{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:05:18Z","timestamp":1787317518337,"version":"build-2736575974"},"reference-count":37,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2024,10,31]],"date-time":"2024-10-31T00:00:00Z","timestamp":1730332800000},"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>In this paper, an efficient computational discretization approach is investigated for nonlinear fourth-order boundary value problems using beam theory. We specifically deal with nonlinear models described by fourth-order boundary value problems. The proposed method is applied on three different types of problems, i.e., the problem when an elastic bearing is non-zero (Case I), the problem under homogeneous boundary conditions of the unknown function and its second derivative (Case II), and the problem with integral boundary conditions (Case III). Moreover, the convergence analysis of the proposed method is provided. Finally, illustrative examples are included to demonstrate the applicability and validity of the technique and the comparison is made with the existing methods to show the efficiency and accuracy of the proposed method.<\/jats:p>","DOI":"10.3390\/axioms13110757","type":"journal-article","created":{"date-parts":[[2024,11,1]],"date-time":"2024-11-01T13:09:27Z","timestamp":1730466567000},"page":"757","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Fast and Accurate Numerical Method for Solving Nonlinear Fourth-Order Boundary Value Problems in the Beam Theory"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9598-4943","authenticated-orcid":false,"given":"Mohammad Ali","family":"Mehrpouya","sequence":"first","affiliation":[{"name":"Department of Mathematics, Tafresh University, Tafresh 39518-79611, Iran"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7351-9223","authenticated-orcid":false,"given":"Rezvan","family":"Salehi","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran P.O. Box 14115-134, Iran"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8375-5553","authenticated-orcid":false,"given":"Patricia J. Y.","family":"Wong","sequence":"additional","affiliation":[{"name":"School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,10,31]]},"reference":[{"key":"ref_1","first-page":"15","article-title":"Vibration analysis of beams","volume":"1","author":"Gawali","year":"2011","journal-title":"World Res. J. Civ. Eng."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"935","DOI":"10.1006\/jsvi.1999.2257","article-title":"Dynamics of transversely vibrating beams using four engineering theories","volume":"225","author":"Han","year":"1999","journal-title":"J. Sound Vib."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1115\/1.4010053","article-title":"The effect of an axial force on the vibration of hinged bars","volume":"17","year":"1950","journal-title":"J. Appl. Mech."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"544","DOI":"10.1016\/j.cnsns.2017.12.002","article-title":"An iterative kernel based method for fourth order nonlinear equation with nonlinear boundary condition","volume":"59","author":"Azarnavid","year":"2018","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_5","first-page":"1","article-title":"Development of a new iterative method and its convergence analysis for nonlinear fourth-order boundary value problems arising in beam analysis","volume":"SI","author":"Tomar","year":"2022","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_6","first-page":"11","article-title":"Iterative solutions for a beam equation with nonlinear boundary conditions of third order","volume":"159","author":"Ma","year":"2004","journal-title":"Appl. Math. Comput."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3834","DOI":"10.1016\/j.na.2009.02.051","article-title":"Monotone positive solutions for a fourth order equation with nonlinear boundary conditions","volume":"71","author":"Alves","year":"2009","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2088","DOI":"10.1137\/110847469","article-title":"A quadratic C0 interior penalty method for linear fourth order boundary value problems with boundary conditions of the Cahn\u2013Hilliard type","volume":"50","author":"Brenner","year":"2012","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/2251-7456-6-1","article-title":"Iterative reproducing kernel method for a beam equation with third-order nonlinear boundary conditions","volume":"6","author":"Geng","year":"2012","journal-title":"Math. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/S0898-1221(98)00047-9","article-title":"Finite-element approximation of a fourth-order differential equation","volume":"35","author":"Shin","year":"1998","journal-title":"Comput. Math. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1016\/j.camwa.2006.08.019","article-title":"Error estimates of finite-element approximations for a fourth-order differential equation","volume":"52","author":"Ohm","year":"2006","journal-title":"Comput. Math. Appl."},{"key":"ref_12","unstructured":"Kim, J., and Shin, J. (2008, January 25\u201329). A finite element approximation of a fourth-order boundary value problem. Proceedings of the Mathematical Optimization Theory and Applications (Proceedings of the Sixth Vietnam-Korea Joint Workshop), Hanoi, Vietnam."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1016\/j.camwa.2010.04.037","article-title":"Iterative method for solving a nonlinear fourth order boundary value problem","volume":"60","author":"Dang","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"887","DOI":"10.1007\/s11075-019-00842-3","article-title":"Existence results and iterative method for a fully fourth-order nonlinear integral boundary value problem","volume":"85","author":"Dang","year":"2020","journal-title":"Numer. Algorithms"},{"key":"ref_15","first-page":"174","article-title":"Existence results and numerical solution of a fourth-order nonlinear differential equation with two integral boundary conditions","volume":"12","author":"Dang","year":"2023","journal-title":"Palest. J. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1016\/j.camwa.2009.04.002","article-title":"Positive solutions for a class of boundary-value problems with integral boundary conditions","volume":"58","author":"Zhang","year":"2009","journal-title":"Comput. Math. Appl."},{"key":"ref_17","first-page":"1","article-title":"Solvability of a fourth-order boundary value problem with integral boundary conditions","volume":"782363","author":"Li","year":"2013","journal-title":"J. Appl. Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1186\/s13661-015-0441-2","article-title":"Monotone positive solution of a fourth-order BVP with integral boundary conditions","volume":"2015","author":"Lv","year":"2015","journal-title":"Bound. Value Probl."},{"key":"ref_19","first-page":"73","article-title":"Positive solutions of a nonlinear fourth-order integral boundary value problem","volume":"54","author":"Benaicha","year":"2016","journal-title":"Ann. West Univ. Timis.-Math. Comput. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1835","DOI":"10.1007\/s00366-020-01125-5","article-title":"Chebyshev\u2013Gauss\u2013Lobatto collocation method for variable-order time fractional generalized Hirota\u2013Satsuma coupled KdV system","volume":"38","author":"Heydari","year":"2022","journal-title":"Eng. Comput."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"925","DOI":"10.1007\/s00366-021-01283-0","article-title":"Jacobi\u2013Gauss\u2013Lobatto collocation approach for non-singular variable-order time fractional generalized Kuramoto\u2013Sivashinsky equation","volume":"38","author":"Heydari","year":"2022","journal-title":"Eng. Comput."},{"key":"ref_22","unstructured":"Canuto, C., Hussaini, M., Quarteroni, A., and Zang, T. (1991). Spectral Methods in Fluid Dynamics, Springer."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Trefethen, L.N. (2000). Spectral Methods in MATLAB, SIAM.","DOI":"10.1137\/1.9780898719598"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1704","DOI":"10.1016\/j.na.2006.08.009","article-title":"The upper and lower solution method for some fourth-order boundary value problems","volume":"67","author":"Bai","year":"2007","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1017\/S0308210506001041","article-title":"Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions","volume":"138","author":"Webb","year":"2008","journal-title":"Proc. R. Soc. Edinb. Sect. A Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/j.jmaa.2004.11.021","article-title":"Two-parameter nonresonance condition for the existence of fourth-order boundary value problems","volume":"308","author":"Li","year":"2005","journal-title":"J. Math. Anal. Appl."},{"key":"ref_27","first-page":"407","article-title":"Positive solutions of fourth-order two point boundary value problems","volume":"148","author":"Liu","year":"2004","journal-title":"Appl. Math. Comput."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2097","DOI":"10.1016\/j.nonrwa.2008.03.017","article-title":"Existence and iteration of monotone positive solutions for an elastic beam equation with a corner","volume":"10","author":"Zhang","year":"2009","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1002\/mma.1670150406","article-title":"Exponential attractors for non-autonomous systems: Long-time behaviour of vibrating beams","volume":"15","author":"Feireisl","year":"1992","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Canuto, C., Hussaini, M.Y., Quarteroni, A., and Zang, T.A. (2007). Spectral Methods: Fundamentals in Single Domains, Springer.","DOI":"10.1007\/978-3-540-30728-0"},{"key":"ref_31","unstructured":"Fornberg, B. (1998). A Practical Guide to Pseudospectral Methods, Cambridge University Press."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Shen, J., Tang, T., and Wang, L. (2011). Spectral Methods: Algorithms, Analysis and Applications, Springer.","DOI":"10.1007\/978-3-540-71041-7"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Gautschi, W. (2004). Orthogonal Polynomials: Computation and Approximation, Oxford University Press.","DOI":"10.1093\/oso\/9780198506720.001.0001"},{"key":"ref_34","unstructured":"Gheorghiu, C.I. (2007). Spectral Methods for Differential Problems, Casa C\u0103rtii de Stiint\u0103."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"909","DOI":"10.1140\/epjp\/s13360-021-01915-w","article-title":"A numerical scheme based on the collocation and optimization methods for accurate solution of sensitive boundary value problems","volume":"136","author":"Mehrpouya","year":"2021","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_36","unstructured":"Atkinson, K. (2008). An Introduction to Numerical Analysis, India Pvt. Limited. [2nd ed.]."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1146","DOI":"10.1080\/00207160.2020.1807521","article-title":"A robust pseudospectral method for numerical solution of nonlinear optimal control problems","volume":"98","author":"Mehrpouya","year":"2021","journal-title":"Int. J. Comput. Math."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/11\/757\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:25:36Z","timestamp":1760113536000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/11\/757"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,31]]},"references-count":37,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2024,11]]}},"alternative-id":["axioms13110757"],"URL":"https:\/\/doi.org\/10.3390\/axioms13110757","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,10,31]]}}}