Reconstruction of the Time-Dependent Coefficient in a Time-Space Fractional Parabolic Equation Subject to Coupled Fractional Robin Boundary Conditions

This study investigates the inverse coefficient problem for a time–space fractional parabolic equation (TSFPE) subject to coupled fractional Robin boundary conditions. By employing a fractional power series expansion, the work provides a constructive framework for analyzing the identification of the time-dependent coefficient associated with the Caputo fractional derivative. The proposed approach offers a methodological contribution within this setting, though the reconstruction of the unknown coefficient is achieved under the assumptions and structural constraints specified in the formulation. The proposed methodology employs a double series expansion in fractional powers of both spatial and temporal variables. This formulation yields a system of recurrence relations for the expansion coefficients, enabling the reconstruction of the unknown coefficient of time in the sense of the Caputo derivative. A distinctive aspect of this work is its comprehensive presentation of detailed mathematical derivations. The principal methodological contribution is the introduction of an alternative analytical approach for identifying unknown coefficients of time in complex TSFPEs. Unlike conventional Taylor-based techniques that fail in the presence of derivative singularities, the proposed method naturally captures non-analytic behaviors and accurately reconstructs the unknown time-dependent coefficients. Numerical example 5, including cases with non-analytic exact solutions, demonstrates the high efficiency, robustness, and superior applicability of the developed approach. A rigorous convergence analysis of the proposed method is provided, substantiating its theoretical validity and accuracy. The effectiveness and high precision of the method are demonstrated through illustrative numerical examples, underscoring its practical applicability and robustness in solving such inverse coefficient problems.