{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T13:39:43Z","timestamp":1781271583664,"version":"3.54.1"},"reference-count":43,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2022,4,18]],"date-time":"2022-04-18T00:00:00Z","timestamp":1650240000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this work, an efficient and robust numerical scheme is proposed to solve the variable coefficients\u2019 fourth-order partial differential equations (FOPDEs) that arise in Euler\u2013Bernoulli beam models. When partial differential equations (PDEs) are of higher order and invoke variable coefficients, then the numerical solution is quite a tedious and challenging problem, which is our main concern in this paper. The current scheme is hybrid in nature in which the second-order finite difference is used for temporal discretization, while spatial derivatives and solutions are approximated via the Haar wavelet. Next, the integration and Haar matrices are used to convert partial differential equations (PDEs) to the system of linear equations, which can be handled easily. Besides this, we derive the theoretical result for stability via the Lax\u2013Richtmyer criterion and verify it computationally. Moreover, we address the computational convergence rate, which is near order two. Several test problems are given to measure the accuracy of the suggested scheme. Computations validate that the present scheme works well for such problems. The calculated results are also compared with the earlier work and the exact solutions. The comparison shows that the outcomes are in good agreement with both the exact solutions and the available results in the literature.<\/jats:p>","DOI":"10.3390\/e24040567","type":"journal-article","created":{"date-parts":[[2022,4,18]],"date-time":"2022-04-18T22:04:02Z","timestamp":1650319442000},"page":"567","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Numerical Solutions of Variable Coefficient Higher-Order Partial Differential Equations Arising in Beam Models"],"prefix":"10.3390","volume":"24","author":[{"given":"Abdul","family":"Ghafoor","sequence":"first","affiliation":[{"name":"Institute of Numerical Sciences, Kohat University of Science and Technology, Kohat 26000, KP, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sirajul","family":"Haq","sequence":"additional","affiliation":[{"name":"Faculty of Engineering Sciences, GIK Institute, Topi 23640, KP, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Manzoor","family":"Hussain","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Sciences and Technology, Women University of Azad Jammu and Kashmir, Bagh 12500, AJK, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8889-3768","authenticated-orcid":false,"given":"Thabet","family":"Abdeljawad","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia"},{"name":"Department of Medical Research, China Medical University, Taichung 40402, Taiwan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Manar A.","family":"Alqudah","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Faculty of Sciences, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,4,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1003","DOI":"10.1080\/01630560701587759","article-title":"An Euler\u2013Bernoulli beam with dynamic frictionless contact: Penalty approximation and existence","volume":"28","author":"Ahn","year":"2007","journal-title":"Numer. Funct. Anal. Optim."},{"key":"ref_2","first-page":"95","article-title":"Parameter estimation for the Euler\u2013Bernoulli beam","volume":"4","author":"Kunisch","year":"1985","journal-title":"Mat. Apficada Comput."},{"key":"ref_3","unstructured":"Timoshenko, S.P., and Gere, J.M. (1961). 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