{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,22]],"date-time":"2026-06-22T19:35:13Z","timestamp":1782156913746,"version":"3.54.5"},"reference-count":17,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2021,6,30]],"date-time":"2021-06-30T00:00:00Z","timestamp":1625011200000},"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>(1) Background: symmetry breaking (self-organized transformation of symmetric stats) is a global phenomenon that arises in an extensive diversity of essentially symmetric physical structures. We investigate the symmetry breaking of time-2D space fractional wave equation in a complex domain; (2) Methods: a fractional differential operator is used together with a symmetric operator to define a new fractional symmetric operator. Then by applying the new operator, we formulate a generalized time-2D space fractional wave equation. We shall utilize the two concepts: subordination and majorization to present our results; (3) Results: we obtain different formulas of analytic solutions using the geometric analysis. The solution suggests univalent (1-1) in the open unit disk. Moreover, under certain conditions, it was starlike and dominated by a chaotic function type sine. In addition, the authors formulated a fractional time wave equation by using the Atangana\u2013Baleanu fractional operators in terms of the Riemann\u2013Liouville and Caputo derivatives.<\/jats:p>","DOI":"10.3390\/axioms10030141","type":"journal-article","created":{"date-parts":[[2021,6,30]],"date-time":"2021-06-30T10:03:19Z","timestamp":1625047399000},"page":"141","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Symmetry Breaking of a Time-2D Space Fractional Wave 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":"Institute of Electrical and Electronics Engineers: 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, 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":[[2021,6,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"109602","DOI":"10.1016\/j.chaos.2020.109602","article-title":"Symmetry breaking of spatial Kerr solitons in fractional dimension","volume":"132","author":"Li","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3815","DOI":"10.1007\/s11071-020-05730-3","article-title":"Symmetry breaking bifurcations in the NLS equation with an asymmetric delta potential","volume":"100","author":"Rusin","year":"2020","journal-title":"Nonlinear Dyn."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1007\/s13160-020-00428-w","article-title":"Numerical analyses for spectral stability of solitary waves near bifurcation points","volume":"38","author":"Yagasaki","year":"2021","journal-title":"Jpn. J. Ind. Appl. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"021019","DOI":"10.1103\/PhysRevX.10.021019","article-title":"Complex spacing ratios: A signature of dissipative quantum chaos","volume":"10","author":"Sa","year":"2020","journal-title":"Phys. Rev."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Miller, S.S., and Mocanu, P.T. (2000). Differential Subordinations: Theory and Applications, CRC Press.","DOI":"10.1201\/9781482289817"},{"key":"ref_6","first-page":"95","article-title":"Majorization by univalent functions","volume":"43","author":"MacGregor","year":"1967","journal-title":"Duke Math. J."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"8070","DOI":"10.1002\/mma.5754","article-title":"A complex analysis approach to Atangana-Baleanu fractional calculus","volume":"44","author":"Fernandez","year":"2019","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"797","DOI":"10.1016\/j.jmaa.2007.03.018","article-title":"On a generalization of Mittag-Leffler function and its properties","volume":"336","author":"Shukla","year":"2007","journal-title":"J. Math. Anal. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"298628","DOI":"10.1155\/2011\/298628","article-title":"Mittag-Leffler functions and their applications","volume":"2011","author":"Haubold","year":"2011","journal-title":"J. Appl. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"586","DOI":"10.1007\/BF02312838","article-title":"Sharpness of certain Campbell and Pommerenke estimates","volume":"63","author":"Godula","year":"1998","journal-title":"Math. Notes"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"420","DOI":"10.4153\/CJM-1973-042-6","article-title":"Majorization-subordination theorems for locally univalent functions, II","volume":"25","author":"Campbell","year":"1973","journal-title":"Can. J. Math."},{"key":"ref_12","first-page":"1","article-title":"On quantum hybrid fractional conformable differential and integral operators in a complex domain","volume":"Volume 115","author":"Ibrahim","year":"2021","journal-title":"Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matem\u00e1ticas"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"4211","DOI":"10.3934\/math.2021249","article-title":"On a combination of fractional differential and integral operators associated with a class of normalized functions","volume":"6","author":"Ibrahim","year":"2021","journal-title":"AIMS Math."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Ibrahim, R.W., Elobaid, R.M., and Obaiys, S.J. (2020). Generalized Briot-Bouquet differential equation based on new differential operator with complex connections. Axioms, 9.","DOI":"10.3390\/axioms9020042"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1186\/s13662-020-02788-6","article-title":"On subclasses of analytic functions based on a quantum symmetric conformable differential operator with application","volume":"2020","author":"Ibrahim","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Ibrahim, R.W., and Baleanu, D. (2021). Analytic Solution of the Langevin Differential Equations Dominated by a Multibrot. Fractal Set. Fractal Fract., 5.","DOI":"10.3390\/fractalfract5020050"},{"key":"ref_17","first-page":"163","article-title":"An invitation to the study of univalent and multivalent functions","volume":"2","author":"Goodman","year":"1979","journal-title":"Intr. Nat. J. Hath. Mh. Sci."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/3\/141\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:24:09Z","timestamp":1760163849000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/3\/141"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,6,30]]},"references-count":17,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2021,9]]}},"alternative-id":["axioms10030141"],"URL":"https:\/\/doi.org\/10.3390\/axioms10030141","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,6,30]]}}}