An Inverse Random Source Problem for a Time Fractional Fourth‐Order Parabolic Equation Under Samarskii–Ionkin Boundary Condition

ABSTRACT This paper is devoted to solve an inverse random source problem for the fourth‐order partial differential equation under Samarskii–Ionkin boundary condition, where the source is driven by a fractional Brownian motion (fBm). The fractional time derivative is described in the Hilfer sense. The regularity of the direct problem, along with the uniqueness, instability, and conditional stability of the inverse source problem, is proved. To address this instability, we apply a generalized quasi‐reversibility regularization method (GQRRM), achieving stable numerical solutions and giving error estimates. Finally, numerical experiments demonstrate the effectiveness of the regularization method.