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Generalized time-fractional kinetic-type equations with multiple parameters

Luca Angelani, Alessandro De Gregorio, Roberto Garra

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.

Generalized time-fractional kinetic-type equations with multiple parameters

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.

Paper Structure

This paper contains 8 sections, 65 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Generalized run-and-tumble model and the fractional kinetic equation
  3. Stochastic solutions to the generalized telegraph-type equation
  4. Particular interesting cases
  5. The generalized telegraph equation
  6. Diffusive equations involving higher order time derivatives
  7. The fractional integral equation ($\eta<0$)
  8. Conclusions

Figures (1)

  • Figure 1: Range of values of exponents $(\eta,\epsilon)$ of the model, see (\ref{['Pks']}), (\ref{['tel']}) and (\ref{['Pxs']}). The coloured area represents the region $\{(\eta,\epsilon) : 0<\eta<2 , 0<\epsilon< \min(1,2-\eta)\}$ where the function $P(x,t)$ can be interpreted as a PDF of a random motion.

Theorems & Definitions (2)

  • Remark 1
  • Remark 2