CHAOTIC BEHAVIOR IN FRACTIONAL HELMHOLTZ AND KELVIN–HELMHOLTZ INSTABILITY PROBLEMS WITH RIESZ OPERATOR

Kolade M. Owolabi, J. F. Gã“mez-Aguilar, Yeliz Karaca, Yong-Min Li, Bahaa Saleh and Ayman A. Aly
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Kolade M. Owolabi: Department of Mathematical Sciences, Federal University of Technology, PMB 704, Akure, Ondo State, Nigeria
J. F. Gã“mez-Aguilar: ��CONACyT-Tecnológico Nacional de México/CENIDET, Interior Internado, Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca, Morelos, México
Yeliz Karaca: ��University of Massachusetts Medical School, Worcester, MA 01655, USA
Yong-Min Li: �Department of Mathematics, Huzhou University, Huzhou 313000, P. R. China¶Institute for Advanced Study Honoring Chen Jian Gong, Hangzhou Normal University, Hangzhou 311121, P. R. China
Bahaa Saleh: ��Department of Mechanical Engineering, College of Engineering, Taif University, P. O. Box 11099, Taif 21944, Saudi Arabia
Ayman A. Aly: ��Department of Mechanical Engineering, College of Engineering, Taif University, P. O. Box 11099, Taif 21944, Saudi Arabia

FRACTALS (fractals), 2022, vol. 30, issue 05, 1-19

Abstract: This paper introduces some important dissipative problems that are recent and still of intermittent interest. The classical dynamics of Helmholtz and Kelvin–Helmholtz instability equations are modeled with the Riesz operator which incorporates the left- and right-sided of the Riemann–Liouville non-integer order operators to mimic naturally the physical patterns of these models arising in hydrodynamics and geophysical fluids. The Laplace and Fourier transform techniques are used to approximate the Riesz fractional operator in a spatial direction. The behaviors of the Helmholtz and Kelvin–Helmholtz equations are observed for some values of fractional power in the regimes, 0 < α ≤ 1 and 1 < α ≤ 2, using different boundary conditions on a square domain in 1D, 2D and 3D (spatial-dimensions). Numerical results reveal some astonishing and very impressive phenomena which arise due to the variations in the initial and source function, as well as fractional parameter α, for subdiffusive and superdiffusive scenarios.

Keywords: Chaotic Oscillations; Pattern Formation; Reaction–Diffusion; Riemann–Liouville Derivative (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1142/S0218348X2240182X

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