Computational techniques for highly oscillatory and chaotic wave problems with fractional-order operator

Abstract

In this paper, we study the dynamic evolution of chaotic and oscillatory waves arising from dissipative dynamical systems of elliptic and parabolic types of partial differential equations. In such a system, the classical second-order partial derivatives are modeled with the Riesz fractional-order operator in one and two dimensions. We employ both finite difference schemes and the Fourier spectral methods for the approximation of fractional derivatives. We examined the accuracy of the schemes by reporting their convergence results. These numerical techniques are applied to solve two practical problems that are of current and recurring interests, namely the fractional multi-wing chaotic system and fractional Helmholtz equation in one and two spatial dimensions. In the computational experiments, it was observed that under certain conditions, nonlinear dynamical system which depends on some variables is able to produce the so-called chaotic patterns. The present example shows a sensitive dependence on the choice of parameters and initial conditions. Some numerical results are presented for different instances of fractional power.