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High order approximations of solutions to initial value problems for linear fractional integro-differential equations

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Abstract

We consider a general class of linear integro-differential equations with Caputo fractional derivatives and weakly singular kernels. First, the underlying initial value problem is reformulated as an integral equation and the possible singular behavior of its exact solution is determined. After that, using a suitable smoothing transformation and spline collocation techniques, the numerical solution of the problem is discussed. Optimal convergence estimates are derived and a superconvergence result of the proposed method is established. The obtained theoretical results are supported by numerical experiments.

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Acknowledgements

We thank both reviewers for their insightful remarks. The research of A. Pedas and M. Vikerpuur was supported by Estonian Research Council Grant PRG864.

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Authors and Affiliations

  1. University of Chester, Parkgate Road, Chester, CH1 4BJ, UK

    Neville J. Ford

  2. Institute of Mathematics and Statistics, University of Tartu, Narva street 18, Tartu, Estonia

    Arvet Pedas & Mikk Vikerpuur

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  1. Neville J. Ford
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  2. Arvet Pedas
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  3. Mikk Vikerpuur
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Correspondence to Mikk Vikerpuur.

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Ford, N.J., Pedas, A. & Vikerpuur, M. High order approximations of solutions to initial value problems for linear fractional integro-differential equations. Fract Calc Appl Anal 26, 2069–2100 (2023). https://doi.org/10.1007/s13540-023-00186-9

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  • DOI: https://doi.org/10.1007/s13540-023-00186-9

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