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A new accurate method for solving fractional relaxation-oscillation with Hilfer derivatives

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Abstract

Fractional relaxation-oscillation equation (FROE) has proved to provide more accurate interpretation of describing materials with viscoelastic properties. However, the current operator considered is not general ones, thus restricted on giving outcome for single fractional operator only. Hilfer derivative is one of the generalized fractional operator that has extra parameter \(\gamma \) which not only able to interpolate between Riemann–Liouville (RL) operator (\(\gamma =0\)) and Caputo operator (\(\gamma =1\)), but also within the range \(0<\gamma <1\). In this paper, a new numerical technique is developed to solve fractional relaxation-oscillation equation in Hilfer sense (HFROE) using operational matrix of integration based on fractional-order alternative Legendre functions (FALFs). By transforming the HFROE into its equivalent Volterra integral equation (VIE), this method reduces the problem into a system of algebraic equation, thus greatly simplifying the problem which easy to solve. Several numerical examples illustrate the accuracy of the method and comparisons are made for the existing method. Finally, solutions’ profile of HFROE within \(0<\gamma <1\) is found out to be lies between the solutions’ profile of HFROE representing RL and Caputo.

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Acknowledgements

This study was supported by the Fundamental Research Grant Scheme (Ref. No. FRGS/1/2022/STG06/UPM/02/2) awarded by the Malaysia Ministry of Education.

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Authors and Affiliations

  1. Institute for Mathematical Research, Universiti Putra Malaysia, Selangor, Malaysia

    Mohd Rashid Admon, Norazak Senu & Zanariah Abdul Majid

  2. Decisions Lab, Mediterranea University of Reggio Calabria, Reggio Calabria, Italy

    Ali Ahmadian

  3. Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon

    Ali Ahmadian & Soheil Salahshour

  4. Department of Mathematics, Near East University, Nicosia, TRNC, Mersin 10, Turkey

    Ali Ahmadian & Soheil Salahshour

  5. Faculty of Engineering and Natural Sciences, Bahcesehir University, Istanbul, Turkey

    Soheil Salahshour

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  1. Mohd Rashid Admon
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Correspondence to Norazak Senu or Ali Ahmadian.

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Communicated by Agnieszka Malinowska.

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Admon, M.R., Senu, N., Ahmadian, A. et al. A new accurate method for solving fractional relaxation-oscillation with Hilfer derivatives. Comp. Appl. Math. 42, 10 (2023). https://doi.org/10.1007/s40314-022-02154-0

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