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Numerical Scheme for Solving Time–Space Vibration String Equation of Fractional Derivative
Article

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

1
Department of Applied Math, Moscow State University of Civil Engineering, 129337 Moscow, Russia
2
Department of Mathematics, Faculty of Science, Zagazig University, Zagazig 44519, Egypt
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(7), 1154; https://doi.org/10.3390/math8071154
Received: 28 May 2020 / Revised: 4 July 2020 / Accepted: 10 July 2020 / Published: 14 July 2020
(This article belongs to the Special Issue Dynamical Systems and Optimal Control)
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. View Full-Text
Keywords: laplace transform; homotopy perturbation method; fractional PDEs; Mittag–Leffler type functions laplace transform; homotopy perturbation method; fractional PDEs; Mittag–Leffler type functions
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MDPI and ACS Style

Aleroev, T.S.; Elsayed, A.M. Analytical and Approximate Solution for Solving the Vibration String Equation with a Fractional Derivative. Mathematics 2020, 8, 1154. https://doi.org/10.3390/math8071154

AMA Style

Aleroev TS, Elsayed AM. Analytical and Approximate Solution for Solving the Vibration String Equation with a Fractional Derivative. Mathematics. 2020; 8(7):1154. https://doi.org/10.3390/math8071154

Chicago/Turabian Style

Aleroev, Temirkhan S.; Elsayed, Asmaa M. 2020. "Analytical and Approximate Solution for Solving the Vibration String Equation with a Fractional Derivative" Mathematics 8, no. 7: 1154. https://doi.org/10.3390/math8071154

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