Nonlinear numerical schemes using specular differentiation for initial value problems of first-order ordinary differential equations
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866909008959373312 |
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| author | Jung, Kiyuob |
| author_facet | Jung, Kiyuob |
| contents | This paper proposes specular differentiation in one-dimensional Euclidean space and provides its fundamental analysis, including a quasi-Fermat theorem and a quasi-Mean Value Theorem. As an application, this paper develops several numerical schemes for solving initial value problems for first-order ordinary differential equations. Based on numerical simulations, we select one scheme and prove its second-order consistency and convergence. By modifying this scheme, we also obtain a numerical scheme with zero local truncation error for ODEs whose solution trajectories are ellipses. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_09900 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Nonlinear numerical schemes using specular differentiation for initial value problems of first-order ordinary differential equations Jung, Kiyuob Numerical Analysis 49J52, 65L05, 65L20, 65L12 This paper proposes specular differentiation in one-dimensional Euclidean space and provides its fundamental analysis, including a quasi-Fermat theorem and a quasi-Mean Value Theorem. As an application, this paper develops several numerical schemes for solving initial value problems for first-order ordinary differential equations. Based on numerical simulations, we select one scheme and prove its second-order consistency and convergence. By modifying this scheme, we also obtain a numerical scheme with zero local truncation error for ODEs whose solution trajectories are ellipses. |
| title | Nonlinear numerical schemes using specular differentiation for initial value problems of first-order ordinary differential equations |
| topic | Numerical Analysis 49J52, 65L05, 65L20, 65L12 |
| url | https://arxiv.org/abs/2601.09900 |