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The main idea consists of two parts. First, we use artificial boundary methods to equivalently reformulate the unbounded problem into an initial-boundary value (IBV) problem. Second, we present two numerical schemes for the IBV problem: a direct scheme and a fast scheme. The direct scheme stands for the direct discretization of the Caputo fractional derivative by using the L1-formula. The fast scheme means that the sum-of-exponentials approximation is used to speed up the evaluation of the Caputo fractional derivative. The resulting fast algorithm significantly reduces the storage requirement and the overall computational cost compared to the direct scheme. Furthermore, the corresponding stability analysis and error estimates of two schemes are established, and numerical examples are given to verify the performance of our approach.<\/jats:p>","DOI":"10.1515\/cmam-2017-0038","type":"journal-article","created":{"date-parts":[[2017,9,2]],"date-time":"2017-09-02T06:02:50Z","timestamp":1504332170000},"page":"77-94","source":"Crossref","is-referenced-by-count":7,"title":["The Numerical Computation of the Time Fractional Schr\u00f6dinger Equation on an Unbounded Domain"],"prefix":"10.1515","volume":"18","author":[{"given":"Dan","family":"Li","sequence":"first","affiliation":[{"name":"Applied and Computational Mathematics , Beijing Computational Science Research Center , Beijing 100193 , P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jiwei","family":"Zhang","sequence":"additional","affiliation":[{"name":"Applied and Computational Mathematics , Beijing Computational Science Research Center , Beijing 100193 , P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhimin","family":"Zhang","sequence":"additional","affiliation":[{"name":"Applied and Computational Mathematics , Beijing Computational Science Research Center , Beijing 100193 , P. R. China ; and Department of Mathematics, Wayne State University, Detroit, MI 48202, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2017,9,2]]},"reference":[{"key":"2023033115122857279_j_cmam-2017-0038_ref_001_w2aab3b7e3406b1b6b1ab2ab1Aa","doi-asserted-by":"crossref","unstructured":"A. A.  Alikhanov,\nA priori estimates for solutions of boundary value problems for equations of fractional order,\nDiffer. Uravn. 46 (2010), no. 5, 658\u2013664.","DOI":"10.1134\/S0012266110050058"},{"key":"2023033115122857279_j_cmam-2017-0038_ref_002_w2aab3b7e3406b1b6b1ab2ab2Aa","doi-asserted-by":"crossref","unstructured":"X.  Antoine and C.  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