Numerical Schemes for Time-Space Fractional Vibration Equations
Zhang, Jingna1; Aleroev, Temirkhan S.2; Tang, Yifa3,4; Huang, Jianfei1
AbstractIn this paper, we present a numerical scheme and an alternating direction implicit (ADI) scheme for the one-dimensional and two-dimensional time-space fractional vibration equations (FVEs), respectively. Firstly, the considered time-space FVEs are equivalently transformed into their partial integro-differential forms with the classical first order integrals and the Riemann-Liouville derivative. This transformation can weaken the smoothness requirement in time when discretizing the partial integro-differential problems. Secondly, we use the Crank-Nicolson technique combined with the midpoint formula, the weighted and shifted Grunwald difference formula and the second order convolution quadrature formula to deal with the temporal discretizations. Meanwhile, the classical central difference formula and fractional central difference formula are applied to approximate the second order derivative and the Riesz derivative in spatial direction, respectively. Further, an ADI scheme is constructed for the two-dimensional case. Then, the convergence and unconditional stability of the proposed schemes are proved rigorously. Both of the schemes are convergent with the second order accuracy in time and space. Finally, two numerical examples are given to support the theoretical results.
KeywordTime-space fractional vibration equations ADI scheme stability convergence
Indexed BySCI
Funding ProjectNational Natural Science Foundation of China[11701502] ; National Natural Science Foundation of China[11871065]
WOS Research AreaMathematics ; Mechanics
WOS SubjectMathematics, Applied ; Mechanics
WOS IDWOS:000640122800004
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Document Type期刊论文
Corresponding AuthorHuang, Jianfei
Affiliation1.Yangzhou Univ, Coll Math Sci, Yangzhou 225002, Jiangsu, Peoples R China
2.Moscow State Univ Civil Engn, Yaroslayskoe Shosse 26, Moscow 129337, Russia
3.Chinese Acad Sci, Acad Math & Syst Sci, ICMSEC, LSEC, Beijing 100190, Peoples R China
4.Univ Chinese Acad Sci, Sch Math Sci, Beijing 100049, Peoples R China
Recommended Citation
GB/T 7714
Zhang, Jingna,Aleroev, Temirkhan S.,Tang, Yifa,et al. Numerical Schemes for Time-Space Fractional Vibration Equations[J]. ADVANCES IN APPLIED MATHEMATICS AND MECHANICS,2021,13(4):806-826.
APA Zhang, Jingna,Aleroev, Temirkhan S.,Tang, Yifa,&Huang, Jianfei.(2021).Numerical Schemes for Time-Space Fractional Vibration Equations.ADVANCES IN APPLIED MATHEMATICS AND MECHANICS,13(4),806-826.
MLA Zhang, Jingna,et al."Numerical Schemes for Time-Space Fractional Vibration Equations".ADVANCES IN APPLIED MATHEMATICS AND MECHANICS 13.4(2021):806-826.
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