Complex conformable derivative
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ICCESEN 2017
Complex conformable derivative Sümeyra Uçar 1 & Nihal Yılmaz Özgür 1 & Beyza Billur İskender Eroğlu 1 Received: 7 November 2018 / Accepted: 27 February 2019 # Saudi Society for Geosciences 2019
Abstract In this study, we introduce a new complex conformable derivative and integral with noninteger order α which coincides the classical complex derivative and integral for α = 1. We examine basic properties of these newly defined derivative and integral such as Cauchy-Riemann equations, conformability and fundamental theorem of complex conformable integral. As an application of the new operators, we solve some types of complex conformable differential equations. Keywords Complex conformable derivative . Complex conformable integral . Conformability
Introduction The standard mathematical models with classical derivatives are not sufficient for some real-world applications. However, fractional calculus is a powerful method to characterise many problems in science and engineering more realistically. The well-known fractional operators are Riemann-Liouville and Caputo which are nonlocal operators with singular kernels. Due to the properties of aforesaid operators, the physical problems modelled with these operators generally solved by numerical techniques (Evirgen 2011; Kilbas et al. 2006; Li and Zeng 2015). In recent times, it has been defined as nonlocal operators with nonsingular kernels such as Caputo Fabrizio and Atangana Baleanu (Atangana and Alkahtani 2016; Caputo and Fabrizio 2015; Atangana and Baleanu 2016). Besides the progress in nonlocal operators, local operators with fractional order are defined such as Khalil et al. (Khalil et al. 2014). Their local operator called as conformable derivative is a limit-based definition for real functions and attracts many researchers attention to study some properties and applications of the derivative (Evirgen 2017; Iskender
This article is part of the Topical Collection on Geo-Resources-EarthEnvironmental Sciences * Sümeyra Uçar [email protected] 1
Department of Mathematics, Balıkesir University, Balıkesir, Turkey
Eroğlu et al. 2017; Yavuz 2018; Yavuz and Özdemir 2018). Classical derivatives and both of local and nonlocal operators with fractional order have been extensively studied for real functions but they are rarely seen in the complex plane (Şan 2018; Li et al. 2009; Gürel et al. 2015). But there are many physical applications in complex plane, for instance electrostatic potential, complex-valued neural networks, battery energy storage system (BESS) and shortterm load forecast (STLF) etc. ((Mandic 2000; Brown and Churchill 2004; Özdemir et al. 2011; Sepas et al. 2017) and the references therein). Especially, analytic functions and conformal mappings concepts of complex functions theory are met in solution process of complex problems. For example, analytic functions are used to solve Dirichlet problems emerging in the two-dimensional modelling of fluid flow, gravitation and heat flow. Also, conformal mapping which transforms complicated reg
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