axioms Article Generalized Quantum Integro-Differential Fractional Operator with Application of 2D-Shallow Water Equation in a Complex Domain Rabha W. Ibrahim 1, * ,† and Dumitru Baleanu 2,3,4,†   Citation: Ibrahim, R.W.; Baleanu, D. Generalized Quantum Integro- Differential Fractional Operator with Application of 2D-Shallow Water Equation in a Complex Domain. Axioms 2021, 10, 342. https:// doi.org/10.3390/axioms10040342 Academic Editors: Hans J. Haubold, Mircea Merca and Serkan Araci Received: 31 October 2021 Accepted: 7 December 2021 Published: 12 December 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 The Institute of Electrical and Electronics Engineers (IEEE), 94086547, Kuala Lumpur 59200, Malaysia 2 Department of Mathematics, Cankaya University, Balgat, Ankara 06530, Turkey; dumitru@cankaya.edu.tr 3 Institute of Space Sciences, R76900 Magurele-Bucharest, Romania 4 Department of Medical Research, China Medical University, Taichung 40402, Taiwan * Correspondence: rabhaibrahim@yahoo.com † These authors contributed equally to this work. Abstract: In this paper, we aim to generalize a fractional integro-differential operator in the open unit disk utilizing Jackson calculus (quantum calculus or q-calculus). Next, by consuming the generalized operator to define a formula of normalized analytic functions, we present a set of integral inequalities using the concepts of subordination and superordination. In addition, as an application, we determine the maximum and minimum solutions of the extended fractional 2D-shallow water equation in a complex domain. Keywords: quantum calculus; fractional calculus; analytic function; subordination; univalent function; open unit disk; differential operator; convolution operator 1. Introduction Elementary series and polynomials, particularly the Mittag–Leffler functions and polynomials and their consequences, can be frequently seen in specific areas of number theory, including the theory of partitions. These functions are valuable in an extensive diversity of fields involving, for instance, finite vector spaces, combinator analysis, lie theory, nonlinear electric circuit theory, particle physics, optical studies, fluid theory, me- chanical engineering, quantum mechanics, cosmology, theory of thermal conduction and measurements (see [1–6]). Quantum power series, especially the Mittag–Leffler functions, are known to have common applications, specifically in numerous areas of function theory, geometric function theory and others. As a substance of detail, q-Mittag–Leffler func- tions are beneficial too in a extensive diversity of arenas. In our study, we employ the definition of the q- Mittag–Leffler functions to modify a fractional integral operator of a complex variable. The 2D-shallow water equations (SWEs) are utilized to designate flow in precipitously well mixed water figures where the straight length scales are much bigger than the fluid depth (long wavelength phenomena) [7]. The SWEs are selected by supposing a hydro- static pressure distribution and a uniform velocity profile in the vertical direction. The SWEs can be used to study numerous physical phenomena of interest, such as storm surges, tidal variations, tsunami waves, and forces performing on off-shore assemblies, and can be joined to transport equations to formulate transport of chemical species. Most of these equations are solved by numerical techniques [8,9]. Our study is based on an approximated analytic solution given in the open unit disk. In this study, we investigate a generalization of fractional integro-differential operators in the open unit disk formulated by the q-calculus. We employ the q-operator to describe a formulation of normalized analytic functions. We consider a set of integral inequalities indicating the notion of differential subordination and superordination. In addition, as an Axioms 2021, 10, 342. https://doi.org/10.3390/axioms10040342 https://www.mdpi.com/journal/axioms