Liouvillian Solutions of Third Order Differential Equations
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866914670974074880 |
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| author | Sanabria, Camilo Combot, Thierry |
| author_facet | Sanabria, Camilo Combot, Thierry |
| contents | Consider a third order linear differential equation $L(f)=0$, where $L\in\mathbb{Q}(z)[\partial_z]$. We design an algorithm computing the Liouvillian solutions of $L(f)=0$. The reducible cases devolve to the classical case of second order operators, and in the irreducible cases, only finitely many differential Galois groups are possible. The differential Galois group is obtained through optimized computations of invariants and semi-invariants, and if solvable, the solutions are returned as pullbacks and gauge transformations of algebraic generalized hypergeometric function ${}_3F_2$. The computation time is practical for reasonable size operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_05143 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Liouvillian Solutions of Third Order Differential Equations Sanabria, Camilo Combot, Thierry Classical Analysis and ODEs 34M15 Consider a third order linear differential equation $L(f)=0$, where $L\in\mathbb{Q}(z)[\partial_z]$. We design an algorithm computing the Liouvillian solutions of $L(f)=0$. The reducible cases devolve to the classical case of second order operators, and in the irreducible cases, only finitely many differential Galois groups are possible. The differential Galois group is obtained through optimized computations of invariants and semi-invariants, and if solvable, the solutions are returned as pullbacks and gauge transformations of algebraic generalized hypergeometric function ${}_3F_2$. The computation time is practical for reasonable size operators. |
| title | Liouvillian Solutions of Third Order Differential Equations |
| topic | Classical Analysis and ODEs 34M15 |
| url | https://arxiv.org/abs/2402.05143 |