Exponential quadrature rules for problems with time-dependent fractional source
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915357917184000 |
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| author | Caliari, Marco Cassini, Fabio |
| author_facet | Caliari, Marco Cassini, Fabio |
| contents | In this manuscript, we propose newly-derived exponential quadrature rules for stiff linear differential equations with time-dependent fractional sources in the form $h(t^r)$, with $0<r<1$ and $h$ a sufficiently smooth function. To construct the methods, the source term is interpolated at $ν$ collocation points by a suitable non-polynomial function, yielding to time marching schemes that we call Exponential Quadrature Rules for Fractional sources (EQRF$ν$). The error analysis is done in the framework of strongly continuous semigroups. Compared to classical exponential quadrature rules, which in our case of interest converge with order $1+r$ at most, we prove that the new methods may reach order $1+νr$ for proper choices of the collocation points. We also show that the proposed integrators can be written in terms of special instances of the Mittag--Leffler functions that we call fractional $φ$ functions. Several numerical experiments demonstrate the theoretical findings and highlight the effectiveness of the approach. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_18395 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exponential quadrature rules for problems with time-dependent fractional source Caliari, Marco Cassini, Fabio Numerical Analysis In this manuscript, we propose newly-derived exponential quadrature rules for stiff linear differential equations with time-dependent fractional sources in the form $h(t^r)$, with $0<r<1$ and $h$ a sufficiently smooth function. To construct the methods, the source term is interpolated at $ν$ collocation points by a suitable non-polynomial function, yielding to time marching schemes that we call Exponential Quadrature Rules for Fractional sources (EQRF$ν$). The error analysis is done in the framework of strongly continuous semigroups. Compared to classical exponential quadrature rules, which in our case of interest converge with order $1+r$ at most, we prove that the new methods may reach order $1+νr$ for proper choices of the collocation points. We also show that the proposed integrators can be written in terms of special instances of the Mittag--Leffler functions that we call fractional $φ$ functions. Several numerical experiments demonstrate the theoretical findings and highlight the effectiveness of the approach. |
| title | Exponential quadrature rules for problems with time-dependent fractional source |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2501.18395 |