Symmetry-Preserving Finite-Difference Schemes and Auto-Bäcklund Transformations for the Schwarz Equation
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917967673950208 |
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| author | Kaptsov, E. I. Dorodnitsyn, V. A. |
| author_facet | Kaptsov, E. I. Dorodnitsyn, V. A. |
| contents | It is demonstrated that one of the equations from the Lie classification list of second-order ODEs is a first integral of the Schwarz equation. As symmetry-preserving finite-difference schemes have been previously constructed for both equations, the preservation of a similar connection between these schemes is studied. It is shown that the schemes for the Schwarz equation and the second-order ODE (with an arbitrary constant $C$) can be related through a Bäcklund-type difference transformation. In addition, previously unexamined aspects of the difference scheme for the second-order ODE are discussed, including its singular solution and the complete set of difference first integrals for the case $C^2=4$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_03369 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Symmetry-Preserving Finite-Difference Schemes and Auto-Bäcklund Transformations for the Schwarz Equation Kaptsov, E. I. Dorodnitsyn, V. A. Numerical Analysis 65L12, 34C14 It is demonstrated that one of the equations from the Lie classification list of second-order ODEs is a first integral of the Schwarz equation. As symmetry-preserving finite-difference schemes have been previously constructed for both equations, the preservation of a similar connection between these schemes is studied. It is shown that the schemes for the Schwarz equation and the second-order ODE (with an arbitrary constant $C$) can be related through a Bäcklund-type difference transformation. In addition, previously unexamined aspects of the difference scheme for the second-order ODE are discussed, including its singular solution and the complete set of difference first integrals for the case $C^2=4$. |
| title | Symmetry-Preserving Finite-Difference Schemes and Auto-Bäcklund Transformations for the Schwarz Equation |
| topic | Numerical Analysis 65L12, 34C14 |
| url | https://arxiv.org/abs/2503.03369 |