TY - JOUR A1 - Sandev, Trifce A1 - Tomovski, Zivorad A1 - Dubbeldam, Johan L. A. A1 - Chechkin, Aleksei V. T1 - Generalized diffusion-wave equation with memory kernel JF - Journal of physics : A, Mathematical and theoretical N2 - We study generalized diffusion-wave equation in which the second order time derivative is replaced by an integro-differential operator. It yields time fractional and distributed order time fractional diffusion-wave equations as particular cases. We consider different memory kernels of the integro-differential operator, derive corresponding fundamental solutions, specify the conditions of their non-negativity and calculate the mean squared displacement for all cases. In particular, we introduce and study generalized diffusion-wave equations with a regularized Prabhakar derivative of single and distributed orders. The equations considered can be used for modeling the broad spectrum of anomalous diffusion processes and various transitions between different diffusion regimes. KW - diffusion-wave equation KW - Mittag-Leffler function KW - anomalous diffusion Y1 - 2018 U6 - https://doi.org/10.1088/1751-8121/aaefa3 SN - 1751-8113 SN - 1751-8121 VL - 52 IS - 1 PB - IOP Publ. Ltd. CY - Bristol 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 - TY - JOUR A1 - Sandev, Trifce A1 - Metzler, Ralf A1 - Tomovski, Zivorad T1 - Correlation functions for the fractional generalized Langevin equation in the presence of internal and external noise JF - Journal of mathematical physics N2 - We study generalized fractional Langevin equations in the presence of a harmonic potential. General expressions for the mean velocity and particle displacement, the mean squared displacement, position and velocity correlation functions, as well as normalized displacement correlation function are derived. We report exact results for the cases of internal and external friction, that is, when the driving noise is either internal and thus the fluctuation-dissipation relation is fulfilled or when the noise is external. The asymptotic behavior of the generalized stochastic oscillator is investigated, and the case of high viscous damping (overdamped limit) is considered. Additional behaviors of the normalized displacement correlation functions different from those for the regular damped harmonic oscillator are observed. In addition, the cases of a constant external force and the force free case are obtained. The validity of the generalized Einstein relation for this process is discussed. The considered fractional generalized Langevin equation may be used to model anomalous diffusive processes including single file-type diffusion. Y1 - 2014 U6 - https://doi.org/10.1063/1.4863478 SN - 0022-2488 SN - 1089-7658 VL - 55 IS - 2 PB - American Institute of Physics CY - Melville 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 - TY - JOUR A1 - Tomovski, Živorad A1 - Metzler, Ralf A1 - Gerhold, Stefan T1 - Fractional characteristic functions, and a fractional calculus approach for moments of random variables JF - Fractional calculus and applied analysis : an international journal for theory and applications N2 - In this paper we introduce a fractional variant of the characteristic function of a random variable. It exists on the whole real line, and is uniformly continuous. We show that fractional moments can be expressed in terms of Riemann-Liouville integrals and derivatives of the fractional characteristic function. The fractional moments are of interest in particular for distributions whose integer moments do not exist. Some illustrative examples for particular distributions are also presented. KW - Fractional calculus (primary) KW - Characteristic function KW - Mittag-Leffler KW - function KW - Fractional moments KW - Mellin transform Y1 - 2022 U6 - https://doi.org/10.1007/s13540-022-00047-x SN - 1314-2224 VL - 25 IS - 4 SP - 1307 EP - 1323 PB - De Gruyter CY - Berlin ; Boston ER -