An inverse random source problem for the time-space fractional diffusion equation driven by fractional Brownian motion
Daxin Nie, Weihua Deng
Abstract
Abstract We study the inverse random source problem for the time-space fractional diffusion equation driven by fractional Brownian motion with Hurst index <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>H</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {H\in(0,1)} . With the aid of a novel estimate, by using the operator 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>H</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {H\in(0,1)} .