{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,4]],"date-time":"2026-08-04T21:58:22Z","timestamp":1785880702652,"version":"3.56.0"},"reference-count":35,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2020,7,19]],"date-time":"2020-07-19T00:00:00Z","timestamp":1595116800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11971142, 11871202, 61673169, 11701176, 11626101, 11601485"],"award-info":[{"award-number":["11971142, 11871202, 61673169, 11701176, 11626101, 11601485"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Fractional differential equations depict nature sufficiently in light of the symmetry properties which describe biological and physical processes. This article is concerned with the numerical treatment of three-term time fractional-order multi-dimensional diffusion equations by using an efficient local meshless method. The space derivative of the models is discretized by the proposed meshless procedure based on the multiquadric radial basis function though the time-fractional part is discretized by Liouville\u2013Caputo fractional derivative. The numerical results are obtained for one-, two- and three-dimensional cases on rectangular and non-rectangular computational domains which verify the validity, efficiency and accuracy of the method.<\/jats:p>","DOI":"10.3390\/sym12071195","type":"journal-article","created":{"date-parts":[[2020,7,22]],"date-time":"2020-07-22T05:10:30Z","timestamp":1595394630000},"page":"1195","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":107,"title":["Solution of Multi-Term Time-Fractional PDE Models Arising in Mathematical Biology and Physics by Local Meshless Method"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4023-3293","authenticated-orcid":false,"given":"Imtiaz","family":"Ahmad","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Swabi, Swabi 23430, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5438-5407","authenticated-orcid":false,"given":"Hijaz","family":"Ahmad","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences, University of Engineering and Technology, Peshawar 25000, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1453-4236","authenticated-orcid":false,"given":"Phatiphat","family":"Thounthong","sequence":"additional","affiliation":[{"name":"Renewable Energy Research Centre, Department of Teacher Training in Electrical Engineering, Faculty of Technical Education, King Mongkut\u2019s University of Technology North Bangkok, 1518 Pracharat 1 Road, Bangsue, Bangkok 10800, Thailand"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0944-2134","authenticated-orcid":false,"given":"Yu-Ming","family":"Chu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Huzhou University, Huzhou 313000, China"},{"name":"Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changasha, University of Science &amp; Technology, Changsha 410114, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1694-7907","authenticated-orcid":false,"given":"Clemente","family":"Cesarano","sequence":"additional","affiliation":[{"name":"Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Roma, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,7,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"704","DOI":"10.1137\/0517050","article-title":"Discretized fractional calculus","volume":"17","author":"Lubich","year":"1986","journal-title":"SIAM J. 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