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We develop an accurate numerical algorithm to solve the two-sided space\u2013time fractional advection\u2013dispersion equation (FADE) based on a spectral shifted Legendre tau (SLT) method in combination with the derived shifted Legendre operational matrices. The fractional derivatives are described in the Caputo sense. We propose a spectral SLT method, both in temporal and spatial discretizations for the two-sided space\u2013time FADE. This technique reduces the two-sided space\u2013time FADE to a system of algebraic equations that simplifies the problem. Numerical results carried out to confirm the spectral accuracy and efficiency of the proposed algorithm. By selecting relatively few Legendre polynomial degrees, we are able to get very accurate approximations, demonstrating the utility of the new approach over other numerical methods.<\/jats:p>","DOI":"10.1177\/1077546314566835","type":"journal-article","created":{"date-parts":[[2015,1,23]],"date-time":"2015-01-23T00:43:55Z","timestamp":1421973835000},"page":"2053-2068","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":38,"title":["Efficient Legendre spectral tau algorithm for solving the two-sided space\u2013time Caputo fractional advection\u2013dispersion equation"],"prefix":"10.1177","volume":"22","author":[{"given":"AH","family":"Bhrawy","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, 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