Computational techniques for highly oscillatory and chaotic wave problems with fractional-order operator
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Computational techniques for highly oscillatory and chaotic wave problems with fractional-order operator Kolade M. Owolabi1,2,a 1 Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam 2 Department of Mathematical Sciences, Federal University of Technology, Akure PMB 704, Ondo State,
Nigeria Received: 31 August 2020 / Accepted: 20 October 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
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.
1 Introduction Nonlinear chaotic models are dynamical systems that are widely celebrated in the literature due to their useful applications in many areas of applied sciences and engineering. Based on chaotic system, an increasing number of research papers has been published over the years, for instance, the chaotic circuit based on memristor [63], Chen and Ueta’s system [11], Lorenz’s system [28], simple chaotic flows [52], Rössler’s system [47], memristive chaotic model with bird- and heart-shaped attractors [5,62], CMOS transistor oscillators [43], digital and circuit realization of chaotic systems [30], analog digital designs [44], chaotic mythical bird system [5], butterfly wings and paradise bird map model [1]. Nowadays, oscillatory and complex chaotic processes have been used for theoretical, practical or experimental purposes in different application areas, which include the pathological image encryption [26], chaotic communication [12], image watermarking [61], chaotic video communication scheme via WAN remote transmission [25] autonomous mobile robots [58], control and synchronization [45], audio and image encryption effects [27,32] and electromechanical oscillators [10], among several others.
a e-mail: [email protected] (corresponding author)
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Over the last few decades the concept of fractional calculus (whic
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