2025/10/04 by S. E. Chorfi, Chorfi, S. E., Fouad Et-tahri +5
Mathematics · #Numerical methods in inverse problems #Differential Equations and Boundary Problems #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.2510.03600
We investigate forward and backward problems associated with abstract time-fractional Schrödinger equations iν∂tαu(t) + A u(t)=0, α∈ (0,1)∪ (1,2) and ν∈\1,α\, where A is a self-adjoint operator with compact resolvent on a Hilbert space H. This kind of equation, which incorporates the Caputo time-fractional derivative of order α, models quantum systems with memory effects and anomalous wave propagation. We first establish the well-posedness of the forward problems in two scenarios: (ν=1, α∈ (0,1)) and (ν=α, α∈ (0,1)∪ (1,2)). Then, we prove well-posedness and stability results for the backward problems depending on the two cases ν=1 and ν=α. Our approach employs the solution's eigenvector expansion along with the properties of the Mittag-Leffler functions, including the distribution of zeros and asymptotic expansions. Finally, we conclude with a discussion of some open problems.