TY  - JOUR
A1  - Sandev, Trifce
A1  - Metzler, Ralf
A1  - Chechkin, Aleksei V.
T1  - From continuous time random walks to the generalized diffusion equation
JF  - Fractional calculus and applied analysis : an international journal for theory and applications
N2  - We 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.
KW  - continuous time random walk (CTRW)
KW  - generalized diffusion equation
KW  - Mittag-Leffler functions
KW  - anomalous diffusion
Y1  - 2018
U6  - https://doi.org/10.1515/fca-2018-0002
SN  - 1311-0454
SN  - 1314-2224
VL  - 21
IS  - 1
SP  - 10
EP  - 28
PB  - De Gruyter
CY  - Berlin
ER  - 
TY  - JOUR
A1  - Sandev, Trifce
A1  - Chechkin, Aleksei V.
A1  - Kantz, Holger
A1  - Metzler, Ralf
T1  - Diffusion and fokker-planck-smoluchowski equations with generalized memory kernel
JF  - Fractional calculus and applied analysis : an international journal for theory and applications
N2  - We consider anomalous stochastic processes based on the renewal continuous time random walk model with different forms for the probability density of waiting times between individual jumps. In the corresponding continuum limit we derive the generalized diffusion and Fokker-Planck-Smoluchowski equations with the corresponding memory kernels. We calculate the qth order moments in the unbiased and biased cases, and demonstrate that the generalized Einstein relation for the considered dynamics remains valid. The relaxation of modes in the case of an external harmonic potential and the convergence of the mean squared displacement to the thermal plateau are analyzed.
KW  - continuous time random walk (CTRW)
KW  - Fokker-Planck-Smoluchowski equation
KW  - Mittag-Leffler functions
KW  - anomalous diffusion
KW  - multi-scaling
Y1  - 2015
U6  - https://doi.org/10.1515/fca-2015-0059
SN  - 1311-0454
SN  - 1314-2224
VL  - 18
IS  - 4
SP  - 1006
EP  - 1038
PB  - De Gruyter
CY  - Berlin
ER  - 
TY  - JOUR
A1  - Tomovski, Zivorad
A1  - Sandev, Trifce
A1  - Metzler, Ralf
A1  - Dubbeldam, Johan
T1  - Generalized space-time fractional diffusion equation with composite fractional time derivative
JF  - Physica : europhysics journal ; A, Statistical mechanics and its applications
N2  - We investigate the solution of space-time fractional diffusion equations with a generalized Riemann-Liouville time fractional derivative and Riesz-Feller space fractional derivative. The Laplace and Fourier transform methods are applied to solve the proposed fractional diffusion equation. The results are represented by using the Mittag-Leffler functions and the Fox H-function. Special cases of the initial and boundary conditions are considered. Numerical scheme and Grunwald-Letnikov approximation are also used to solve the space-time fractional diffusion equation. The fractional moments of the fundamental solution of the considered space-time fractional diffusion equation are obtained. Many known results are special cases of those obtained in this paper. We investigate also the solution of a space-time fractional diffusion equations with a singular term of the form delta(x). t-beta/Gamma(1-beta) (beta > 0).
KW  - Fractional diffusion equation
KW  - Composite fractional derivative
KW  - Riesz-Feller fractional derivative
KW  - Mittag-Leffler functions
KW  - Fox H-function
KW  - Fractional moments
KW  - Asymptotic expansions
KW  - Grunwald-Letnikov approximation
Y1  - 2012
U6  - https://doi.org/10.1016/j.physa.2011.12.035
SN  - 0378-4371
SN  - 1873-2119
VL  - 391
IS  - 8
SP  - 2527
EP  - 2542
PB  - Elsevier
CY  - Amsterdam
ER  -