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  - 
TY  - JOUR
A1  - Sandev, Trifce
A1  - Metzler, Ralf
A1  - Tomovski, Zivorad
T1  - Velocity and displacement correlation functions for fractional generalized Langevin equations
JF  - Fractional calculus and applied analysis : an international journal for theory and applications
N2  - We study analytically a generalized fractional Langevin equation. General formulas for calculation of variances and the mean square displacement are derived. Cases with a three parameter Mittag-Leffler frictional memory kernel are considered. Exact results in terms of the Mittag-Leffler type functions for the relaxation functions, average velocity and average particle displacement are obtained. The mean square displacement and variances are investigated analytically. Asymptotic behaviors of the particle in the short and long time limit are found. The model considered in this paper may be used for modeling anomalous diffusive processes in complex media including phenomena similar to single file diffusion or possible generalizations thereof. We show the importance of the initial conditions on the anomalous diffusive behavior of the particle.
KW  - fractional generalized Langevin equation
KW  - frictional memory kernel
KW  - variances
KW  - mean square displacement
KW  - anomalous diffusion
Y1  - 2012
U6  - https://doi.org/10.2478/s13540-012-0031-2
SN  - 1311-0454
VL  - 15
IS  - 3
SP  - 426
EP  - 450
PB  - Versita
CY  - Warsaw
ER  -