{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,19]],"date-time":"2026-01-19T08:16:48Z","timestamp":1768810608188,"version":"3.49.0"},"reference-count":37,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2020,5,1]],"date-time":"2020-05-01T00:00:00Z","timestamp":1588291200000},"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>Fractional differential equations describe nature adequately because of the symmetry properties which describe physical and biological processes. In this article, a fourth-order new implicit difference scheme is formulated and applied to solve the two-dimensional time-fractional modified sub-diffusion equation involving two times Riemann\u2013Liouville fractional derivatives. The stability of the fourth-order implicit difference scheme is investigated using the von Neumann technique. The proposed scheme is shown to be unconditionally stable. Numerical examples are given to illustrate the feasibility of the proposed scheme.<\/jats:p>","DOI":"10.3390\/sym12050691","type":"journal-article","created":{"date-parts":[[2020,5,5]],"date-time":"2020-05-05T06:41:20Z","timestamp":1588660880000},"page":"691","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":25,"title":["Fourth-Order Difference Approximation for Time-Fractional Modified Sub-Diffusion Equation"],"prefix":"10.3390","volume":"12","author":[{"given":"Umair","family":"Ali","sequence":"first","affiliation":[{"name":"Department of Mathematics, Al-Fajar University, Mari Indus 42350, Mianwali, Pakistan"}],"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\/0000-0001-7470-3567","authenticated-orcid":false,"given":"Muhammad","family":"Usman","sequence":"additional","affiliation":[{"name":"BIC-ESAT, College of Engineering, Peking University, Beijing 100871, China"}],"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, Pulau Pinang 11800, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ilyas","family":"Khan","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City 72915, Vietnam"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5769-4320","authenticated-orcid":false,"given":"Kottakkaran Sooppy","family":"Nisar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Arts and Sciences, Prince Sattam bin Abdulaziz University, Wadi Al-Dawaser 11991, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,5,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Ali, U., Abdullah, F.A., and Mohyud-Din, S.T. 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