A unified way to solve IVPs and IBVPs for the time-fractional diffusion-wave equation

TL;DR

This paper presents a unified approach to solving initial and initial-boundary value problems for the time-fractional diffusion-wave equation, covering both Caputo and Riemann-Liouville derivatives, and introduces a new auxiliary function with applications to probability distributions.

Contribution

It introduces a unified method for solving various fractional diffusion-wave equations and explores a new auxiliary function with probabilistic applications.

Findings

Unified solution framework for IVPs and IBVPs

Introduction of a two-parameter auxiliary function

New family of probability distributions including normal

Abstract

The time-fractional diffusion-wave equation is revisited, where the time derivative is of order $2 \nu$ and $0 < \nu \le 1$. The behaviour of the equation is "diffusion-like" (respectively, "wave-like") when $0 < \nu \le \frac{1}{2}$ (respectively, $\frac{1}{2} < \nu \le 1$). Two types of time-fractional derivatives are considered, namely the Caputo and Riemann-Liouville derivatives. Initial value problems and initial-boundary value problems are investigated and handled in a unified way using an embedding method. A two-parameter auxiliary function is introduced and its properties are investigated. The time-fractional diffusion equation is used to generate a new family of probability distributions, and that includes the normal distribution as a particular case.