Page 115 - Mathematical Techniques of Fractional Order Systems

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Chapter 4

Exact Solution of Linear

Fractional Distributed Order
Systems With Exponential Order

Weight Functions

Hamed Taghavian and Mohammad Saleh Tavazoei
Sharif University of Technology, Tehran, Iran

4.1 INTRODUCTION
Fractional calculus is now omnipresent in different realms of science and
technology, such as in control systems theory (Azar et al., 2017; Dumlu and
Erenturk, 2014), signal processing (Tavazoei, 2015; Aslam and Raja, 2015),
and differential equations (Cao et al., 2016; Bekir et al., 2017; Jin et al.,
2016). In fact fractional differential operators help in modeling physical pro-
cesses with long-lasting memory and anamolous behavior. As famous
instances of fractional differential equations, one could refer to anomolous
diffusion equations (Lv and Xu, 2016; Vabishchevich, 2016) describing the
diffusion behavior occurring mainly in biological systems and fractional
relaxation equations (Garrappa et al., 2014; Garra et al., 2014) describing the
delayed reaction of a system to matter condensation.
Nowadays fractional operators have undergone an even further generali-
zation by the advent of distributed order operators. It has been shown that
using these operators is essential for modeling some physical phenomena in
an accurate way, among which one could refer to ultraslow diffusion
equations whose square displacement growth occurs at a logarithmic rate and
distributed order space diffusion equations describing accelerating superdif-
fusion phenomena (Chechkin et al., 2002). In fact alongside diffusion
equations (Garra et al., 2014; Chechkin et al., 2002; Sandev et al., 2015;
Mainardi et al., 2008; Chechkin et al., 2003), differential equations involving
derivatives of distributed order are encountered in different physical pro-
blems, including wave equations (Gorenflo et al., 2013) and relaxation
phenomena (Meerschaert and Toaldo, 2015). In addition, distributed order

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