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Analytical and Approximate Solution for Solving the Vibration String Equation with a Fractional Derivative

Author

Listed:
  • Temirkhan S. Aleroev

    (Department of Applied Math, Moscow State University of Civil Engineering, 129337 Moscow, Russia)

  • Asmaa M. Elsayed

    (Department of Applied Math, Moscow State University of Civil Engineering, 129337 Moscow, Russia
    Department of Mathematics, Faculty of Science, Zagazig University, Zagazig 44519, Egypt)

Abstract

This paper is proposed for solving a partial differential equation of second order with a fractional derivative with respect to time (the vibration string equation), where the fractional derivative order is in the range from zero to two. We propose a numerical solution that is based on the Laplace transform method with the homotopy perturbation method. The method of the separation of variables (the Fourier method) is constructed for the analytic solution. The derived solutions are represented by Mittag–LefLeffler type functions. Orthogonality and convergence of the solution are discussed. Finally, we present an example to illustrate the methods.

Suggested Citation

  • Temirkhan S. Aleroev & Asmaa M. Elsayed, 2020. "Analytical and Approximate Solution for Solving the Vibration String Equation with a Fractional Derivative," Mathematics, MDPI, vol. 8(7), pages 1-9, July.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:7:p:1154-:d:384308
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    References listed on IDEAS

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    1. Saxena, R.K. & Mathai, A.M. & Haubold, H.J., 2004. "On generalized fractional kinetic equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 344(3), pages 657-664.
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    Cited by:

    1. Victor N. Orlov & Asmaa M. Elsayed & Elsayed I. Mahmoud, 2022. "Two Linearized Schemes for One-Dimensional Time and Space Fractional Differential Equations," Mathematics, MDPI, vol. 10(19), pages 1-16, October.
    2. Elsayed I. Mahmoud & Temirkhan S. Aleroev, 2022. "Boundary Value Problem of Space-Time Fractional Advection Diffusion Equation," Mathematics, MDPI, vol. 10(17), pages 1-12, September.

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