Related papers: Numerical simulations of anomalous diffusion
Modeling of phenomena such as anomalous transport via fractional-order differential equations has been established as an effective alternative to partial differential equations, due to the inherent ability to describe large-scale behavior…
Uchaikin suggested a mathematical model of an anomalous diffusion in a space was suggested. This model origins in an investigation of processes in complex systems with variable structure: glasses, liquid crystals, biopolymers, proteins and…
Time fractional advection-dispersion equations arise as generalizations of classical integer order advection-dispersion equations and are increasingly used to model fluid flow problems through porous media. In this paper we develop an…
The aim of this paper is to develop and analyze numerical schemes for approximately solving the backward problem of subdiffusion equation involving a fractional derivative in time with order $\alpha\in(0,1)$. After using quasi-boundary…
In this paper, we deal with analysis of the initial-boundary value problems for the semilinear time-fractional diffusion equations, while the case of the linear equations was considered in the first part of the present work. These equations…
Over the last two decades, anomalous diffusion processes in which the mean squares variance grows slower or faster than that in a Gaussian process have found many applications. At a macroscopic level, these processes are adequately…
In the recent literature, the g-subdiffusion equation involving Caputo fractional derivatives with respect to another function has been studied in relation to anomalous diffusions with a continuous transition between different subdiffusive…
This study addresses the inverse source problem for the fractional diffusion-wave equation, characterized by a source comprising spatial and temporal components. The investigation is primarily concerned with practical scenarios where data…
A Carleman estimate and the unique continuation of solutions for an anomalous diffusion equation with fractional time derivative of order $0<\alpha<1$ are given. The estimate is derived via some subelliptic estimate for an operator…
We present a series of results focused on the decay in time of solutions of classical and anomalous diffusive equations in a bounded domain. The size of the solution is measured in a Lebesgue space, and the setting comprises time-fractional…
This review article aims to stress and reunite some of the analytic formalism of the anomalous diffusive processes that have succeeded in their description. Also, it has the objective to discuss which of the new directions they have taken…
Recently, the fractional Fokker-Planck equations (FFPEs) with multiple internal states are built for the particles undergoing anomalous diffusion with different waiting time distributions for different internal states, which describe the…
The main objective of this paper is analysis of the initial-boundary value problems for the linear time-fractional diffusion equations with a uniformly elliptic spatial differential operator of the second order and the Caputo type…
In this paper invariant subspace method has been employed for solving linear and non-linear fractional partial differential equations involving Caputo derivative. A variety of illustrative examples are solved to demonstrate the…
In this paper we propose an algorithm for the numerical solution of arbitrary differential equations of fractional order. The algorithm is obtained by using the following decomposition of the differential equation into a system of…
The generalized diffusion equations with fractional order derivatives have shown be quite efficient to describe the diffusion in complex systems, with the advantage of producing exact expressions for the underlying diffusive properties.…
We study an initial-boundary value problem of variable-order time-fractional diffusion equations in one space dimension. Based on the wellposedness of the proposed model and the smoothing properties of its solutions, which are shown to be…
Solutions of boundary value problems for a diffusion equation of fractional and variable order in differential and difference settings are studied. It is shown that the method of energy inequalities is applicable to obtaining a priori…
The application of the approximation-operational approach to solving linear differential equations of fractional order with variable coefficients is considered. It is shown that the method can also be applied to solving differential…
A numerical method to solve the fractional diffusion equation, which could also be easily extended to many other fractional dynamics equations, is considered. These fractional equations have been proposed in order to describe anomalous…