{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,18]],"date-time":"2026-01-18T12:04:40Z","timestamp":1768737880252,"version":"3.49.0"},"reference-count":41,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2020,9,1]],"date-time":"2020-09-01T00:00:00Z","timestamp":1598918400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The variable-order (VO) fractional calculus can be seen as a natural extension of the constant-order, which can be utilized in physical and biological applications. In this study, we derive a new numerical approximation for the VO fractional Riemann\u2013Liouville integral formula and developed an implicit difference scheme (IDS) for the variable-order fractional sub-diffusion equation (VO-FSDE). The derived approximation used in the VO time fractional derivative with the central difference approximation for the space derivative. Investigated the unconditional stability by the van Neumann method, consistency, and convergence analysis of the proposed scheme. Finally, a numerical example is presented to verify the theoretical analysis and effectiveness of the proposed scheme.<\/jats:p>","DOI":"10.3390\/sym12091437","type":"journal-article","created":{"date-parts":[[2020,9,1]],"date-time":"2020-09-01T14:21:10Z","timestamp":1598970070000},"page":"1437","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":23,"title":["An Efficient Numerical Scheme for Variable-Order Fractional Sub-Diffusion Equation"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5840-2289","authenticated-orcid":false,"given":"Umair","family":"Ali","sequence":"first","affiliation":[{"name":"Department of Mathematics, AL-Fajar University, Mari Indus 42350, Pakistan"},{"name":"School of Mathematical Sciences, Universiti Sains Malaysia, USM Penang 11800, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1490-0339","authenticated-orcid":false,"given":"Muhammad","family":"Sohail","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Statistics, Institute of Space Technology, Islamabad 44000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5215-9617","authenticated-orcid":false,"given":"Farah Aini","family":"Abdullah","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Universiti Sains Malaysia, USM Penang 11800, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,9,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Li, X., and Wong, P.J. 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