Solutions of differential equations in Freud-weighted Sobolev spaces
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866914319879372800 |
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| author | Breden, Maxime Chu, Hugo |
| author_facet | Breden, Maxime Chu, Hugo |
| contents | We lay some mathematically rigorous foundations for the resolution of differential equations with respect to semi-classical bases and topologies, namely Freud-Sobolev polynomials and spaces. In this quest, we uncover an elegant theory melding various topics in Numerical and Functional Analysis: Poincaré inequalities, Sobolev orthogonal polynomials, Painlevé equations and more. Brought together, these ingredients allow us to quantify the compactness of Sobolev embeddings on Freud-weighted spaces and finally resolve some differential equations in this topology. As an application, we rigorously and tightly enclose solutions of the Gross-Pitaevskii equation with sextic potential. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_13672 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Solutions of differential equations in Freud-weighted Sobolev spaces Breden, Maxime Chu, Hugo Numerical Analysis Analysis of PDEs Classical Analysis and ODEs Functional Analysis 35B45, 42C05, 46B50, 46E20, 46E35, 47B36, 41A81, 35J61, 65G20, 65Q30, 65N15 We lay some mathematically rigorous foundations for the resolution of differential equations with respect to semi-classical bases and topologies, namely Freud-Sobolev polynomials and spaces. In this quest, we uncover an elegant theory melding various topics in Numerical and Functional Analysis: Poincaré inequalities, Sobolev orthogonal polynomials, Painlevé equations and more. Brought together, these ingredients allow us to quantify the compactness of Sobolev embeddings on Freud-weighted spaces and finally resolve some differential equations in this topology. As an application, we rigorously and tightly enclose solutions of the Gross-Pitaevskii equation with sextic potential. |
| title | Solutions of differential equations in Freud-weighted Sobolev spaces |
| topic | Numerical Analysis Analysis of PDEs Classical Analysis and ODEs Functional Analysis 35B45, 42C05, 46B50, 46E20, 46E35, 47B36, 41A81, 35J61, 65G20, 65Q30, 65N15 |
| url | https://arxiv.org/abs/2501.13672 |