INVERSE SOURCE PROBLEMS FOR GENERALIZED FRACTIONAL BOUNDARY VALUE MODELS WITH AB–CAPUTO LAPLACIAN OPERATORS

Abstract

This paper investigates two classes of inverse source problems associated with multi-parameter space–time fractional boundary value models involving the Atangana–Baleanu–Caputo (ABC) Laplacian operator. The first model concerns the recovery of a space-dependent source term from over-specified final data, while the second model addresses the reconstruction of a time-dependent source factor from additional integral conditions. A sequence of lemmas establishes the analytical properties of the generalized fractional kernels and their Laplace representations. Building on these results, existence, uniqueness, and ill-posedness theorems are derived for both inverse models under appropriate regularity assumptions. Several illustrative examples demonstrate how the theoretical framework can be applied to recover unknown sources in boundary value problems with fractional dynamics. The results highlight the flexibility of ABC-based operators in capturing memory effects and offer a foundation for future numerical schemes and practical applications in anomalous diffusion, viscoelasticity, and related fields.