Generalized time-fractional kinetic-type equations with multiple parameters

TL;DR

This paper introduces a broad class of generalized fractional kinetic equations with multiple parameters, providing explicit solutions and linking them to stochastic processes, thus expanding the modeling capabilities of kinetic and telegraph-type equations.

Contribution

It presents a new multi-parameter generalization of kinetic equations, derives explicit solutions in the Laplace domain, and connects these solutions to stochastic process interpretations.

Findings

Explicit solutions for generalized fractional kinetic equations

Connection to stochastic processes via inverse subordinators

Special cases include generalized telegraph and diffusion equations

Abstract

In this paper we study a new generalization of the kinetic equation emerging in run-and-tumble models. We show that this generalization leads to a wide class of generalized fractional kinetic (GFK) and telegraph-type equations depending by two (or three) parameters. We provide an explicit expression of the solution in the Laplace domain and show that, for a particular choice of the parameters, the fundamental solution of the GFK equation can be interpreted as the probability density function of a stochastic process obtained by a suitable transformation of the inverse of a subordinator. Then, we discuss some particular interesting cases, such as generalized telegraph models, diffusion fractional equations involving higher order time derivatives and fractional integral equations.