English

New inverse problems for a time-switched system of wave and diffusion equations

Analysis of PDEs 2026-05-26 v1 Mathematical Physics math.MP

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 α(1,2)\alpha \in (1,2)) governs the dynamics on the interval [0,a)[0,a), and a fractional diffusion equation (with Caputo derivative of order β(0,1)\beta \in (0,1) taken with respect to the switching point t=at=a) governs the dynamics on (a,b](a,b]. The two problems differ in which part of the transmitting condition at the interface t=at=a is regarded as unknown. In both cases the overdetermination data consist of a single spatial measurement of the solution at a fixed time ξ(a,b)\xi \in (a,b). Using the spectral expansion method with respect to the classical Sturm-Liouville eigensystem on [0,1][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)u(t,x) and the unknown interface functions h(x)h(x) and hˉ(x)\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 MM-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.

Keywords

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

R2 v1 2026-07-22T07:29:24.956Z