{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:50:37Z","timestamp":1760241037222,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2019,11,14]],"date-time":"2019-11-14T00:00:00Z","timestamp":1573689600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100006753","name":"Centro de Matem\u00e1tica Universidade do Porto","doi-asserted-by":"publisher","award":["UID\/ MAT\/ 00144\/ 2019"],"award-info":[{"award-number":["UID\/ MAT\/ 00144\/ 2019"]}],"id":[{"id":"10.13039\/501100006753","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["MCA"],"abstract":"<jats:p>Differential eigenvalue problems arise in many fields of Mathematics and Physics, often arriving, as auxiliary problems, when solving partial differential equations. In this work, we present a method for eigenvalues computation following the Tau method philosophy and using Tau Toolbox tools. This Matlab toolbox was recently presented and here we explore its potential use and suitability for this problem. The first step is to translate the eigenvalue differential problem into an algebraic approximated eigenvalues problem. In a second step, making use of symbolic computations, we arrive at the exact polynomial expression of the determinant of the algebraic problem matrix, allowing us to get high accuracy approximations of differential eigenvalues.<\/jats:p>","DOI":"10.3390\/mca24040096","type":"journal-article","created":{"date-parts":[[2019,11,15]],"date-time":"2019-11-15T11:25:56Z","timestamp":1573817156000},"page":"96","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Almost Exact Computation of Eigenvalues in Approximate Differential Problems"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3512-5930","authenticated-orcid":false,"given":"Jos\u00e9 M. A.","family":"Matos","sequence":"first","affiliation":[{"name":"Instituto Superior de Engenharia do Porto, Rua Dr. Ant\u00f3nio Bernardino de Almeida, 431, 4249-015 Porto, Portugal"},{"name":"Centro de Matem\u00e1tica da Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1451-4777","authenticated-orcid":false,"given":"Maria Jo\u00e3o","family":"Rodrigues","sequence":"additional","affiliation":[{"name":"Centro de Matem\u00e1tica da Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal"},{"name":"Faculdade de Ci\u00eancias da Universidade do Porto, Rua do Campo Alegre, s\/n, 4169-007 Porto, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2019,11,14]]},"reference":[{"key":"ref_1","first-page":"1247","article-title":"An efficient method for computing eigenelements of Sturm\u2013Liouville fourth-order boundary value problems","volume":"182","author":"Attili","year":"2006","journal-title":"Appl. Math. Comput."},{"key":"ref_2","first-page":"73","article-title":"Computation of Eigenvalues of the Fourth Order Sturm\u2013Liouville BVP by Galerkin Weighted Residual Method","volume":"9","author":"Farzana","year":"2015","journal-title":"J. Adv. Math. Comput. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1137\/0153002","article-title":"Pseudospectra of the Orr\u2013Sommerfeld Operator","volume":"53","author":"Reddy","year":"1993","journal-title":"SIAM J. Appl. Math."},{"key":"ref_4","unstructured":"Pop, I.S., and Gheorghiu, C.I. (August, January 29). A Chebyshev\u2013Galerkin Method for Fourth Order Problems. Proceedings of the International Conference on Approximation and Optimization, Cluj-Napoca, Romania."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"689","DOI":"10.1017\/S0022112071002842","article-title":"Accurate solution of the Orr\u2013Sommerfeld stability equation","volume":"50","author":"Orzag","year":"1971","journal-title":"J. Fluid Mech."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"611","DOI":"10.3934\/cpaa.2010.9.611","article-title":"Eigenmodes and eigenfrequencies of vibrating elliptic membranes: A Klein oscillation theorem and numerical calculations","volume":"9","author":"Neves","year":"2009","journal-title":"Commun. Pure Appl. Anal."},{"key":"ref_7","first-page":"11990","article-title":"Spectral collocation solutions to multiparameter Mathieu\u2019s system","volume":"218","author":"Gheorghiu","year":"2012","journal-title":"Appl. Math. Comput."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"3064","DOI":"10.1016\/j.cam.2010.04.023","article-title":"Accurate solutions of fourth order Sturm\u2013Liouville problems","volume":"234","author":"Chanane","year":"2010","journal-title":"J. Comput. Appl. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1007\/BF02259906","article-title":"Numerical solution of differential eigenvalue problems with an operational approach to the Tau method","volume":"31","author":"Ortiz","year":"1983","journal-title":"Computing"},{"key":"ref_10","unstructured":"Matos, J.M.A., Rodrigues, M.J., and Matos, J.C. (2017). Explicit formulae for derivatives and primitives of orthogonal polynomials. arXiv."},{"key":"ref_11","unstructured":"Matos, J.M.A., Rodrigues, M.J., and Matos, J.C. (2019). Explicit Formulae for Intergral-Differential Operational Matrices, submitted for publication."},{"key":"ref_12","unstructured":"Gheorghiu, C.I. (2007). Spectral Methods for Differential Problems, Institute of Numerical Analysis."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1137\/070704228","article-title":"Gegenbauer Tau Methods with and without Spurious Eigenvalues","volume":"47","author":"Charalambides","year":"2008","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1007\/s11786-016-0269-x","article-title":"Towards a Lanczos Tau-Method Toolkit for Differential Problems","volume":"10","author":"Trindade","year":"2016","journal-title":"Math. Comput. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1007\/s11786-018-0331-y","article-title":"Dealing with functional coefficients within Tau method","volume":"12","author":"Trindade","year":"2018","journal-title":"Math. Comput. Sci."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Constanda, C., Dalla Riva, M., Lamberti, P.D., and Musolino, P. (2017). Spectral Lanczos\u2019 Tau Method for Systems of Nonlinear Integro-Differential Equations. Integral Methods in Science and Engineering, Volume 1: Theoretical Techniques, Springer.","DOI":"10.1007\/978-3-319-59384-5"},{"key":"ref_17","unstructured":"Chihara, T.S. (2011). An Introduction to Orthogonal Polynomials, Dover Publications."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1007\/BF01385633","article-title":"Some Results About the Pseudospectral Approximation of One-Dimensional Fourth-Order Problems","volume":"58","author":"Funaro","year":"1990","journal-title":"Numer. Math."},{"key":"ref_19","unstructured":"Abramowitz, M., and Stegun, I.A. (1974). Mathieu Functions. Handbook of Mathematical Functions, Dover Publications."}],"container-title":["Mathematical and Computational Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2297-8747\/24\/4\/96\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:34:35Z","timestamp":1760189675000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2297-8747\/24\/4\/96"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,11,14]]},"references-count":19,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2019,12]]}},"alternative-id":["mca24040096"],"URL":"https:\/\/doi.org\/10.3390\/mca24040096","relation":{},"ISSN":["2297-8747"],"issn-type":[{"type":"electronic","value":"2297-8747"}],"subject":[],"published":{"date-parts":[[2019,11,14]]}}}