New inverse problems for a time-switched system of wave and diffusion equations
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| Format: | Preprint |
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2026
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| author | Karimov, E. T. Murolimova, N. A. |
| author_facet | Karimov, E. T. Murolimova, N. A. |
| contents | We study two new classes of inverse problems for a time-switched system in which a fractional wave equation (with Caputo derivative of order $α\in (1,2)$) governs the dynamics on the interval $[0,a)$, and a fractional diffusion equation (with Caputo derivative of order $β\in (0,1)$ 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 $ξ\in (a,b)$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_24189 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | New inverse problems for a time-switched system of wave and diffusion equations Karimov, E. T. Murolimova, N. A. Analysis of PDEs Mathematical Physics 35M10, 35R30 We study two new classes of inverse problems for a time-switched system in which a fractional wave equation (with Caputo derivative of order $α\in (1,2)$) governs the dynamics on the interval $[0,a)$, and a fractional diffusion equation (with Caputo derivative of order $β\in (0,1)$ 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 $ξ\in (a,b)$. 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. |
| title | New inverse problems for a time-switched system of wave and diffusion equations |
| topic | Analysis of PDEs Mathematical Physics 35M10, 35R30 |
| url | https://arxiv.org/abs/2605.24189 |