scholarly journals From continuous time random walks to the generalized diffusion equation

2018 ◽  
Vol 21 (1) ◽  
pp. 10-28 ◽  
Author(s):  
Trifce Sandev ◽  
Ralf Metzler ◽  
Aleksei Chechkin
Keyword(s):  
Diffusion Equation ◽  
Continuous Time ◽  
Fractional Diffusion ◽  
Distribution Functions ◽  
Diffusion Equations ◽  
Fractional Diffusion Equations ◽  
Probability Distribution Functions ◽  
Special Cases ◽  
Continuous Time Random Walks ◽  
Random Walk Theory

AbstractWe obtain a generalized diffusion equation in modified or Riemann-Liouville form from continuous time random walk theory. The waiting time probability density function and mean squared displacement for different forms of the equation are explicitly calculated. We show examples of generalized diffusion equations in normal or Caputo form that encode the same probability distribution functions as those obtained from the generalized diffusion equation in modified form. The obtained equations are general and many known fractional diffusion equations are included as special cases.

scholarly journals Clustered continuous-time random walks: diffusion and relaxation consequences

2012 ◽  
Vol 468 (2142) ◽  
pp. 1615-1628 ◽  
Author(s):  
Karina Weron ◽  
Aleksander Stanislavsky ◽  
Agnieszka Jurlewicz ◽  
Mark M. Meerschaert ◽  
Hans-Peter Scheffler
Keyword(s):  
Random Walks ◽  
Power Law ◽  
Continuous Time ◽  
Waiting Times ◽  
Scaling Limit ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
The Novel ◽  
Fractional Diffusion Equations ◽  
Continuous Time Random Walks

We present a class of continuous-time random walks (CTRWs), in which random jumps are separated by random waiting times. The novel feature of these CTRWs is that the jumps are clustered. This introduces a coupled effect, with longer waiting times separating larger jump clusters. We show that the CTRW scaling limits are time-changed processes. Their densities solve two different fractional diffusion equations, depending on whether the waiting time is coupled to the preceding jump, or the following one. These fractional diffusion equations can be used to model all types of experimentally observed two power-law relaxation patterns. The parameters of the scaling limit process determine the power-law exponents and loss peak frequencies.

Numerical Algorithm of the Fractional Diffusion Equation Based on Sinc-Chebyshev Collocation

2014 ◽  
Vol 926-930 ◽  
pp. 3105-3108
Author(s):  
Zhi Mao ◽  
Ting Ting Wang
Keyword(s):  
Diffusion Equation ◽  
Numerical Algorithm ◽  
Variable Coefficient ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Algebraic Equations ◽  
Fractional Diffusion Equations ◽  
Linear Algebraic Equations ◽  
Collocation Technique ◽  
Sinc Functions

Fractional diffusion equations have recently been applied in various area of engineering. In this paper, a new numerical algorithm for solving the fractional diffusion equations with a variable coefficient is proposed. Based on the collocation technique where the shifted Chebyshev polynomials in time and the sinc functions in space are utilized respectively, the problem is reduced to the solution of a system of linear algebraic equations. The procedure is tested and the efficiency of the proposed algorithm is confirmed through the numerical example.

scholarly journals Identification of points sources via time fractional diffusion equation

Filomat ◽  
2018 ◽  
Vol 32 (18) ◽  
pp. 6189-6201 ◽  
Author(s):  
A. Ghanmi ◽  
R. Mdimagh ◽  
I.B. Saad
Keyword(s):  
Inverse Problem ◽  
Diffusion Equation ◽  
Stability Result ◽  
Fractional Diffusion ◽  
Cauchy Data ◽  
Diffusion Equations ◽  
Lipschitz Stability ◽  
Single Measurement ◽  
Fractional Diffusion Equations ◽  
Time Fractional Diffusion Equation

This article investigates the source identification in the fractional diffusion equations, by performing a single measurement of the Cauchy data on the accessible boundary. The main results of this work consist in giving an identifiability result and establishing a local Lipschitz stability result. To solve the inverse problem of identifying fractional sources from such observations, a non-iterative algebraical method based on the Reciprocity Gap functional is proposed.

Continuous time random walk models associated with distributed order diffusion equations

2015 ◽  
Vol 18 (3) ◽  
Author(s):  
Sabir Umarov
Keyword(s):  
Random Walk ◽  
Continuous Time ◽  
Diffusion Processes ◽  
Stability Index ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Analytic Method ◽  
Fractional Diffusion Equations ◽  
The Stability ◽  
Distributed Order

AbstractIn this paper continuous time and discrete random walk models approximating diffusion processes associated with time-fractional and spacedistributed order differential equations are studied. Stochastic processes associated with the considered equations represent time-changed processes, where the time-change process is the inverse to a Levy’s stable subordinator with the stability index β ∈ (0, 1). In the paper the convergence of modeled continuous time and discrete random walks to time-changed processes associated with distributed order fractional diffusion equations are proved using an analytic method.

HOW DOES THE DIFFUSION EQUATION ON FRACTALS LOOK?

Fractals ◽  
2004 ◽  
Vol 12 (02) ◽  
pp. 149-156 ◽  
Author(s):  
H. EDUARDO ROMAN
Keyword(s):  
Diffusion Equation ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Fractional Diffusion Equations ◽  
Satisfactory Answer

Different forms of diffusion equations on fractals proposed in the literature are reviewed and critically discussed. Variants of the known fractional diffusion equations are suggested here and worked out analytically. On the basis of these results we conclude that the quest: "what is the form of the diffusion equation on fractals," is still open, but we are possibly close to obtaining a satisfactory answer.

scholarly journals A new fractional derivative operator and its application to diffusion equation

2021 ◽  
Author(s):  
Ruchi Sharma ◽  
Pranay Goswami ◽  
RAVI DUBEY ◽  
Fethi Belgacem
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Derivative Operator ◽  
Fractional Diffusion Equations

In this paper, we introduced a new fractional derivative operator based on Lonezo Hartely function, which is called G-function. With the help of the operator, we solved a fractional diffusion equations. Some applications related to the operator is also discussed as form of corollaries.

scholarly journals Averaged Control for Fractional ODEs and Fractional Diffusion Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Darko Mitrovic ◽  
Andrej Novak ◽  
Tarik Uzunović
Keyword(s):  
Banach Space ◽  
Diffusion Equation ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Diffusion Equations ◽  
Fractional Diffusion Equations ◽  
Fractional Type

We generalize results concerning averaged controllability on fractional type equations: system of fractional ODEs and the fractional diffusion equation. The proofs are accomplished by introducing appropriate Banach space in which we prove observability inequalities.

scholarly journals REVISITING THE DERIVATION OF THE FRACTIONAL DIFFUSION EQUATION

Fractals ◽  
2003 ◽  
Vol 11 (supp01) ◽  
pp. 281-289 ◽  
Author(s):  
ENRICO SCALAS ◽  
RUDOLF GORENFLO ◽  
FRANCESCO MAINARDI ◽  
MARCO RABERTO
Keyword(s):  
Cauchy Problem ◽  
Limit Theorem ◽  
Diffusion Equation ◽  
Continuous Time ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Diffusion Limit ◽  
Straightforward Application ◽  
The Cauchy Problem ◽  
Continuous Time Random Walks

The fractional diffusion equation is derived from the master equation of continuous time random walks (CTRWs) via a straightforward application of the Gnedenko-Kolmogorov limit theorem. The Cauchy problem for the fractional diffusion equation is solved in various important and general cases. The meaning of the proper diffusion limit for CTRWs is discussed.

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