{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:18:05Z","timestamp":1760059085708,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2025,5,21]],"date-time":"2025-05-21T00:00:00Z","timestamp":1747785600000},"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 research, a groundbreaking framework for the octic B-spline collocation method in n-dimensional spaces is presented. This work is an extension of previous works that involved the creation of B-spline functions in n-dimensional space for the purpose of solving mathematical models in n-dimensions. The octic B-spline collocation approach in n-dimensional space is an extension of the standard B-spline collocation approach to higher dimensions. It involves using eighth order (octic) B-splines, which have higher smoothness and continuity properties than lower-order B-splines. To demonstrate the effectiveness and precision of the suggested method, a selection of test problems in two- and three-dimensional space is utilized. For making comparisons, we make use of a wide variety of numerical problems, which are described in this paper.<\/jats:p>","DOI":"10.3390\/axioms14050388","type":"journal-article","created":{"date-parts":[[2025,5,21]],"date-time":"2025-05-21T06:31:27Z","timestamp":1747809087000},"page":"388","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["New n-Dimensional Finite Element Technique for Solving Boundary Value Problems in n-Dimensional Space"],"prefix":"10.3390","volume":"14","author":[{"given":"Weam G.","family":"Alharbi","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk 71491, Saudi Arabia"}]},{"given":"Kamal R.","family":"Raslan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr-City 11884, Cairo, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7801-2760","authenticated-orcid":false,"given":"Khalid K.","family":"Ali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr-City 11884, Cairo, Egypt"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"480","DOI":"10.1016\/j.physleta.2010.11.017","article-title":"Nonlinear dynamics of DNA\u2013Riccati generalized solitary wave solutions","volume":"375","author":"Alka","year":"2011","journal-title":"Phys. Lett."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"105215","DOI":"10.1088\/1402-4896\/abb739","article-title":"Structure of optical soliton solution for nonlinear resonant space-time Schr\u00f6dinger equation in conformable sense with full nonlinearity term","volume":"95","author":"Alabedalhadi","year":"2020","journal-title":"Phys. Scr."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/j.proeng.2014.06.310","article-title":"Generalized finite difference method for solving two-dimensions Burgers\u2019 equations","volume":"79","author":"Fana","year":"2014","journal-title":"Procedia Eng."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"5134","DOI":"10.1016\/j.apm.2011.11.078","article-title":"Solving 2D and 3D Poisson equations andbiharmonic equations by the Haar wavelet method","volume":"36","author":"Shi","year":"2012","journal-title":"Appl. Math. Model."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1007\/s40096-017-0220-6","article-title":"Wavelet methods for solving three-dimensional partial differential equations","volume":"11","author":"Singh","year":"2017","journal-title":"Math. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1016\/0045-7825(94)00760-K","article-title":"A two dimensional cubic B-spline finite element: Used in a study of MHD-duct flow","volume":"124","author":"Gardner","year":"1995","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1007\/s40096-020-00331-y","article-title":"Swarn Singh and Suruchi Singh, Numerical solution of second-order two-dimensional hyperbolic equation by bi-cubic B-spline collocation method","volume":"14","author":"Arora","year":"2020","journal-title":"Math. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/00207160.2015.1085976","article-title":"Numerical solutions of two-dimensional unsteady convection-diffusion problems using modified bicubic B-spline finite elements","volume":"94","author":"Mittal","year":"2017","journal-title":"Int. J. Comput. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2879","DOI":"10.1016\/j.asej.2017.12.001","article-title":"Solving 2D-Poisson equation using modified cubic B-spline differentialquadrature method","volume":"9","author":"Elsherbeny","year":"2018","journal-title":"Ain Shams Eng. J."},{"key":"ref_10","first-page":"23","article-title":"The modified Bi-quintic B-splines for solving the two-dimensional unsteady Burgers\u2019 equation","volume":"1","author":"Kutluay","year":"2012","journal-title":"Eur. Int. J. Sci. Technol."},{"key":"ref_11","first-page":"26","article-title":"Derivation of the modified bi-quintic b-spline base functions: An application to Poisson equation","volume":"3","author":"Kutluay","year":"2013","journal-title":"Am. J. Comput. Appl. Math."},{"key":"ref_12","first-page":"26","article-title":"The Modified Bi-quintic B-spline Base Functions: An Application to Diffusion Equation","volume":"5","author":"Kutluay","year":"2017","journal-title":"Int. J. Partial. Differ. Equ. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1057","DOI":"10.1002\/num.22566","article-title":"On n-dimensional quadratic B-splines","volume":"37","author":"Raslan","year":"2021","journal-title":"Numer. Methods Partial. Differ. Equ."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"432","DOI":"10.1515\/nleng-2020-0027","article-title":"A new structure formulations for cubic B-spline collocation method in three and four-dimensions","volume":"9","author":"Raslan","year":"2020","journal-title":"Nonlinear Eng."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"442","DOI":"10.1186\/s13662-021-03596-2","article-title":"A new structure to n-dimensional trigonometric cubic B-spline functions for solving n-dimensional partial differential equations","volume":"2021","author":"Raslan","year":"2021","journal-title":"Adv. Differ. Equ."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Raslan, K.R., Ali, K.K., and Shaalan, M.A. (2022). n-Dimensional quartic B-spline collocation method to solve different types of n-dimensional partial differential equations. J. Ocean. Eng. Sci., 1\u20139.","DOI":"10.1016\/j.joes.2022.06.034"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1515\/nleng-2022-0016","article-title":"n-Dimensional quintic B-spline functions for solving n-dimensional partial differential equations","volume":"11","author":"Raslan","year":"2022","journal-title":"Nonlinear Eng."},{"key":"ref_18","first-page":"30","article-title":"A Novel Generalized n-dimensional Sixtic B-Spline Function to Solving n-dimensional Mathematical Models","volume":"3886554","author":"Ali","year":"2024","journal-title":"J. Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"20220298","DOI":"10.1515\/nleng-2022-0298","article-title":"Derivation of septic B-spline function in n-dimensional to solve n-dimensional partial differential equations","volume":"12","author":"Raslan","year":"2023","journal-title":"Nonlinear Eng."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1007\/s40819-022-01437-8","article-title":"Octic B-spline Collocation Scheme for Numerical Investigation of Fifth Order Boundary Value Problems","volume":"8","author":"Jena","year":"2022","journal-title":"Int. Appl. Comput. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1576","DOI":"10.1016\/j.camwa.2017.02.006","article-title":"Spline-based DQM for multi-dimensional PDEs: Application to biharmonic and Poisson Equations in 2D and 3D","volume":"73","author":"Mohammad","year":"2017","journal-title":"Comput. Math. Appl."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/5\/388\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:36:24Z","timestamp":1760031384000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/5\/388"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,21]]},"references-count":21,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2025,5]]}},"alternative-id":["axioms14050388"],"URL":"https:\/\/doi.org\/10.3390\/axioms14050388","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,5,21]]}}}