{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,17]],"date-time":"2026-04-17T14:17:26Z","timestamp":1776435446461,"version":"3.51.2"},"reference-count":47,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2018,4,26]],"date-time":"2018-04-26T00:00:00Z","timestamp":1524700800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this article, a numerical scheme is formulated and analysed to solve the time-space fractional advection\u2013diffusion equation, where the Riesz derivative and the Caputo derivative are considered in spatial and temporal directions, respectively. The Riesz space derivative is approximated by the second-order fractional weighted and shifted Gr\u00fcnwald\u2013Letnikov formula. Based on the equivalence between the fractional differential equation and the integral equation, we have transformed the fractional differential equation into an equivalent integral equation. Then, the integral is approximated by the trapezoidal formula. Further, the stability and convergence analysis are discussed rigorously. The resulting scheme is formally proved with the second order accuracy both in space and time. Numerical experiments are also presented to verify the theoretical analysis.<\/jats:p>","DOI":"10.3390\/e20050321","type":"journal-article","created":{"date-parts":[[2018,4,27]],"date-time":"2018-04-27T06:52:23Z","timestamp":1524811943000},"page":"321","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":33,"title":["Finite Difference Method for Time-Space Fractional Advection\u2013Diffusion Equations with Riesz Derivative"],"prefix":"10.3390","volume":"20","author":[{"given":"Sadia","family":"Arshad","sequence":"first","affiliation":[{"name":"The State Key Laboratory of Scientific and Engineering Computing (LSEC), The Institute of Computational Mathematics and Scientific\/Engineering Computing (ICMSEC), Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China"},{"name":"COMSATS Institute of Information Technology, Lahore 54500, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0286-7244","authenticated-orcid":false,"given":"Dumitru","family":"Baleanu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Cankaya University, Ankara 06530, Turkey"},{"name":"Institute of Space Sciences, Magurele-Bucharest 077125, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jianfei","family":"Huang","sequence":"additional","affiliation":[{"name":"College of Mathematical Sciences, Yangzhou University, Yangzhou 225002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0979-5746","authenticated-orcid":false,"given":"Maysaa Mohamed","family":"Al Qurashi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, King Saud University, P.O. Box 22452, Riyadh 11495, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yifa","family":"Tang","sequence":"additional","affiliation":[{"name":"The State Key Laboratory of Scientific and Engineering Computing (LSEC), The Institute of Computational Mathematics and Scientific\/Engineering Computing (ICMSEC), Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China"},{"name":"School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yue","family":"Zhao","sequence":"additional","affiliation":[{"name":"The State Key Laboratory of Scientific and Engineering Computing (LSEC), The Institute of Computational Mathematics and Scientific\/Engineering Computing (ICMSEC), Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China"},{"name":"School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,4,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1007\/s11071-010-9724-4","article-title":"Entropy analysis of integer and fractional dynamical systems","volume":"62","author":"Machado","year":"2010","journal-title":"Nonlinear Dyn."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1515\/jnet.1998.23.2.166","article-title":"Fractional diffusion and entropy production","volume":"23","author":"Hoffmann","year":"1998","journal-title":"J. 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