Hilbert Space Analysis of an Inverse Problem for a Caputo Time-Fractional Evolution Equation with Spatial Derivatives and Involution

In this study, we develop a two-variable fractional differential equation with respect to the temporal variable and the spatial variable , where the derivative is fractional in time using the Caputo derivative and classical in space. The equation is derived from the differential equation of orthogonal Chebyshev polynomials, with a perturbation introduced via involution, representing a generalization of standard fractional models.

The equation is solved using the separation of variables method, representing the solution as an infinite series with Chebyshev polynomials forming a basis in the weighted Hilbert space . This leads to a spectral problem in the spatial variable , from which the eigenvalues and eigenfunctions are determined. The main problem is then addressed through the series expansion, resulting in a linear fractional equation previously studied in the literature, yielding an explicit analytical solution suitable for theoretical analysis.

To ensure stability and differentiability, temporal boundary conditions are imposed, which are verified through the convergence of the solution series. Finally, the uniqueness of the solution is established based on the completeness of the Hilbert basis and the initial condition at t=0 .

This methodology can be generalized to other polynomial bases, such as Legendre or Hermite polynomials, and extended to construct other fractional equations using alternative derivatives, including Riemann–Liouville or Letnikov derivatives, offering a flexible framework for modeling complex time-space fractional systems.