Exponentially convergent method for time-fractional evolution equation
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866915076824367104 |
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| author | Vasylyk, V. Makarov, V. L. |
| author_facet | Vasylyk, V. Makarov, V. L. |
| contents | An exponentially convergent numerical method for solving a differential equation with a right-hand fractional Riemann-Liouville time-derivative and an unbounded operator coefficient in Banach space is proposed and analysed for a homogeneous/inhomogeneous equation of the Hardy-Tichmarsh type.
We employ a solution representation by the Danford-Cauchy integral on hyperbola that envelopes spectrum of the operator coefficient with a subsequent application of an exponentially convergent quadrature.
To do that, parameters of the hyperbola are chosen so that the integration function has an analytical extension into a strip around the real axis and then apply the Sinc-quadrature.
We show the exponential accuracy and illustrate the results by a numerical example confirming the {\it a priori} estimate.
Existence conditions for the solution of the inhomogeneous equation are established. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_17521 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Exponentially convergent method for time-fractional evolution equation Vasylyk, V. Makarov, V. L. Numerical Analysis 65J10 An exponentially convergent numerical method for solving a differential equation with a right-hand fractional Riemann-Liouville time-derivative and an unbounded operator coefficient in Banach space is proposed and analysed for a homogeneous/inhomogeneous equation of the Hardy-Tichmarsh type. We employ a solution representation by the Danford-Cauchy integral on hyperbola that envelopes spectrum of the operator coefficient with a subsequent application of an exponentially convergent quadrature. To do that, parameters of the hyperbola are chosen so that the integration function has an analytical extension into a strip around the real axis and then apply the Sinc-quadrature. We show the exponential accuracy and illustrate the results by a numerical example confirming the {\it a priori} estimate. Existence conditions for the solution of the inhomogeneous equation are established. |
| title | Exponentially convergent method for time-fractional evolution equation |
| topic | Numerical Analysis 65J10 |
| url | https://arxiv.org/abs/2412.17521 |