A Novel Approach to the Inverse Problem of Identifying Fractional Derivative Orders in Partial Differential Equations

Determining the order of fractional derivatives in partial differential equations (PDEs) presents a significant challenge in applied mathematics. Fractional derivatives play a crucial role in accurately modeling complex physical phenomena, including anomalous diffusion, wave propagation, viscoelasticity, and various fluid dynamic processes. Unlike classical derivatives, fractional-order models account for memory and nonlocal effects, making them highly relevant for real-world systems. However, the direct measurement of the fractional order is typically impractical due to the lack of appropriate instruments or experimental techniques. As a result, this leads to an inverse problem—finding the unknown fractional order from available, often indirect, observational data of the solution to the governing PDE.

Recent studies have made substantial progress in analyzing such inverse problems, particularly for time-fractional subdiffusion equations, where uniqueness results have been rigorously established. Nevertheless, many physical processes cannot be fully captured by subdiffusion models alone. This presentation explores a newly developed inverse problem formulation that extends the analysis to a broader class of equations, including fractional-wave equations, Rayleigh-Stokes-type models, and mixed-type fractional PDEs.

Our approach not only ensures the uniqueness of the estimated fractional order but also proves the existence of solutions under suitable conditions. This provides a more complete and versatile theoretical framework with strong potential for applications in engineering, physics, and beyond.