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 $\alpha \in (1,2)$) governs the dynamics on the interval $[0,a)$, and a fractional diffusion equation (with Caputo derivative of order $\beta \in (0,1)$ taken with respect to the switching point $t=a$) governs the dynamics on $(a,b]$. The two problems differ in which part of the transmitting condition at the interface $t=a$ is regarded as unknown. In both cases the overdetermination data consist of a single spatial measurement of the solution at a fixed time $\xi \in (a,b)$. Using the spectral expansion method with respect to the classical Sturm-Liouville eigensystem on $[0,1]$, 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 $u(t,x)$ and the unknown interface functions $h(x)$ and $\bar{h}(x)$ 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 $M$-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.
| Comments: | 16 pages, 1 figure |
| Subjects: | Analysis of PDEs (math.AP); Mathematical Physics (math-ph) |
| MSC classes: | 35M10, 35R30 |
| Cite as: | arXiv:2605.24189 [math.AP] |
| (or arXiv:2605.24189v1 [math.AP] for this version) | |
| https://doi.org/10.48550/arXiv.2605.24189 arXiv-issued DOI via DataCite (pending registration) |
From: Erkinjon Karimov [view email]
[v1]
Fri, 22 May 2026 20:26:59 UTC (1,137 KB)
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