An inverse random source problem for the time-space fractional diffusion equation driven by fractional Brownian motion

We study the inverse random source problem for the time-space fractional diffusion equation driven by fractional Brownian motion with Hurst index H is an element of (0, 1). With the aid of a novel estimate, by using the opera-tor approach we propose regularity analyses for the direct problem. Then we provide a reconstruction scheme for the source terms f and g up to sign. Next, combining the properties of Mittag-Leffler function, the complete uniqueness and instability analyses are provided. It is worth mentioning that all the analyses are unified for H is an element of (0, 1).

Automated summary

This paper studies the inverse random source problem for the time-space fractional diffusion equation driven by fractional Brownian motion with Hurst index H in the interval (0,1). Using a novel estimate and the operator approach, regularity analyses for the direct problem are proposed. A reconstruction scheme for the source terms f and g up to sign is provided. Furthermore, complete uniqueness and instability analyses are presented by combining the properties of the Mittag-Leffler function. It is noteworthy that the analyses are unified for any H in the interval (0,1).