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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.02145 |
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| _version_ | 1866914567753302016 |
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| author | Özsarı, Türker Mantzavinos, Dionyssios Kalimeris, Konstantinos |
| author_facet | Özsarı, Türker Mantzavinos, Dionyssios Kalimeris, Konstantinos |
| contents | We show that, for certain evolution partial differential equations, the solution on a finite interval $(0,\ell)$ can be reconstructed as a superposition of restrictions to $(0,\ell)$ of solutions to two associated partial differential equations posed on the half-lines $(0,\infty)$ and $(-\infty,\ell)$. Determining the appropriate data for these half-line problems amounts to solving an inverse problem, which we formulate via the unified transform of Fokas (also known as the Fokas method) and address via a fixed point argument in $L^2$-based Sobolev spaces, including fractional ones through interpolation techniques. We illustrate our approach through two canonical examples, the heat equation and the Korteweg-de Vries (KdV) equation, and provide numerical simulations for the former example. We further demonstrate that the new approach extends to more general evolution partial differential equations, including those with time-dependent coefficients. A key outcome of this work is that spatial and temporal regularity estimates for problems on a finite interval can be directly derived from the corresponding estimates on the half-line. These results can, in turn, be used to establish local well-posedness for related nonlinear problems, as the essential ingredients are the linear estimates within nonlinear frameworks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_02145 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A new approach for the analysis of evolution partial differential equations on a finite interval Özsarı, Türker Mantzavinos, Dionyssios Kalimeris, Konstantinos Analysis of PDEs 35G16, 35G31, 35Q53, 35K05 We show that, for certain evolution partial differential equations, the solution on a finite interval $(0,\ell)$ can be reconstructed as a superposition of restrictions to $(0,\ell)$ of solutions to two associated partial differential equations posed on the half-lines $(0,\infty)$ and $(-\infty,\ell)$. Determining the appropriate data for these half-line problems amounts to solving an inverse problem, which we formulate via the unified transform of Fokas (also known as the Fokas method) and address via a fixed point argument in $L^2$-based Sobolev spaces, including fractional ones through interpolation techniques. We illustrate our approach through two canonical examples, the heat equation and the Korteweg-de Vries (KdV) equation, and provide numerical simulations for the former example. We further demonstrate that the new approach extends to more general evolution partial differential equations, including those with time-dependent coefficients. A key outcome of this work is that spatial and temporal regularity estimates for problems on a finite interval can be directly derived from the corresponding estimates on the half-line. These results can, in turn, be used to establish local well-posedness for related nonlinear problems, as the essential ingredients are the linear estimates within nonlinear frameworks. |
| title | A new approach for the analysis of evolution partial differential equations on a finite interval |
| topic | Analysis of PDEs 35G16, 35G31, 35Q53, 35K05 |
| url | https://arxiv.org/abs/2511.02145 |