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An efficient numerical scheme to solve fractional diffusion-wave and fractional Klein–Gordon equations in fluid mechanics

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  • Hashemizadeh, E.
  • Ebrahimzadeh, A.

Abstract

The numerous applications of time fractional partial differential equations in different fields of science especially in fluid mechanics necessitate the presentation of an efficient numerical method to solve them. In this paper, Galerkin method and operational matrix of fractional Riemann–Liouville integration for shifted Legendre polynomials has been applied to solve these equations. Some definitions for fractional calculus along with some basic properties of shifted Legendre polynomials have also been put forth. When approximations are substituted into the fractional partial differential equations, a set of algebraic equations would be resulted. The convergence of the suggested method was also depicted. In the end, the linear time fractional Klein–Gordon equation, dissipative Klein–Gordon equations and diffusion-wave equations were utilized as three examples so as to study the performance of the numerical scheme.

Suggested Citation

  • Hashemizadeh, E. & Ebrahimzadeh, A., 2018. "An efficient numerical scheme to solve fractional diffusion-wave and fractional Klein–Gordon equations in fluid mechanics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 503(C), pages 1189-1203.
    • Handle: RePEc:eee:phsmap:v:503:y:2018:i:c:p:1189-1203
    DOI: 10.1016/j.physa.2018.08.086
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    References listed on IDEAS

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    1. Baillie, Richard T., 1996. "Long memory processes and fractional integration in econometrics," Journal of Econometrics, Elsevier, vol. 73(1), pages 5-59, July.
    2. Inc, Mustafa & Yusuf, Abdullahi & Isa Aliyu, Aliyu & Baleanu, Dumitru, 2018. "Time-fractional Cahn–Allen and time-fractional Klein–Gordon equations: Lie symmetry analysis, explicit solutions and convergence analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 493(C), pages 94-106.
    3. Daria Kondrashova & Rustem Valiullin & Jörg Kärger & Armin Bunde, 2017. "Structure-correlated diffusion anisotropy in nanoporous channel networks by Monte Carlo simulations and percolation theory," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 90(7), pages 1-6, July.
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    Cited by:

    1. Li Jiang & Tao Wang & Qing-Xue Huang, 2023. "Resonance Analysis of Horizontal Nonlinear Vibrations of Roll Systems for Cold Rolling Mills under Double-Frequency Excitations," Mathematics, MDPI, vol. 11(7), pages 1-15, March.

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