Asymptotic expansions for the solution of a linear PDE with a multifrequency highly oscillatory potential
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866909255997587456 |
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| author | Perczyński, Rafał Augustynowicz, Antoni |
| author_facet | Perczyński, Rafał Augustynowicz, Antoni |
| contents | Highly oscillatory differential equations present significant challenges in numerical treatments. The Modulated Fourier Expansion (MFE), used as an ansatz, is a commonly employed tool as a numerical approximation method. In this article, the Modulated Fourier Expansion is analytically derived for a linear partial differential equation with a multifrequency highly oscillatory potential. The solution of the equation is expressed as a convergent Neumann series within the appropriate Sobolev space. The proposed approach enables, firstly, to derive a general formula for the error associated with the approximation of the solution by MFE, and secondly, to determine the coefficients for this expansion -- without the need to solve numerically the system of differential equations to find the coefficients of MFE. Numerical experiments illustrate the theoretical investigations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_14650 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Asymptotic expansions for the solution of a linear PDE with a multifrequency highly oscillatory potential Perczyński, Rafał Augustynowicz, Antoni Numerical Analysis 65M38, 65M70, 65M99 Highly oscillatory differential equations present significant challenges in numerical treatments. The Modulated Fourier Expansion (MFE), used as an ansatz, is a commonly employed tool as a numerical approximation method. In this article, the Modulated Fourier Expansion is analytically derived for a linear partial differential equation with a multifrequency highly oscillatory potential. The solution of the equation is expressed as a convergent Neumann series within the appropriate Sobolev space. The proposed approach enables, firstly, to derive a general formula for the error associated with the approximation of the solution by MFE, and secondly, to determine the coefficients for this expansion -- without the need to solve numerically the system of differential equations to find the coefficients of MFE. Numerical experiments illustrate the theoretical investigations. |
| title | Asymptotic expansions for the solution of a linear PDE with a multifrequency highly oscillatory potential |
| topic | Numerical Analysis 65M38, 65M70, 65M99 |
| url | https://arxiv.org/abs/2310.14650 |