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This paper is devoted to studying non-commensurate fractional order planar systems. Our contributions are to derive sufficient conditions for the global attractivity of non-trivial solutions to fractional-order inhomogeneous linear planar systems and for the Mittag-Leffler stability of an equilibrium point to fractional order nonlinear planar systems. To achieve these goals, our approach is as follows. Firstly, based on Cauchy’s argument principle in complex analysis, we obtain various explicit sufficient conditions for the asymptotic stability of linear systems whose coefficient matrices are constant. Secondly, by using Hankel type contours, we derive some important estimates of special functions arising from a variation of constants formula of solutions to inhomogeneous linear systems. Then, by proposing carefully chosen weighted norms combined with the Banach fixed point theorem for appropriate Banach spaces, we get the desired conclusions. Finally, numerical examples are provided to illustrate the effect of the main theoretical results.
An Improvement of a Nonclassical Numerical Method for the Computation of Fractional Derivatives
(2009)
Standard methods for the numerical calculation of fractional derivatives can be slow and memory consuming due to the nonlocality of the differential operators. Yuan and Agrawal (2002, “A Numerical Scheme for Dynamic Systems Containing Fractional Derivatives,” ASME J. Vibr. Acoust., 124, pp. 321–324) have proposed a more efficient approach for operators whose order is between 0 and 1 that differs substantially from the traditional concepts. It seems, however, that the accuracy of the results can be poor. We modify the approach, adapting it better to the properties of the problem, and show that this leads to a significantly improved quality. Our idea also works for operators of order greater than 1.
The area of fractional calculus (FC) has been fast developing and is presently being applied in all scientific fields. Therefore, it is of key relevance to assess the present state of development and to foresee, if possible, the future evolution, or, at least, the challenges identified in the scope of advanced research works. This paper gives a vision about the directions for further research as well as some open problems of FC. A number of topics in mathematics, numerical algorithms and physics are analyzed, giving a systematic perspective for future research.
Alle deutschen Hochschulen präsentieren sich im Internet über eine Webseite. Datenschutzaspekte spielen für Besucher dieser Seiten, wie zukünftige Studierende und Forschungspartner, eine immer wichtigere Rolle. Der vorliegende Beitrag untersucht die Existenz und den Inhalt der nach der deutschen Gesetzgebung verpflichtenden Datenschutzerklärungen (DSE). Ferner wird untersucht, ob sich die DSE mit den tatsächlichen Inhalten der Webseiten deckt. Die Methodik beinhaltet eine manuelle Überprüfung ausgewählter Webseiten und eine automatisierte Prüfung mittels eines modifizierten PyQt-Browsers mit dem Fokus auf der Erkennung von Trackern. Die Auswertung der Daten zeigt, dass viele DSE entweder fehlen, falsche Informationen beinhalten oder unvollständig sind. Der Artikel endet mit einer Diskussion der Ursachen und Empfehlungen zur Verbesserung der Missstände.
For the investigation of high frequency electromagnetic waves scattered at metallic bodies it is necessary to use algorithms with high accuracy. Moreover, ordinary differential equation systems play an important role in this context. Thus, the authors present an ODE-Solver based on the method of Lie Series.
The algorithm is computing in arbitrary precision. An adaptive step width control enables improvement in computation time and precision. Furthermore, the authors show tests of one classical ODE-Problem plus geodesics on an ellipsoid.