Generalized time-fractional kinetic-type equations with multiple parameters

In this paper, we study a new generalization of the kinetic equation emerging in run-and-tumble models [see, e.g., Angelani et al., J. Stat. Phys. 191, 129 (2024) for a time-fractional version of the kinetic equation]. We show that this generalization leads to a wide class of generalized fractional kinetic (GFK) and telegraph-type equations that depend on 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 particularly interesting cases, such as generalized telegraph models, fractional diffusion equations involving higher order time derivatives, and fractional integral equations.