{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T10:10:01Z","timestamp":1747131001860,"version":"3.40.5"},"reference-count":20,"publisher":"Walter de Gruyter GmbH","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2018,10,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In this paper we propose a hybrid method for solving inhomogeneous elliptic PDEs based on the unified transform. This approach relies on the derivation of the global relation, containing certain integral transforms of the given boundary data as well as of the unknown boundary values. Herewith, the approximate global relation for the Poisson equation is solved numerically using a collocation method on the complex \u03bb-plane, based on Legendre expansions. The corresponding numerical results are presented using closed-form expressions and numerical approximations for different types of boundary and source data, indicating the applicability of the considered approach. Additionally, the full solution is computed in a recursive manner by splitting the domain into smaller concentric polygons, and by using a spatial-stepping scheme followed by an interpolation step. Furthermore, numerical results are also given for the solution of the Poisson and the inhomogeneous Helmholtz equations on several convex polygons. Additional results are provided for the case of nonconvex polygons as well as for the case of a problem with discontinuities across an interface. The proposed approach provides a framework for solving inhomogeneous elliptic PDEs using the unified transform.<\/jats:p>","DOI":"10.1515\/cmam-2017-0053","type":"journal-article","created":{"date-parts":[[2017,12,5]],"date-time":"2017-12-05T22:15:51Z","timestamp":1512512151000},"page":"653-672","source":"Crossref","is-referenced-by-count":1,"title":["A Hybrid Method for Solving Inhomogeneous Elliptic PDEs Based on Fokas Method"],"prefix":"10.1515","volume":"18","author":[{"given":"Eleftherios-Nektarios G.","family":"Grylonakis","sequence":"first","affiliation":[{"name":"Department of Electrical and Computer Engineering , School of Engineering , Democritus University of Thrace , University Campus, Kimmeria, GR 67100 Xanthi , Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christos K.","family":"Filelis-Papadopoulos","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering , School of Engineering , Democritus University of Thrace , University Campus, Kimmeria, GR 67100 Xanthi , Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"George A.","family":"Gravvanis","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering , School of Engineering , Democritus University of Thrace , University Campus, Kimmeria, GR 67100 Xanthi , Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2017,12,5]]},"reference":[{"key":"2025051309552805410_j_cmam-2017-0053_ref_001_w2aab3b7e2938b1b6b1ab2ab1Aa","doi-asserted-by":"crossref","unstructured":"A. C. L.  Ashton and A. S.  Fokas,\nElliptic equations with low regularity boundary data via the unified method,\nComplex Var. Elliptic Equ. 60 (2015), no. 5, 596\u2013619.","DOI":"10.1080\/17476933.2014.964227"},{"key":"2025051309552805410_j_cmam-2017-0053_ref_002_w2aab3b7e2938b1b6b1ab2ab2Aa","unstructured":"C. F.  Chan Man, D.  De Kee and P. N.  Kaloni,\nAdvanced Mathematics for Engineering and Science,\nWorld Scientific, Hackensack,, 2003."},{"key":"2025051309552805410_j_cmam-2017-0053_ref_003_w2aab3b7e2938b1b6b1ab2ab3Aa","doi-asserted-by":"crossref","unstructured":"C.-I. R.  Davis and B.  Fornberg,\nA spectrally accurate numerical implementation of the Fokas transform method for Helmholtz-type PDEs,\nComplex Var. Elliptic Equ. 59 (2014), no. 4, 564\u2013577.","DOI":"10.1080\/17476933.2013.766883"},{"key":"2025051309552805410_j_cmam-2017-0053_ref_004_w2aab3b7e2938b1b6b1ab2ab4Aa","doi-asserted-by":"crossref","unstructured":"A. S.  Fokas,\nA unified transform method for solving linear and certain nonlinear PDEs,\nProc. Roy. Soc. London Ser. A 453 (1997), no. 1962, 1411\u20131443.","DOI":"10.1098\/rspa.1997.0077"},{"key":"2025051309552805410_j_cmam-2017-0053_ref_005_w2aab3b7e2938b1b6b1ab2ab5Aa","doi-asserted-by":"crossref","unstructured":"A. S.  Fokas,\nTwo-dimensional linear partial differential equations in a convex polygon,\nR. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457 (2001), no. 2006, 371\u2013393.","DOI":"10.1098\/rspa.2000.0671"},{"key":"2025051309552805410_j_cmam-2017-0053_ref_006_w2aab3b7e2938b1b6b1ab2ab6Aa","doi-asserted-by":"crossref","unstructured":"A. S.  Fokas,\nA new transform method for evolution partial differential equations,\nIMA J. Appl. Math. 67 (2002), no. 6, 559\u2013590.","DOI":"10.1093\/imamat\/67.6.559"},{"key":"2025051309552805410_j_cmam-2017-0053_ref_007_w2aab3b7e2938b1b6b1ab2ab7Aa","doi-asserted-by":"crossref","unstructured":"A. S.  Fokas,\nBoundary-value problems for linear PDEs with variable coefficients,\nProc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 (2004), no. 2044, 1131\u20131151.","DOI":"10.1098\/rspa.2003.1208"},{"key":"2025051309552805410_j_cmam-2017-0053_ref_008_w2aab3b7e2938b1b6b1ab2ab8Aa","doi-asserted-by":"crossref","unstructured":"A. S.  Fokas,\nA Unified Approach to Boundary Value Problems,\nCBMS-NSF Regional Conf. Ser. in Appl. Math. 78,\nSociety for Industrial and Applied Mathematics, Philadelphia, 2008.","DOI":"10.1137\/1.9780898717068"},{"key":"2025051309552805410_j_cmam-2017-0053_ref_009_w2aab3b7e2938b1b6b1ab2ab9Aa","doi-asserted-by":"crossref","unstructured":"A. S.  Fokas, A.  Iserles and S. A.  Smitheman,\nThe unified transform in polygonal domains via the explicit Fourier transform of Legendre polynomials,\nUnified Transform for Boundary Value Problems,\nSociety for Industrial and Applied Mathematics, Philadelphia (2015), 163\u2013171.","DOI":"10.1137\/1.9781611973822.ch7"},{"key":"2025051309552805410_j_cmam-2017-0053_ref_010_w2aab3b7e2938b1b6b1ab2ac10Aa","doi-asserted-by":"crossref","unstructured":"B.  Fornberg and N.  Flyer,\nA numerical implementation of Fokas boundary integral approach: Laplace\u2019s equation on a polygonal domain,\nProc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467 (2011), no. 2134, 2983\u20133003.","DOI":"10.1098\/rspa.2011.0032"},{"key":"2025051309552805410_j_cmam-2017-0053_ref_011_w2aab3b7e2938b1b6b1ab2ac11Aa","doi-asserted-by":"crossref","unstructured":"S. R.  Fulton, A. S.  Fokas and C. A.  Xenophontos,\nAn analytical method for linear elliptic PDEs and its numerical implementation,\nJ. Comput. Appl. Math. 167 (2004), no. 2, 465\u2013483.","DOI":"10.1016\/j.cam.2003.10.012"},{"key":"2025051309552805410_j_cmam-2017-0053_ref_012_w2aab3b7e2938b1b6b1ab2ac12Aa","unstructured":"E.-N. G.  Grylonakis, C. K.  Filelis-Papadopoulos and G. A.  Gravvanis,\nA note on solving the generalized Dirichlet to Neumann map on irregular polygons using generic factored approximate sparse inverses,\nCMES Comput. Model. Eng. Sci. 109 (2015), no. 6, 505\u2013517."},{"key":"2025051309552805410_j_cmam-2017-0053_ref_013_w2aab3b7e2938b1b6b1ab2ac13Aa","doi-asserted-by":"crossref","unstructured":"P.  Hashemzadeh, A. S.  Fokas and S. A.  Smitheman,\nA numerical technique for linear elliptic partial differential equations in polygonal domains,\nProc. A. 471 (2015), no. 2175, Article ID 20140747.","DOI":"10.1098\/rspa.2014.0747"},{"key":"2025051309552805410_j_cmam-2017-0053_ref_014_w2aab3b7e2938b1b6b1ab2ac14Aa","unstructured":"A. S.  Kronrod,\nNodes and Weights of Quadrature Formulas. Sixteen-Place Tables,\nConsultants Bureau, New York, 1965."},{"key":"2025051309552805410_j_cmam-2017-0053_ref_015_w2aab3b7e2938b1b6b1ab2ac15Aa","doi-asserted-by":"crossref","unstructured":"A. N.  Marques, J.-C.  Nave and R. R.  Rosales,\nA correction function method for Poisson problems with interface jump conditions,\nJ. Comput. Phys. 230 (2011), no. 20, 7567\u20137597.","DOI":"10.1016\/j.jcp.2011.06.014"},{"key":"2025051309552805410_j_cmam-2017-0053_ref_016_w2aab3b7e2938b1b6b1ab2ac16Aa","doi-asserted-by":"crossref","unstructured":"L. F.  Shampine,\nMATLAB program for quadrature in 2D,\nAppl. Math. Comput. 202 (2008), no. 1, 266\u2013274.","DOI":"10.1016\/j.amc.2008.02.012"},{"key":"2025051309552805410_j_cmam-2017-0053_ref_017_w2aab3b7e2938b1b6b1ab2ac17Aa","doi-asserted-by":"crossref","unstructured":"A. G.  Sifalakis, A. S.  Fokas, S. R.  Fulton and Y. G.  Saridakis,\nThe generalized Dirichlet\u2013Neumann map for linear elliptic PDEs and its numerical implementation,\nJ. Comput. Appl. Math. 219 (2008), no. 1, 9\u201334.","DOI":"10.1016\/j.cam.2007.07.012"},{"key":"2025051309552805410_j_cmam-2017-0053_ref_018_w2aab3b7e2938b1b6b1ab2ac18Aa","doi-asserted-by":"crossref","unstructured":"E. A.  Spence and A. S.  Fokas,\nA new transform method II: The global relation and boundary-value problems in polar coordinates,\nProc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 466 (2010), no. 2120, 2283\u20132307.","DOI":"10.1098\/rspa.2009.0513"},{"key":"2025051309552805410_j_cmam-2017-0053_ref_019_w2aab3b7e2938b1b6b1ab2ac19Aa","doi-asserted-by":"crossref","unstructured":"E.  S\u00fcli and D. F.  Mayers,\nAn Introduction to Numerical Analysis,\nCambridge University Press, Cambridge, 2003.","DOI":"10.1017\/CBO9780511801181"},{"key":"2025051309552805410_j_cmam-2017-0053_ref_020_w2aab3b7e2938b1b6b1ab2ac20Aa","doi-asserted-by":"crossref","unstructured":"Z.  Yosibash,\nNumerical analysis on singular solutions of the Poisson equation in two-dimensions,\nComput. Mech. 20 (1997), no. 4, 320\u2013330.","DOI":"10.1007\/s004660050254"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/18\/4\/article-p653.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2017-0053\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2017-0053\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T09:56:09Z","timestamp":1747130169000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2017-0053\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,12,5]]},"references-count":20,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2017,12,5]]},"published-print":{"date-parts":[[2018,10,1]]}},"alternative-id":["10.1515\/cmam-2017-0053"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2017-0053","relation":{},"ISSN":["1609-9389","1609-4840"],"issn-type":[{"type":"electronic","value":"1609-9389"},{"type":"print","value":"1609-4840"}],"subject":[],"published":{"date-parts":[[2017,12,5]]}}}