{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,13]],"date-time":"2026-04-13T13:14:00Z","timestamp":1776086040569,"version":"3.50.1"},"reference-count":46,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2024,4,11]],"date-time":"2024-04-11T00:00:00Z","timestamp":1712793600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>A B-spline function is a series of flexible elements that are managed by a set of control points to produce smooth curves. By using a variety of points, these functions make it possible to build and maintain complicated shapes. Any spline function of a certain degree can be expressed as a linear combination of the B-spline basis of that degree. The flexibility, symmetry and high-order accuracy of the B-spline functions make it possible to tackle the best solutions. In this study, extended cubic B-spline (ECBS) functions are utilized for the numerical solutions of the generalized nonlinear time-fractional Klein\u2013Gordon Equation (TFKGE). Initially, the Caputo time-fractional derivative (CTFD) is approximated using standard finite difference techniques, and the space derivatives are discretized by utilizing ECBS functions. The stability and convergence analysis are discussed for the given numerical scheme. The presented technique is tested on a variety of problems, and the approximate results are compared with the existing computational schemes.<\/jats:p>","DOI":"10.3390\/computation12040080","type":"journal-article","created":{"date-parts":[[2024,4,11]],"date-time":"2024-04-11T03:29:04Z","timestamp":1712806144000},"page":"80","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Application of an Extended Cubic B-Spline to Find the Numerical Solution of the Generalized Nonlinear Time-Fractional Klein\u2013Gordon Equation in Mathematical Physics"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1567-0264","authenticated-orcid":false,"given":"Miguel","family":"Vivas-Cortez","sequence":"first","affiliation":[{"name":"School of Physical Sciences and Mathematics, Faculty of Exact and Natural Sciences, Pontifical Catholic University of Ecuador, Av. 12 de Octubre 1076 y Roca, Apartado, Quito 17-01-2184, Ecuador"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5247-2913","authenticated-orcid":false,"given":"M. J.","family":"Huntul","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Jazan University, P.O. Box. 114, Jazan 45142, Saudi Arabia"}]},{"given":"Maria","family":"Khalid","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-4084-442X","authenticated-orcid":false,"given":"Madiha","family":"Shafiq","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0491-1528","authenticated-orcid":false,"given":"Muhammad","family":"Abbas","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4442-7498","authenticated-orcid":false,"given":"Muhammad Kashif","family":"Iqbal","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government College University, Faisalabad 38000, Pakistan"}]}],"member":"1968","published-online":{"date-parts":[[2024,4,11]]},"reference":[{"key":"ref_1","first-page":"83","article-title":"Approximate solutions for solving the Klein-Gordon and sine-Gordon equations","volume":"22","author":"Yousif","year":"2017","journal-title":"J. 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