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Existence Results for Langevin Equation Involving Atangana-Baleanu Fractional Operators

Author

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  • Dumitru Baleanu

    (Department of Mathematics, Cankaya University, 06790 Ankara, Turkey
    Institute of Space Sciences, 077125 Magurele-Bucharest, Romania
    Department of Medical Research, China Medical University, Taichung 40447, Taiwan)

  • Rahmat Darzi

    (Department of Mathematics, Neka Branch, Islamic Azad University, Neka, Iran)

  • Bahram Agheli

    (Department of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran)

Abstract

A new form of nonlinear Langevin equation (NLE), featuring two derivatives of non-integer orders, is studied in this research. An existence conclusion due to the nonlinear alternative of Leray-Schauder type (LSN) for the solution is offered first and, following that, the uniqueness of solution using Banach contraction principle (BCP) is demonstrated. Eventually, the derivatives of non-integer orders are elaborated in Atangana-Baleanu sense.

Suggested Citation

  • Dumitru Baleanu & Rahmat Darzi & Bahram Agheli, 2020. "Existence Results for Langevin Equation Involving Atangana-Baleanu Fractional Operators," Mathematics, MDPI, vol. 8(3), pages 1-12, March.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:3:p:408-:d:331853
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    References listed on IDEAS

    as
    1. Jean-Philippe Bouchaud & Rama Cont, 1998. "A Langevin approach to stock market fluctuations and crashes," Science & Finance (CFM) working paper archive 500027, Science & Finance, Capital Fund Management.
    2. Fazli, Hossein & Nieto, Juan J., 2018. "Fractional Langevin equation with anti-periodic boundary conditions," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 332-337.
    3. Eab, C.H. & Lim, S.C., 2010. "Fractional generalized Langevin equation approach to single-file diffusion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(13), pages 2510-2521.
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