Related papers: Drift, diffusion, and third order derivatives in F…
Anomalous relaxation and diffusion processes have been widely characterized by fractional derivative models, where the definition of the fractional-order derivative remains a historical debate due to the singular memory kernel that…
In this work, we investigate a variational formulation for a time-fractional Fokker-Planck equation which arises in the study of complex physical systems involving anomalously slow diffusion. The model involves a fractional-order Caputo…
A class of linear evolutionary equations with material laws involving fractional time-derivatives is considered. The main result is well-posedness and causality for this problem class. The approach is illustrated with two examples: a…
The~numerical solutions to a non-linear Fractional Fokker--Planck (FFP) equation are studied estimating the generalized diffusion coefficients. The~aim is to model anomalous diffusion using an FFP description with fractional velocity…
Deriving evolution equations accounting for both anomalous diffusion and reactions is notoriously difficult, even in the simplest cases. In contrast to normal diffusion, reaction kinetics cannot be incorporated into evolution equations…
In the present paper, we address a class of the fractional derivatives of constant and variable orders for the first time. Fractional-order relaxation equations of constants and variable orders in the sense of Caputo type are modeled from…
Dynamics of platicons caused by the third-order dispersion is studied. It is shown that under the influence of the third-order dispersion platicons obtain angular velocity depending both on dispersion and on detuning value. A method of…
This paper is devoted to existence and uniqueness results for classes of nonlinear diffusion equations (or systems) which may be viewed as regular perturbations of Wasserstein gradient flows. First, in the case. where the drift is a…
Mathematical structure of the reflection coefficients for the one-dimensional Fokker-Planck equation is studied. A new formalism using differential operators is introduced and applied to the analysis in high- and low-energy regions.…
The well known scaling of the Edwards-Wilkinson equation is essentially determined by dimensional analysis. Once a drift term is added, more sophisticated reasoning is required, which initially suggests that the drift term dominates over…
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…
By replacing the spatial derivative with the Dunkl derivative, we generalize the Fokker-Planck equation in (1+1) dimensions. We obtain the Dunkl-Fokker-Planck eigenvalues equation and solve it for the harmonic oscillator plus a…
The Fokker-Planck equation for the probability $f(r,t)$ to find a random walker at position $r$ at time $t$ is derived for the case that the the probability to make jumps depends nonlinearly on $f(r,t)$. The result is a generalized form of…
The Fokker-Planck equation for a heavy particle in a granular fluid is derived from the Liouville equation. The host fluid is assumed to be in its homogeneous cooling state and all interactions are idealized as smooth, inelastic hard…
We present a method how to estimate from experimental data of a turbulent velocity field the drift and the diffusion coefficient of a Fokker-Planck equation. It is shown that solutions of this Fokker-Planck equation reproduce with high…
In this work we present the logarithmic diffusion equation as a limit case when the index that characterizes a nonlinear Fokker-Planck equation, in its diffusive term, goes to zero. A linear drift and a source term are considered in this…
We present a short overview of the recent results in the theory of diffusion and wave equations with generalised derivative operators. We give generic examples of such generalised diffusion and wave equations, which include time-fractional,…
The Fokker-Planck equation needs to be modified when the diffusion parameters are not constant, in order to match it with the backward equation. It even becomes simpler, and the integration of the underlying stochastic differential equation…
Motivated by empirical data, we develop a statistical description of the queue dynamics for large tick assets based on a two-dimensional Fokker-Planck (diffusion) equation, that explicitly includes state dependence, i.e. the fact that the…
This article is the exploration of the viewpoint within which propelled particles in a steady-state are regarded as a system with quenched disorder. The analogy is exact when the rate of the drift orientation vanishes and the linear…