{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,8]],"date-time":"2025-11-08T22:53:11Z","timestamp":1762642391992,"version":"build-2065373602"},"reference-count":56,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,7,28]],"date-time":"2020-07-28T00:00:00Z","timestamp":1595894400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["79707"],"award-info":[{"award-number":["79707"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Using the unified transform, also known as the Fokas method, we analyse the modified Helmholtz equation in the regular hexagon with symmetric Dirichlet boundary conditions; namely, the boundary value problem where the trace of the solution is given by the same function on each side of the hexagon. We show that if this function is odd, then this problem can be solved in closed form; numerical verification is also provided.<\/jats:p>","DOI":"10.3390\/axioms9030089","type":"journal-article","created":{"date-parts":[[2020,7,28]],"date-time":"2020-07-28T10:16:49Z","timestamp":1595931409000},"page":"89","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["The Modified Helmholtz Equation on a Regular Hexagon\u2014The Symmetric Dirichlet Problem"],"prefix":"10.3390","volume":"9","author":[{"given":"Konstantinos","family":"Kalimeris","sequence":"first","affiliation":[{"name":"Research Center of Pure and Applied Mathematics, Academy of Athens, 11527 Athens, Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Athanassios S.","family":"Fokas","sequence":"additional","affiliation":[{"name":"Research Center of Pure and Applied Mathematics, Academy of Athens, 11527 Athens, Greece"},{"name":"Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK"},{"name":"Viterbi School of Engineering, University of Southern California, Los Angeles, CA 90089-2560, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,7,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1411","DOI":"10.1098\/rspa.1997.0077","article-title":"A unified transform method for solving linear and certain nonlinear PDEs","volume":"453","author":"Fokas","year":"1997","journal-title":"Proc. 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