{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,12]],"date-time":"2026-09-12T05:29:11Z","timestamp":1789190951317,"version":"build-2803163510"},"reference-count":26,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,12,12]],"date-time":"2021-12-12T00:00:00Z","timestamp":1639267200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.3390\/axioms10040342","type":"journal-article","created":{"date-parts":[[2021,12,13]],"date-time":"2021-12-13T01:28:11Z","timestamp":1639358891000},"page":"342","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Generalized Quantum Integro-Differential Fractional Operator with Application of 2D-Shallow Water Equation in a Complex Domain"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9341-025X","authenticated-orcid":false,"given":"Rabha W.","family":"Ibrahim","sequence":"first","affiliation":[{"name":"The Institute of Electrical and Electronics Engineers (IEEE), 94086547, Kuala Lumpur 59200, Malaysia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0286-7244","authenticated-orcid":false,"given":"Dumitru","family":"Baleanu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Cankaya University, Balgat, Ankara 06530, Turkey"},{"name":"Institute of Space Sciences, R76900 Magurele-Bucharest, Romania"},{"name":"Department of Medical Research, China Medical University, Taichung 40402, Taiwan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"4284","DOI":"10.22436\/jnsa.010.08.23","article-title":"Generalized convolution properties based on the modified Mittag-Leffler function","volume":"10","author":"Srivastava","year":"2017","journal-title":"J. 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