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

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 <= 1 and 1 < alpha <= 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 a, for subdiffusive and superdiffusive scenarios.