Generalized Fractional Ostrowski's Type Inequalities Involving Riemann-Liouville Fractional Integration

Generalized Fractional Ostrowski's Type Inequalities Involving Riemann-Liouville Fractional Integration

Ather Qayyum (Institute of Southern Punjab Multan, Pakistan) and Muhammad Shoaib (Higher Colleges of Technology, UAE)
DOI: 10.4018/978-1-7998-3122-8.ch011

Abstract

Fractional calculus has applications in many practical problems such as electromagnetic waves, visco-elastic systems, quantum evolution of complex systems, diffusion waves, physics, engineering, finance, social sciences, economics, mathematical biology, and chaos theory. Theories of inequality are growing rapidly; the fractional version of Ostrowski's type inequality is one of them. In this chapter, the author firstly presents some generalized Montgomery identities for Riemann-Liouville fractional integrals, which are designed by using a new and special type of Peano kernels. Secondly, inequalities via convex function are also discussed. In addition, some general fractional representation formulae are studied for a function in terms of the fractional Riemann-Liouville integrals of different orders and its ordinary derivatives. Moreover, by utilizing Montgomery identities, some new fractional versions of Ostrowski's type integral inequalities are established. Some well-known results are deduced as special cases from the results are developed here.
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Introduction

Fractional calculus was introduced by J. Liouville in the beginning of the 19th century. However, it wasn’t put into practical use until the last quarter of the 19th century when it was first used by Heaviside in the analysis of electrical transmission lines. Since then, the interest in fractional calculus has been growing continually owing to its applications in different branches of sciences. 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, 2009; A El-Sayed et al, 1996, 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 also useful applications in some interdisciplinary fields of science (Ben Adda F, 1997; Caputo, 1967 ; Ichise et al, 1971; Mainardi, 2010; Samet et al, 2018; Kusnezov et al, 1999; Dragomir, 2005; Mitrinović et al, 1994; Anastassiou et al, 2009; Podlubny, 1999 & Song et al, 2010; Laskin, 2000; Heaviside, 1971; Hilfer 2000; Bagley et al, 1991; Qayyum et al, 2015; Qayyum et al, 2019; Taghavian & Tavazoei, 2017; Taghavian & Tavazoei, 2018; Taghavian & Tavazoei, 2019).

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