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

Asmaa M. Elsayed and Viktor N. Orlov
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Asmaa M. Elsayed: Department of Applied Math, Moscow State University of Civil Engineering, Yaroslavskoe Shosse, 26, 129337 Moscow, Russia
Viktor N. Orlov: Department of Applied Math, Moscow State University of Civil Engineering, Yaroslavskoe Shosse, 26, 129337 Moscow, Russia

Mathematics, 2020, vol. 8, issue 7, 1-10

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.

Keywords: time–space fractional vibration equations; weighted and shifted Grünwald difference operator; alternating direction implicit scheme; stability and convergence (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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