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Then, the unconditional stability and the superlinear convergence with the order [Formula: see text] of the proposed scheme are strictly proved and discussed. Due to the nonlocal property of fractional operators, the new scheme is time-consuming for long-time simulations. Thus, a fast implement of the proposed scheme is presented based on the sum-of-exponentials (SOE) approximation for the kernel [Formula: see text] on the interval [Formula: see text] in the Riemann\u2013Liouville integral, where [Formula: see text] is the stepsize. Some numerical experiments are provided to support the theoretical results of the new scheme and demonstrate the computational performance of its fast implement.<\/jats:p>","DOI":"10.1142\/s1793962319410010","type":"journal-article","created":{"date-parts":[[2018,9,24]],"date-time":"2018-09-24T05:52:32Z","timestamp":1537768352000},"page":"1941001","source":"Crossref","is-referenced-by-count":3,"title":["A superlinear convergence scheme for nonlinear fractional differential equations and its fast implement"],"prefix":"10.1142","volume":"10","author":[{"given":"Haobo","family":"Gong","sequence":"first","affiliation":[{"name":"College of Mathematical Sciences, Yangzhou University, Yangzhou 225002, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jingna","family":"Zhang","sequence":"additional","affiliation":[{"name":"College of Mathematical Sciences, Yangzhou University, Yangzhou 225002, P. R. 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