{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T14:41:50Z","timestamp":1753886510456,"version":"3.41.2"},"reference-count":5,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2010,3,25]],"date-time":"2010-03-25T00:00:00Z","timestamp":1269475200000},"content-version":"vor","delay-in-days":83,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100006752","name":"University of Porto","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100006752","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>We deal with the following fractional generalization of the Laplace equation for rectangular domains (<jats:italic>x<\/jats:italic>, <jats:italic>y<\/jats:italic>) \u2208 (<jats:italic>x<\/jats:italic><jats:sub>0<\/jats:sub>, <jats:italic>X<\/jats:italic><jats:sub>0<\/jats:sub>) \u00d7 (<jats:italic>y<\/jats:italic><jats:sub>0<\/jats:sub>, <jats:italic>Y<\/jats:italic><jats:sub>0<\/jats:sub>) \u2282 <jats:italic>\u211d<\/jats:italic><jats:sub>+<\/jats:sub> \u00d7 <jats:italic>\u211d<\/jats:italic><jats:sub>+<\/jats:sub>, which is associated with the Riemann\u2010Liouville fractional derivatives <jats:italic>\u0394<\/jats:italic><jats:sup><jats:italic>\u03b1<\/jats:italic>,<jats:italic>\u03b2<\/jats:italic><\/jats:sup><jats:italic>u<\/jats:italic>(<jats:italic>x<\/jats:italic>, <jats:italic>y<\/jats:italic>) = <jats:italic>\u03bb<\/jats:italic><jats:italic>u<\/jats:italic>(<jats:italic>x<\/jats:italic>, <jats:italic>y<\/jats:italic>), , where <jats:italic>\u03bb<\/jats:italic> \u2208 <jats:italic>\u2102<\/jats:italic>, (<jats:italic>\u03b1<\/jats:italic>, <jats:italic>\u03b2<\/jats:italic>) \u2208 [0, 1] \u00d7 [0, 1]. Reducing the left\u2010hand side of this equation to\nthe sum of fractional integrals by <jats:italic>x<\/jats:italic> and <jats:italic>y<\/jats:italic>, we then use the operational technique for\nthe conventional right\u2010sided Laplace transformation and its extension to generalized\nfunctions to describe a complete family of eigenfunctions and fundamental solutions\nof the operator <jats:italic>\u0394<\/jats:italic><jats:sup><jats:italic>\u03b1<\/jats:italic>,<jats:italic>\u03b2<\/jats:italic><\/jats:sup> in classes of functions represented by the left\u2010sided fractional\nintegral of a summable function or just admitting a summable fractional derivative. \nA symbolic operational form of the solutions in terms of the Mittag\u2010Leffler functions\nis exhibited. The case of the separation of variables is also considered. An analog\nof the fractional logarithmic solution is presented. Classical particular cases of\nsolutions are demonstrated.<\/jats:p>","DOI":"10.1155\/2010\/541934","type":"journal-article","created":{"date-parts":[[2010,3,25]],"date-time":"2010-03-25T15:30:29Z","timestamp":1269531029000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Eigenfunctions and Fundamental Solutions of the Fractional Two\u2010Parameter Laplacian"],"prefix":"10.1155","volume":"2010","author":[{"given":"Semyon","family":"Yakubovich","sequence":"first","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2010,3,25]]},"reference":[{"volume-title":"Fractional Integrals and Derivatives: Theory and Applications","year":"1993","author":"Samko S. G.","key":"e_1_2_5_1_2"},{"key":"e_1_2_5_2_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-011-1196-6"},{"volume-title":"Integral Transforms and Operational Calculus","year":"1965","author":"Ditkin V. A.","key":"e_1_2_5_3_2"},{"volume-title":"Generalized Integral Transformations","year":"1987","author":"Zemanian A. H.","key":"e_1_2_5_4_2"},{"volume-title":"Integral Transforms and Representations of Functions in a Complex Domain","year":"1966","author":"Dzhrbashyan M. M.","key":"e_1_2_5_5_2"}],"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2010\/541934.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2010\/541934.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/2010\/541934","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,18]],"date-time":"2024-06-18T16:25:37Z","timestamp":1718727937000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/2010\/541934"}},"subtitle":[],"editor":[{"given":"Nak","family":"Cho","sequence":"additional","affiliation":[]}],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":5,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1155\/2010\/541934"],"URL":"https:\/\/doi.org\/10.1155\/2010\/541934","archive":["Portico"],"relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"type":"print","value":"0161-1712"},{"type":"electronic","value":"1687-0425"}],"subject":[],"published":{"date-parts":[[2010,1]]},"assertion":[{"value":"2009-11-05","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2010-02-22","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2010-03-25","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"541934"}}