Subordination approach to multi-term time-fractional diffusion-wave equations

TL;DR

This paper investigates multi-term time-fractional diffusion-wave equations with Caputo derivatives, establishing fundamental solution properties, a subordination principle, and providing explicit representations and numerical validation.

Contribution

It introduces a subordination approach for multi-term fractional equations, proving solution existence, nonnegativity, and deriving explicit probability density functions.

Findings

Fundamental solutions are nonnegative and exhibit specific propagation speeds.

A subordination principle links solutions to cosine families and probability densities.

Explicit formulas for probability densities enable regularity analysis and numerical validation.

Abstract

This paper is concerned with the fractional evolution equation with a discrete distribution of Caputo time-derivatives such that the largest and the smallest orders, $\alpha$ and $\alpha_m$, satisfy the conditions $1<\alpha\le 2$ and $\alpha-\alpha_m\le 1$. First, based on a study of the related propagation function, the nonnegativity of the fundamental solutions to the spatially one-dimensional Cauchy and signaling problems is proven and propagation speed of a disturbance is discussed. Next, we study the equation with a general linear spatial differential operator defined in a Banach space and suppose it generates a cosine family. A subordination principle is established, which implies the existence of a unique solution and gives an integral representation of the solution operator in terms of the corresponding cosine family and a probability density function. Explicit representation of the probability density function is derived. The subordination principle is applied for obtaining regularity results. The analytical findings are supported by numerical work.