Reconstructing a Random Source for a Stochastic Equation With Biharmonic Operator With Fractional White Noise

At present, inverse random problems are receiving increasing attention, especially the case of inverse random sources. Here, we are interested in an inverse random source problem with biharmonic operator. The same as the existing studies of inverse random source problems, some theoretical results of the corresponding direct and inverse problems have been investigated in this paper. The difference is that the source term is driven by time-fractional Brownian motion, and the spatial function in the source term needs to be determined. It has been found that, for different problems, the properties of the integrand in the corresponding stochastic integrals are distinct, which has a significant impact on the entire theoretical analysis. For the problem considered here, the lack of monotonicity in the integrand of the stochastic integrals makes the analysis more challenging. Additionally, biharmonic operators will cause some difficulties in numerical calculations.