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Some Integral Transform Results for Hilfer–Prabhakar Fractional Derivative and Analysis of Free-Electron Laser Equation

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Abstract

In this paper, Natural transforms of Prabhakar integral, Hilfer–Prabhakar (HP) fractional derivative and regularized Caputo form of HP fractional derivative (HPFD) are computed. Furthermore, we investigate the solution of a fractional free-electron laser equation and fractional Cauchy problems involving the HPFD via the Natural transform. The solution of Cauchy problems modeled with Hilfer–Prabhakar fractional derivatives and its regularized Caputo version is achieved in the form of generalized three parameter Mittag–Leffler function by operations of Natural and Fourier transforms.

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This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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Authors and Affiliations

  1. Faculty of Mathematical and Statistical Sciences, Shri Ramswaroop Memorial University, Barabanki, Uttar Pradesh, 225003, India

    Ved Prakash Dubey

  2. Department of Mathematics, JECRC University, Jaipur, Rajasthan, 303905, India

    Jagdev Singh

  3. Department of Mathematics, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul, 02447, Korea

    Jagdev Singh

  4. Department of Computer Science and Mathematics, Lebanese American University, P.O. Box 13-5053, Beirut, Lebanon

    Jagdev Singh

  5. Department of Physics, L.N.D. College (B.R. Ambedkar Bihar University, Muzaffarpur), Motihari, Bihar, 845401, India

    Sarvesh Dubey

  6. Department of Mathematics, University of Rajasthan, Jaipur, Rajasthan, 302004, India

    Devendra Kumar

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  1. Ved Prakash Dubey
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  2. Jagdev Singh
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  3. Sarvesh Dubey
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  4. Devendra Kumar
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Contributions

All authors contributed to the study conception and design. VPD contributed to writing—original draft and methodology. JS was involved in the supervision and conceptualization. SD assisted in the formal analysis and validation. DK contributed to the visualization and investigation. All authors read and approved the final manuscript.

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Correspondence to Jagdev Singh.

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Dubey, V.P., Singh, J., Dubey, S. et al. Some Integral Transform Results for Hilfer–Prabhakar Fractional Derivative and Analysis of Free-Electron Laser Equation. Iran J Sci 47, 1333–1342 (2023). https://doi.org/10.1007/s40995-023-01493-9

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  • DOI: https://doi.org/10.1007/s40995-023-01493-9

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