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Comput."],"published-print":{"date-parts":[[2017,1]]},"abstract":"<jats:p>We develop spectral collocation methods for fractional differential equations with variable order with two end-point singularities. Specifically, we derive three-term recurrence relations for both integrals and derivatives of the weighted Jacobi polynomials of the form $(1+x)^{\\mu_1}(1-x)^{\\mu_2}P_{j}^{a,b}(x) \\,({a,b,\\mu_1,\\mu_2&gt;-1})$, which leads to the desired differentiation matrices. We apply the new differentiation matrices to construct collocation methods to solve fractional boundary value problems and fractional partial differential equations with two end-point singularities. We demonstrate that the singular basis enhances greatly the accuracy of the numerical solutions by properly tuning the parameters $\\mu_1$ and $\\mu_2$, even for cases for which we do not know explicitly the form of singularities in the solution at the boundaries. Comparison with other existing methods shows the superior accuracy of the proposed spectral collocation method.<\/jats:p>","DOI":"10.1137\/16m1076083","type":"journal-article","created":{"date-parts":[[2017,2,28]],"date-time":"2017-02-28T11:36:08Z","timestamp":1488281768000},"page":"A360-A383","source":"Crossref","is-referenced-by-count":68,"title":["A Generalized  Spectral Collocation Method with Tunable  Accuracy for Fractional Differential Equations with End-Point Singularities"],"prefix":"10.1137","volume":"39","author":[{"given":"Fanhai","family":"Zeng","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhiping","family":"Mao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"George Em","family":"Karniadakis","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,2,28]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-0427(00)00267-3"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-30726-6"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1090\/mcom3035"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/17.3.479"},{"key":"atypb5","series-title":"Lecture Notes in Math.","volume-title":"The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type","author":"Diethelm K.","year":"2004"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2014.06.007"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-2014-02905-0"},{"key":"atypb8","volume-title":"Regularity of the Solution to 1-D Fractional Order Diffusion Equations, arXiv:1608.00128v1","author":"Ervin V. 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