Subdiffusion with particle immobilization process described by differential equation with Riemann--Liouville type fractional time derivative

TL;DR

This paper derives a fractional differential equation for subdiffusion with particle immobilization using the continuous time random walk model, providing a method to compute the inverse Laplace transform and analyzing the long-term stationary state.

Contribution

It introduces a new fractional differential equation with a Riemann-Liouville derivative for modeling subdiffusion with immobilization, including a novel inverse Laplace transform calculation method.

Findings

The process reaches a stationary exponential distribution over time.

The derived equation accurately models particle immobilization in subdiffusion.

A new approach for calculating the inverse Laplace transform kernel is proposed.

Abstract

An equation describing subdiffusion with possible immobilization of particles is derived by means of the continuous time random walk model. The equation contains a fractional time derivative of Riemann--Liouville type which is a differential-integral operator with the kernel defined by the Laplace transform. We propose the method for calculating the inverse Laplace transform providing the kernel in the time domain. In the long time limit the subdiffusion--immobilization process reaches a stationary state in which the probability density of a particle distribution is an exponential function.