{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,22]],"date-time":"2025-12-22T04:18:22Z","timestamp":1766377102547,"version":"build-2065373602"},"reference-count":29,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,10,26]],"date-time":"2022-10-26T00:00:00Z","timestamp":1666742400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ajman University","award":["2022-IRG-HBS-8"],"award-info":[{"award-number":["2022-IRG-HBS-8"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The notion of k-symbol special functions has recently been introduced. This new concept offers many interesting geometric properties for these special functions including logarithmic convexity. The aim of the present paper is to exploit essentially two-dimensional wave propagation in the earth-ionosphere wave path using k-symbol Airy functions (KAFs) in the open unit disk. It is shown that the standard wave-mode working formula may be determined by orthogonality considerations without the use of intricate justifications of the complex plane. By taking into account the symmetry-convex depiction of the KAFs, the formula combination is derived.<\/jats:p>","DOI":"10.3390\/axioms11110590","type":"journal-article","created":{"date-parts":[[2022,10,26]],"date-time":"2022-10-26T05:11:33Z","timestamp":1666761093000},"page":"590","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Geometric Study of 2D-Wave Equations in View of K-Symbol Airy Functions"],"prefix":"10.3390","volume":"11","author":[{"given":"Samir B.","family":"Hadid","sequence":"first","affiliation":[{"name":"Department of Mathematics and Sciences, College of Humanities and Sciences, Ajman University, Ajman P.O. Box 346 00000, United Arab Emirates"},{"name":"Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman P.O. Box 346 00000, United Arab Emirates"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9341-025X","authenticated-orcid":false,"given":"Rabha W.","family":"Ibrahim","sequence":"additional","affiliation":[{"name":"Mathematics Research Center, Department of Mathematics, Near East University, Near East Boulevard, TRNC Mersin 10, Nicosia 99138, Turkey"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,26]]},"reference":[{"key":"ref_1","first-page":"179","article-title":"On hypergeometric functions and Pochhammer k-symbol","volume":"15","author":"Diaz","year":"2007","journal-title":"Divulg. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"365","DOI":"10.2991\/jnmp.2006.13.3.4","article-title":"Feynman-Jackson integrals","volume":"13","author":"Diaz","year":"2006","journal-title":"J. Nonlinear Math. 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