Well-posedness of the stochastic time-fractional diffusion and wave equations and inverse random source problems

TL;DR

This paper investigates the mathematical properties of stochastic time-fractional diffusion and wave equations, establishing existence, uniqueness, and regularity of solutions, and applies these results to prove the uniqueness of an inverse source problem.

Contribution

It provides new theoretical results on the well-posedness of stochastic time-fractional equations and addresses an inverse source problem in this context.

Findings

Proved existence and uniqueness of stochastic weak solutions.

Established regularity estimates for solutions.

Demonstrated the uniqueness of the inverse source problem.

Abstract

In this paper, we are concerned with the stochastic time-fractional diffusion-wave equations in a Hilbert space. The main objective of this paper is to establish properties of the stochastic weak solutions of the initial-boundary value problem, such as the existence, uniqueness and regularity estimates. Moreover, we apply the obtained theories to an inverse source problem. The uniqueness of this inverse problem under the boundary measurements is proved.