In this chapter, the authors report their work on the application of fractional derivative to the study of the memristor dynamic where the effects of the parasitic fractional elements of the memristor have been studied. The fractional differential equations of the memristor and the memristor-based circuits under the effects of the parasitic fractional elements have been formulated and solved both analytically and numerically. Such effects of the parasitic fractional elements have been studied via the simulations based on the obtained solutions where many interesting results have been proposed in the work. For example, it has been found that the parasitic fractional elements cause both charge and flux decay of the memristor and the impasse point breaking of the phase portraits between flux and charge of the memristor-based circuits similarly to the conventional parasitic elements. The effects of the order and the nonlinearity of the parasitic fractional elements have also been reported.
TopIntroduction
Fractional calculus has important applications in science and technology, e.g. in biological systems (Ahmed & Elgazzar, 2007; Matouk et al, 2015; Elsadany & Matouk, 2015; Matouk et al, 2015; Matouk & Elsadany, 2016; Selvam et al, 2017; Selvam et al, 2018a-b; Ameen & Novati, 2017; Al-Khedhairi et al, 2018), economic and financial systems (Laskin, 2000; Chen, 2008; Chen et al, 2011; Zhen et al, 2011; Hegazi et al, 2013b; Khan & Kumar, 2016), social models (Ahmad & El-Khazali, 2007; Ali, 2019), physical models (Heaviside, 1971; El-Sayed, 1996; Kusnezov et al, 1999; Hilfer, 2000; Khan & Kumar, 2018; Saqib et al, 2018a-d, Khan & Kumar, 2019a; Al-Khedhairi et al, 2019a; Al-Khedhairi et al, 2019b) and engineering systems (Sun et al, 1984; Petras, 2009; Matouk, 2009a; Matouk, 2009b; Matouk 2010; Matouk, 2011a; Matouk, 2011b; Hegazi & Matouk, 2011; Hegazi et al, 2011; Hegazi & Matouk, 2013; Hegazi et al, 2013a; Matouk & Elsadany, 2014; Banu & Nasir, 2015; Nasir & Singh, 2015; Matouk, 2016; El-Sayed et al, 2016a; El-Sayed et al, 2016b; Khan & Kumar, 2019b). Furthermore, fractional derivatives have useful applications in some interdisciplinary fields of science (Ichise et al, 1971; Bagley & Torvik, 1983; Bagley & Calico, 1991; Ben Adda F, 1997; Magin, 2006; Tavazoei et al, 2009; Tavazoei, 2010; Mainardi, 2010; Abdel Latif, 2011; El-Sayed & Salman, 2013; Latif et al, 2013; Abdel Kader et al, 2017; Abdel Kader et al, 2018; Abdel Latif et al, 2018; Ali & Ameen, 2019; Saqib et al, 2019a-b; Taghavian & Tavazoei, 2017; Taghavian & Tavazoei, 2018; Taghavian & Tavazoei, 2019). Fractional differential operators have also been applied in the studies on memristor (Machado, 2013; Yi-Fei & Yuan, 2013; Fouda & Radwan, 2013; Fouda & Radwan, 2015; Yu et al, 2015; Shi & Hu, 2017; Banchuin, 2018; Banchuin, 2019). Throughout this chapter our work on the application of fractional derivative to the study of memristor dynamic under the effects of parasitic fractional elements (Banchuin, 2019) will be reported by starting from some back ground on memristor.