Numerical Approximation of the logarithmic Laplacian via sinc-basis
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916949852684288 |
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| author | Dondl, Patrick Striet, Ludwig |
| author_facet | Dondl, Patrick Striet, Ludwig |
| contents | In recent works, the authors of this chapter have shown with co-authors how a basis consisting of dilated and shifted $\text{sinc}$-functions can be used to solve fractional partial differential equations. As a model problem, the fractional Dirichlet problem with homogeneous exterior value conditions was solved. In this work, we briefly recap the algorithms developed there and that -- from a computational point of view -- they can be used to solve nonlocal equations given through different operators as well. As an example, we numerically solve the Dirichlet problem for the logarithmic Laplacian $\log(-Δ)$ which has the Fourier symbol $\log(\left|ω\right|^2)$ and compute its Eigenvalues on disks with different radii in $\mathbb R^2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_11693 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Numerical Approximation of the logarithmic Laplacian via sinc-basis Dondl, Patrick Striet, Ludwig Numerical Analysis 35R11, 65N35 G.1 In recent works, the authors of this chapter have shown with co-authors how a basis consisting of dilated and shifted $\text{sinc}$-functions can be used to solve fractional partial differential equations. As a model problem, the fractional Dirichlet problem with homogeneous exterior value conditions was solved. In this work, we briefly recap the algorithms developed there and that -- from a computational point of view -- they can be used to solve nonlocal equations given through different operators as well. As an example, we numerically solve the Dirichlet problem for the logarithmic Laplacian $\log(-Δ)$ which has the Fourier symbol $\log(\left|ω\right|^2)$ and compute its Eigenvalues on disks with different radii in $\mathbb R^2$. |
| title | Numerical Approximation of the logarithmic Laplacian via sinc-basis |
| topic | Numerical Analysis 35R11, 65N35 G.1 |
| url | https://arxiv.org/abs/2509.11693 |