Chaotic behavior in fractional Helmholtz and Kelvin-Helmholtz instability problems with Riesz operator. (English) Zbl 1496.65125
Summary: 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, \(0<\alpha\leq1\) and \(1<\alpha\leq2\), 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 \(\alpha\), for subdiffusive and superdiffusive scenarios.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 65N06 | Finite difference methods for boundary value problems involving PDEs |
| 26A33 | Fractional derivatives and integrals |
| 35R11 | Fractional partial differential equations |
| 35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |
| 35B36 | Pattern formations in context of PDEs |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 44A10 | Laplace transform |
| 65T50 | Numerical methods for discrete and fast Fourier transforms |
| 76D50 | Stratification effects in viscous fluids |
| 86A05 | Hydrology, hydrography, oceanography |
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