2024 - Sustainable Industrial Processing Summit
SIPS 2024 Volume 11. Rowlands Intl. Symp / Mathematics
| Editors: | F. Kongoli, A. Bountis, M. Johnson, S. Karam, L. Kauffman, P. Mandell, M. Mikalajunas, W. Miller, G. Ord, R.M. Santilli, E. Suhir, E. Trell, T. Vougiouklis |
| Publisher: | Flogen Star OUTREACH |
| Publication Year: | 2024 |
| Pages: | 372 pages |
| ISBN: | 978-1-998384-24-2 (CD) |
| ISSN: | 2291-1227 (Metals and Materials Processing in a Clean Environment Series) |
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THE DISSIPATIVE EFFECT OF CAPUTO–TIME–FRACTIONAL DERIVATIVES FOR THE SOLUTIONS OF NONLINEAR WAVE EQUATIONS
Anastassios Bountis1; Julia Cantisán Gómez2; Jesús Cuevas–Maraver3; J. E. Macıas-Dıaz4; P. Kevrekidis5;1UNIVERSITY OF PATRAS, Patras, Greece; 2UNIVERSIDAD REY JUAN CARLOS, Madrid, Spain; 3UNIVERSIDAD DE SEVILLA, Sevilla, Spain; 4UNIVERSIDAD AUTONOMA DE AGUASCALIENTES, Aguascalientes, Mexico; 5UNIVERSITY OF MASSACHUSETTS, Amherst, United States;
Type of Paper: Regular
Id Paper: 373
Topic: 38
Abstract:
We would like to draw attention in the present paper to a curious mathematical observation concerning fractional differential equations describing physical systems, whose time evolution for integer derivatives has a time-honored conservative form. This observation, although known to the general mathematical community [1, 2, 3], has not, in our view, been satisfactorily addressed. More specifically, we follow the recent exploration of Caputo-Riesz time-space-fractional nonlinear wave equation [4], in which the authors introduced an energy-type functional and proposed a finite-difference scheme to approximate the solutions of the continuous model. The relevant Klein-Gordon equation is considered, where we explore the sine-Gordon nonlinearity with smooth initial data. For the Riesz and Caputo derivative coefficients α=β=2, we naturally retrieve the exact, analytical form of breather waves expected from the literature.
Focusing on the Caputo temporal derivative variation within 1<β<2 values for α=2, however, we observe artificial dissipative effects, which lead to complete breather disappearance, over a time scale depending on the value of β. We compare such findings to single degree-of-freedom linear and nonlinear oscillators in the presence of Caputo temporal derivatives and also consider anti-damping mechanisms to counter the relevant effect. These findings also motivate some interesting directions for further study, e.g., regarding the consideration of topological solitary waves, such as kinks/antikinks and their dynamical evolution in this model.