@article{SandevChechkinKorabeletal.2015,
  author    = {Sandev, Trifce and Chechkin, Aleksei V. and Korabel, Nickolay and Kantz, Holger and Sokolov, Igor M. and Metzler, Ralf},
  title     = {Distributed-order diffusion equations and multifractality: Models and solutions},
  series = {Physical review : E, Statistical, nonlinear and soft matter physics},
  volume    = {92},
  journal   = {Physical review : E, Statistical, nonlinear and soft matter physics},
  number    = {4},
  publisher = {American Physical Society},
  address   = {College Park},
  issn      = {1539-3755},
  doi       = {10.1103/PhysRevE.92.042117},
  pages     = {19},
  year      = {2015},
  abstract  = {We study distributed-order time fractional diffusion equations characterized by multifractal memory kernels, in contrast to the simple power-law kernel of common time fractional diffusion equations. Based on the physical approach to anomalous diffusion provided by the seminal Scher-Montroll-Weiss continuous time random walk, we analyze both natural and modified-form distributed-order time fractional diffusion equations and compare the two approaches. The mean squared displacement is obtained and its limiting behavior analyzed. We derive the connection between the Wiener process, described by the conventional Langevin equation and the dynamics encoded by the distributed-order time fractional diffusion equation in terms of a generalized subordination of time. A detailed analysis of the multifractal properties of distributed-order diffusion equations is provided.},
  language  = {en}
}
@article{PadashSandevKantzetal.2022,
  author    = {Padash, Amin and Sandev, Trifce and Kantz, Holger and Metzler, Ralf and Chechkin, Aleksei},
  title     = {Asymmetric Levy flights are more efficient in random search},
  series = {Fractal and fractional},
  volume    = {6},
  journal   = {Fractal and fractional},
  number    = {5},
  publisher = {MDPI},
  address   = {Basel},
  issn      = {2504-3110},
  doi       = {10.3390/fractalfract6050260},
  pages     = {23},
  year      = {2022},
  abstract  = {We study the first-arrival (first-hitting) dynamics and efficiency of a one-dimensional random search model performing asymmetric Levy flights by leveraging the Fokker-Planck equation with a delta-sink and an asymmetric space-fractional derivative operator with stable index alpha and asymmetry (skewness) parameter beta. We find exact analytical results for the probability density of first-arrival times and the search efficiency, and we analyse their behaviour within the limits of short and long times. We find that when the starting point of the searcher is to the right of the target, random search by Brownian motion is more efficient than Levy flights with beta <= 0 (with a rightward bias) for short initial distances, while for beta>0 (with a leftward bias) Levy flights with alpha -> 1 are more efficient. When increasing the initial distance of the searcher to the target, Levy flight search (except for alpha=1 with beta=0) is more efficient than the Brownian search. Moreover, the asymmetry in jumps leads to essentially higher efficiency of the Levy search compared to symmetric Levy flights at both short and long distances, and the effect is more pronounced for stable indices alpha close to unity.},
  language  = {en}
}
@article{SandevChechkinKantzetal.2015,
  author    = {Sandev, Trifce and Chechkin, Aleksei V. and Kantz, Holger and Metzler, Ralf},
  title     = {Diffusion and fokker-planck-smoluchowski equations with generalized memory kernel},
  series = {Fractional calculus and applied analysis : an international journal for theory and applications},
  volume    = {18},
  journal   = {Fractional calculus and applied analysis : an international journal for theory and applications},
  number    = {4},
  publisher = {De Gruyter},
  address   = {Berlin},
  issn      = {1311-0454},
  doi       = {10.1515/fca-2015-0059},
  pages     = {1006 -- 1038},
  year      = {2015},
  abstract  = {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.},
  language  = {en}
}
@article{SandevIominKantzetal.2016,
  author    = {Sandev, Trifce and Iomin, Alexander and Kantz, Holger and Metzler, Ralf and Chechkin, Aleksei V.},
  title     = {Comb Model with Slow and Ultraslow Diffusion},
  series = {Mathematical modelling of natural phenomena},
  volume    = {11},
  journal   = {Mathematical modelling of natural phenomena},
  publisher = {EDP Sciences},
  address   = {Les Ulis},
  issn      = {0973-5348},
  doi       = {10.1051/mmnp/201611302},
  pages     = {18 -- 33},
  year      = {2016},
  abstract  = {We consider a generalized diffusion equation in two dimensions for modeling diffusion on a comb-like structures. We analyze the probability distribution functions and we derive the mean squared displacement in x and y directions. Different forms of the memory kernels (Dirac delta, power-law, and distributed order) are considered. It is shown that anomalous diffusion may occur along both x and y directions. Ultraslow diffusion and some more general diffusive processes are observed as well. We give the corresponding continuous time random walk model for the considered two dimensional diffusion-like equation on a comb, and we derive the probability distribution functions which subordinate the process governed by this equation to the Wiener process.},
  language  = {en}
}