{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,17]],"date-time":"2026-01-17T23:48:29Z","timestamp":1768693709564,"version":"3.49.0"},"reference-count":59,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,3,5]],"date-time":"2024-03-05T00:00:00Z","timestamp":1709596800000},"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 is defined by the degree and quantity of knots, and it is observed to provide a higher level of flexibility in curve and surface layout. The extended cubic B-spline (ExCBS) functions with new approximation for second derivative and finite difference technique are incorporated in this study to solve the time-fractional Allen\u2013Cahn equation (TFACE). Initially, Caputo\u2019s formula is used to discretize the time-fractional derivative, while a new ExCBS is used for the spatial derivative\u2019s discretization. Convergence analysis is carried out and the stability of the proposed method is also analyzed. The scheme\u2019s applicability and feasibility are demonstrated through numerical analysis.<\/jats:p>","DOI":"10.3390\/computation12030051","type":"journal-article","created":{"date-parts":[[2024,3,5]],"date-time":"2024-03-05T11:26:59Z","timestamp":1709638019000},"page":"51","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Extension of Cubic B-Spline for Solving the Time-Fractional Allen\u2013Cahn Equation in the Context of Mathematical Physics"],"prefix":"10.3390","volume":"12","author":[{"given":"Mubeen","family":"Fatima","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0634-2370","authenticated-orcid":false,"given":"Ravi P.","family":"Agarwal","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Texas A & M University-Kingsville, Kingsville, TX 78363, USA"}]},{"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\/0000-0001-6837-8075","authenticated-orcid":false,"given":"Pshtiwan Othman","family":"Mohammed","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Education, University of Sulaimani, Sulaymaniyah 46001, Iraq"},{"name":"Research and Development Center, University of Sulaimani, Sulaymaniyah 46001, Iraq"}]},{"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\/0000-0001-8833-6585","authenticated-orcid":false,"given":"Nejmeddine","family":"Chorfi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, P.O. 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