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An unsteady problem is considered for a space-fractional diffusion equation in a bounded domain. A first-order evolutionary equation containing a fractional power of an elliptic operator of second order is studied for general boundary…
We present a numerical method that consistently implements thermal fluctuations and hydrodynamic interactions to the motion of Brownian particles dispersed in incompressible host fluids. In this method, the thermal fluctuations are…
Brownian motion in the three dimensional Lobachevsky space or hyperbolic space is considered in the paper written by F.I.Karpelevich, V.N.Tutubalin and M.G.Shur. A solution for radial symmetric diffusion equation in the three dimensional…
Fractional, anomalous diffusion in space-periodic potentials is investigated. The analytical solution for the effective, fractional diffusion coefficient in an arbitrary periodic potential is obtained in closed form in terms of two…
The effective diffusion of Brownian particles in periodic potential has been a central topic in nonequilibrium statistical physcis. A classical result is the Lifson formula which provides the effective diffusion constant in periodic…
The paper investigates solutions of the fractional hyperbolic diffusion equation in its most general form with two fractional derivatives of distinct orders. The solutions are given as spatial-temporal homogeneous and isotropic random…
Fractional diffusion has become a fundamental tool for the modeling of multiscale and heterogeneous phenomena. However, due to its nonlocal nature, its accurate numerical approximation is delicate. We survey our research program on the…
We consider the fully non-local diffusion equations with non-negative $L^1$-data. Based on the approximation and energy methods, we prove the existence and uniqueness of non-negative entropy solutions for such problems. In particular, our…
We analyze numerically a forward-backward diffusion equation with a cubic-like diffusion function, -emerging in the framework of phase transitions modeling- and its "entropy" formulation determined by considering it as the singular limit of…
A system of interacting Brownian particles subject to short-range repulsive potentials is considered. A continuum description in the form of a nonlinear diffusion equation is derived systematically in the dilute limit using the method of…
By means of a particle model that includes interactions only via the local particle concentration, we show that hyperballistic diffusion may result. This is done by findng the exact solution of the corresponding non-linear diffusion…
In this paper we investigate a sub-diffusion equation for simulating the anomalous diffusion phenomenon in real physical environment. Based on an equivalent transformation of the original sub-diffusion equation followed by the use of a…
In this paper we present numerical methods - finite differences and finite elements - for solution of partial differential equation of fractional order in time for one-dimensional space. This equation describes anomalous diffusion which is…
The diffusion of chiral active Brownian particles in three-dimensional space is studied analytically, by consideration of the corresponding Fokker-Planck equation for the probability density of finding a particle at position…
We analyze the diffusion of a Brownian particle in a fluid under stationary flow. By using the scheme of non-equilibrium thermodynamics in phase space, we obtain the Fokker-Planck equation which is compared with others derived from kinetic…
The Brownian motion of a particle in a one-dimensional periodic potential subjected to a uniform external force F is studied. Using the formula for the diffusion coefficient D obtained by other authors and an alternative one derived from…
The one-dimensional overdamped Brownian motion in a symmetric periodic potential modulated by external time-reversible noise is analyzed. The calculation of the effective diffusion coefficient is reduced to the mean first passage time…
In this work, we are interested in building the fully discrete scheme for stochastic fractional diffusion equation driven by fractional Brownian sheet which is temporally and spatially fractional with Hurst parameters $H_{1}, H_{2}…
Recent progress in experimental techniques such as single particle tracking allows to analyze both nonequilibrium properties and approach to equilibrium. There are examples showing that processes occurring at finite timescales are…
The fundamental solutions of diffusion equation for the local-equilibrium and nonlocal models are considered as the limiting cases of the solution of a problem related to consideration of the Brownian particles random walks. The differences…