New inverse problems for a time-switched system of wave and diffusion equations
Abstract
We study two new classes of inverse problems for a time-switched system in which a fractional wave equation (with Caputo derivative of order ) governs the dynamics on the interval , and a fractional diffusion equation (with Caputo derivative of order taken with respect to the switching point ) governs the dynamics on . The two problems differ in which part of the transmitting condition at the interface is regarded as unknown. In both cases the overdetermination data consist of a single spatial measurement of the solution at a fixed time . Using the spectral expansion method with respect to the classical Sturm-Liouville eigensystem on , we reduce each problem to a sequence of coupled scalar Cauchy problems involving the two-parameter Mittag-Leffler function. Explicit series representations for the solution and the unknown interface functions and are derived. Uniform convergence of the resulting infinite series and their relevant derivatives is established through four auxiliary lemmas, using the decay estimates for the Mittag-Leffler function, integration-by-parts arguments, the Cauchy--Schwarz inequality, and the Weierstrass -test. A uniqueness and existence theorem is stated for Problem~1 under explicit Sobolev-type regularity conditions on the data, with an analogous result for Problem~2.
Cite
@article{arxiv.2605.24189,
title = {New inverse problems for a time-switched system of wave and diffusion equations},
author = {E. T. Karimov and N. A. Murolimova},
journal= {arXiv preprint arXiv:2605.24189},
year = {2026}
}
Comments
16 pages, 1 figure