# Generalized diffusion-wave equation with memory kernel

## TL;DR

This paper introduces a generalized diffusion-wave equation with various memory kernels, including fractional derivatives, to model diverse anomalous diffusion processes and transitions between diffusion regimes.

## Contribution

It develops a unified framework for generalized diffusion-wave equations with different memory kernels, deriving fundamental solutions and analyzing their properties.

## Key findings

- Derived fundamental solutions for various kernels.
- Established conditions for non-negativity of solutions.
- Calculated mean squared displacement for all cases.

## Abstract

We study generalized diffusion-wave equation in which the second order time derivative is replaced by integro-differential operator. It yields time fractional and distributed order time fractional diffusion-wave equations as particular cases. We consider different memory kernels of the integro-differential operator, derive corresponding fundamental solutions, specify the conditions of their non-negativity and calculate mean squared displacement for all cases. In particular, we introduce and study generalized diffusion-wave equations with regularized Prabhakar derivative of single and distributed orders. The equations considered can be used for modeling broad spectrum of anomalous diffusion processes and various transitions between different diffusion regimes.