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

Author

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  • Asmaa M. Elsayed

    (Department of Applied Math, Moscow State University of Civil Engineering, Yaroslavskoe Shosse, 26, 129337 Moscow, Russia
    Department of Mathematics, Faculty of Science, Zagazig University, Zagazig 44519, Egypt)

  • Viktor N. Orlov

    (Department of Applied Math, Moscow State University of Civil Engineering, Yaroslavskoe Shosse, 26, 129337 Moscow, Russia)

Abstract

In this paper, we present a numerical scheme and alternating direction implicit scheme for the one-dimensional time–space fractional vibration equation. Firstly, the considered time–space fractional vibration equation is equivalently transformed into their partial integro-differential forms by using the integral operator. Secondly, we use the Crank–Nicholson scheme based on the weighted and shifted Grünwald–difference formula to discretize the Riemann–Liouville and Caputo derivative, also use the midpoint formula to discretize the first order derivative. Meanwhile, the classical central difference formula is applied to approximate the second order derivative. The convergence and unconditional stability of the suggested scheme are obtained. Finally, we present an example to illustrate the method.

Suggested Citation

  • Asmaa M. Elsayed & Viktor N. Orlov, 2020. "Numerical Scheme for Solving Time–Space Vibration String Equation of Fractional Derivative," Mathematics, MDPI, vol. 8(7), pages 1-10, July.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:7:p:1069-:d:379336
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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.

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