Finite-velocity diffusion on a comb

A Cattaneo equation for a comb structure is considered. We present a rigorous analysis of the obtained fractional diffusion equation, and corresponding solutions for the probability distribution function are obtained in the form of the Fox $H$-function and its infinite series. The mean square displacement along the backbone is obtained as well in terms of the infinite series of the Fox $H$-function. The obtained solutions describe the transition from normal diffusion to subdiffusion, which results from the comb geometry.Comment: 7 page

Effects of a fractional friction with power-law memory kernel on string vibrations

AbstractIn this paper we give an analytical treatment of a wave equation for a vibrating string in the presence of a fractional friction with power-law memory kernel. The exact solution is obtained in terms of the Mittag-Leffler type functions and a generalized integral operator containing a four parameter Mittag-Leffler function in the kernel. The method of separation of the variables, Laplace transform method and Sturm–Liouville problem are used to solve the equation analytically. The asymptotic behaviors of the solution of a special case fractional differential equation are obtained directly from the analytical solution of the equation and by using the Tauberian theorems. The proposed model may be used for describing processes where the memory effects of complex media could not be neglected

Anomalous diffusion on a fractal mesh

An exact analytical analysis of anomalous diffusion on a fractal mesh is presented. The fractal mesh structure is a direct product of two fractal sets which belong to a main branch of backbones and side branch of fingers. The fractal sets of both backbones and fingers are constructed on the entire (infinite) $y$ and $x$ axises. To this end we suggested a special algorithm of this special construction. The transport properties of the fractal mesh is studied, in particular, subdiffusion along the backbones is obtained with the dispersion relation $\langle x^2(t)\rangle\sim t^{\beta}$, where the transport exponent $\beta<1$ is determined by the fractal dimensions of both backbone and fingers. Superdiffusion with $\beta>1$ has been observed as well when the environment is controlled by means of a memory kernel

Heterogeneous diffusion in comb and fractal grid structures

We give an exact analytical results for diffusion with a power-law position dependent diffusion coefficient along the main channel (backbone) on a comb and grid comb structures. For the mean square displacement along the backbone of the comb we obtain behavior $\langle x^2(t)\rangle\sim t^{1/(2-\alpha)}$, where $\alpha$ is the power-law exponent of the position dependent diffusion coefficient $D(x)\sim |x|^{\alpha}$. Depending on the value of $\alpha$ we observe different regimes, from anomalous subdiffusion, superdiffusion, and hyperdiffusion. For the case of the fractal grid we observe the mean square displacement, which depends on the fractal dimension of the structure of the backbones, i.e., $\langle x^2(t)\rangle\sim t^{(1+\nu)/(2-\alpha)}$, where $0<\nu<1$ is the fractal dimension of the backbones structure. The reduced probability distribution functions for both cases are obtained by help of the Fox $H$-functions

Random resetting in search problems

By periodically returning a search process to a known or random state, random resetting possesses the potential to unveil new trajectories, sidestep potential obstacles, and consequently enhance the efficiency of locating desired targets. In this chapter, we highlight the pivotal theoretical contributions that have enriched our understanding of random resetting within an abundance of stochastic processes, ranging from standard diffusion to its fractional counterpart. We also touch upon the general criteria required for resetting to improve the search process, particularly when distribution describing the time needed to reach the target is broader compared to a normal one. Building on this foundation, we delve into real-world applications where resetting optimizes the efficiency of reaching the desired outcome, spanning topics from home range search, ion transport to the intricate dynamics of income. Conclusively, the results presented in this chapter offer a cohesive perspective on the multifaceted influence of random resetting across diverse fields.Comment: Prepared as an invited chapter for the THE TARGET PROBLEM (Eds. D. S. Grebenkov, R. Metzler, G. Oshanin