fractional-order differences of the Grünwald-Letnikov type are suggested. These lattice fractional diffusion equations contain difference operators that describe long-range jumps from one lattice site to another. In continuum limit, the suggested lattice diffusion equations with noninteger order differences give the diffusion equations with the Grünwald-Letnikov fractional derivatives for continuum. We propose a consistent derivation of the fractional diffusion equation with the fractional derivatives of Grünwald-Letnikov type. The suggested lattice diffusion equations can be considered as a new microstructural basis of space-fractional diffusion in nonlocal media.

scholarly journals The Maximum Principle for Variable-Order Fractional Diffusion Equations and the Estimates of Higher Variable-Order Fractional Derivatives

2020 ◽  
Vol 8 ◽  
Author(s):  
Guangming Xue ◽  
Funing Lin ◽  
Guangwang Su
Keyword(s):  
Maximum Principle ◽  
Fractional Derivatives ◽  
Arbitrary Point ◽  
Initial Boundary ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Maximum Principles ◽  
Variable Order ◽  
The Maximum Principle ◽  
Fractional Diffusion Equations

In this paper, the maximum principle of variable-order fractional diffusion equations and the estimates of fractional derivatives with higher variable order are investigated. Firstly, we deduce the fractional derivative of a function of higher variable order at an arbitrary point. We also give an estimate of the error. Some important inequalities for fractional derivatives of variable order at arbitrary points and extreme points are presented. Then, the maximum principles of Riesz-Caputo fractional differential equations in terms of the multi-term space-time variable order are proved. Finally, under the initial-boundary value conditions, it is verified via the proposed principle that the solutions are unique, and their continuous dependance holds.

Maximum principles for a class of generalized time-fractional diffusion equations

2020 ◽  
Vol 23 (3) ◽  
pp. 822-836
Author(s):  
Shengda Zeng ◽  
Stanisław Migórski ◽  
Van Thien Nguyen ◽  
Yunru Bai
Keyword(s):  
Fractional Derivatives ◽  
Extreme Points ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Maximum Principles ◽  
Variable Order ◽  
Fractional Diffusion Equations ◽  
Caputo Derivatives ◽  
Time Space ◽  
Generalized Time

AbstractTwo significant inequalities for generalized time fractional derivatives at extreme points are obtained. Then, we apply the inequalities to establish the maximum principles for multi-term time-space fractional variable-order operators. Finally, we employ the principles to investigate two kinds of diffusion equations involving generalized time-fractional Caputo derivatives and space-fractional Riesz-Caputo derivatives.

scholarly journals Second-order numerical methods for the tempered fractional diffusion equations

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Zeshan Qiu ◽  
Xuenian Cao
Keyword(s):  
Numerical Methods ◽  
Numerical Experiments ◽  
Fractional Derivatives ◽  
Fractional Diffusion ◽  
Second Order ◽  
Diffusion Equations ◽  
Difference Operators ◽  
Numerical Schemes ◽  
Fractional Diffusion Equations ◽  
Left And Right

AbstractIn this paper, a class of second-order tempered difference operators for the left and right Riemann–Liouville tempered fractional derivatives is constructed. And a class of second-order numerical methods is presented for solving the space tempered fractional diffusion equations, where the space tempered fractional derivatives are evaluated by the proposed tempered difference operators, and in the time direction is discreted by the Crank–Nicolson method. Numerical schemes are proved to be unconditionally stable and convergent with order $O(h^{2}+\tau ^{2})$O(h2+τ2). Numerical experiments demonstrate the effectiveness of the numerical schemes.

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