Investigación

 

Investigation of electrical RC circuit within the framework of fractional calculus

 

H. Ertik^a, A.E. Çalik^b, H. Şirin^c, M. Şen and B. Öder^d

 

^a Department of Mathematics Education, Alanya Faculty of Education, Akdeniz University, Alanya, Antalya, 07425, TURKEY.

^b Department of Physics, Faculty of Arts and Sciences, Dumlupinar University Kütahya, 43100, TURKEY, e-mail: aengin.calik@dpu.edu.tr.

^c Department of Physics, Faculty of Science, Ege University Bornova, Ízmir, 35100, TURKEY.

^dDepartment of Physics, Institute of Science and Technology, Dumlupinar University Kütahya, 43100, TURKEY.

 

Received 20 October 2014;
accepted 11 December 2014

 

Abstract

In this paper, charging and discharging processes of different capacitors in electrical RC circuit are considered theoretically and experimentally. The non-local behaviors in these processes, arising from the time fractality, are investigated via fractional calculus. In this context, the time fractional differential equation related to electrical RC circuit is proposed by making use of Caputo fractional derivative. The resulting solution exhibits a feature in between power law and exponential law forms, and is obtained in terms of Mittag-Leffler function which describes physical systems with memory. The order of fractional derivative characterizes the fractality of time and being considered in the interval 0 < α ≤ 1. The traditional conclusions are recovered for α = 1, where time becomes homogenous and system has Markovian nature. By using time fractional approach, the discrepancies between the experimentally measured data and the theoretical calculations have been removed.

Keywords: RC circuit; fractional calculus; Caputo fractional derivative; Mittag-Leffler function; Planck units.

 

PACS: 02.30.Hq; 07.50.Ek

 

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