{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T21:15:42Z","timestamp":1781298942029,"version":"3.54.1"},"reference-count":33,"publisher":"American Mathematical Society (AMS)","issue":"300","license":[{"start":{"date-parts":[[2016,10,1]],"date-time":"2016-10-01T00:00:00Z","timestamp":1475280000000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["91130002"],"award-info":[{"award-number":["91130002"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11371298"],"award-info":[{"award-number":["11371298"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>In this paper, we consider spectral approximation of fractional differential equations (FDEs). A main ingredient of our approach is to define a new class of generalized Jacobi functions (GJFs), which is intrinsically related to fractional calculus and can serve as natural basis functions for properly designed spectral methods for FDEs. We establish spectral approximation results for these GJFs in weighted Sobolev spaces involving fractional derivatives. We construct efficient GJF-Petrov-Galerkin methods for a class of prototypical fractional initial value problems (FIVPs) and fractional boundary value problems (FBVPs) of general order, and we show that with an appropriate choice of the parameters in GJFs, the resulting linear systems are sparse and well-conditioned. Moreover, we derive error estimates with convergence rates only depending on the smoothness of data, so true spectral accuracy can be attained if the data are smooth enough. The ideas and results presented in this paper will be useful in dealing with more general FDEs involving Riemann-Liouville or Caputo fractional derivatives.<\/p>","DOI":"10.1090\/mcom3035","type":"journal-article","created":{"date-parts":[[2015,2,4]],"date-time":"2015-02-04T10:52:56Z","timestamp":1423047176000},"page":"1603-1638","source":"Crossref","is-referenced-by-count":211,"title":["Generalized Jacobi functions and their applications to fractional differential equations"],"prefix":"10.1090","volume":"85","author":[{"given":"Sheng","family":"Chen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jie","family":"Shen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Li-Lian","family":"Wang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2015,10,1]]},"reference":[{"key":"1","unstructured":"M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972."},{"key":"2","series-title":"Pure and Applied Mathematics, Vol. 65","volume-title":"Sobolev spaces","author":"Adams, Robert A.","year":"1975"},{"key":"3","series-title":"Encyclopedia of Mathematics and its Applications","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781107325937","volume-title":"Special functions","volume":"71","author":"Andrews, George E.","year":"1999","ISBN":"https:\/\/id.crossref.org\/isbn\/0521623219"},{"key":"4","first-page":"1","article-title":"Survey lectures on the mathematical foundations of the finite element method","author":"Babu\u0161ka, Ivo","year":"1972"},{"key":"5","unstructured":"H. Bateman, The solution of linear differential equations by means of definite integrals, Trans. Camb. Phil. Soc., 21:171\u2013196, 1909."},{"key":"6","doi-asserted-by":"crossref","unstructured":"L. Cagliero and T. H. Koornwinder, Explicit matrix inverses for lower triangular matrices with entries involving Jacobi polynomials, J. Approx. Theor. 193 (2015), 20\u201338.","DOI":"10.1016\/j.jat.2014.03.016"},{"key":"7","series-title":"Lecture Notes in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-14574-2","volume-title":"The analysis of fractional differential equations","volume":"2004","author":"Diethelm, Kai","year":"2010","ISBN":"https:\/\/id.crossref.org\/isbn\/9783642145735"},{"issue":"4","key":"8","doi-asserted-by":"publisher","first-page":"667","DOI":"10.1137\/110833294","article-title":"Analysis and approximation of nonlocal diffusion problems with volume constraints","volume":"54","author":"Du, Qiang","year":"2012","journal-title":"SIAM Rev.","ISSN":"https:\/\/id.crossref.org\/issn\/1095-7200","issn-type":"print"},{"issue":"2","key":"9","doi-asserted-by":"publisher","first-page":"572","DOI":"10.1137\/050642757","article-title":"Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation","volume":"45","author":"Ervin, Vincent J.","year":"2007","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"3","key":"10","doi-asserted-by":"publisher","first-page":"558","DOI":"10.1002\/num.20112","article-title":"Variational formulation for the stationary fractional advection dispersion equation","volume":"22","author":"Ervin, Vincent J.","year":"2006","journal-title":"Numer. Methods Partial Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0749-159X","issn-type":"print"},{"issue":"2","key":"11","doi-asserted-by":"publisher","first-page":"256","DOI":"10.1002\/num.20169","article-title":"Variational solution of fractional advection dispersion equations on bounded domains in \u211d^{\ud835\udd55}","volume":"23","author":"Ervin, Vincent J.","year":"2007","journal-title":"Numer. Methods Partial Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0749-159X","issn-type":"print"},{"issue":"1-3","key":"12","doi-asserted-by":"publisher","first-page":"305","DOI":"10.1007\/s10915-005-9055-7","article-title":"Optimal spectral-Galerkin methods using generalized Jacobi polynomials","volume":"27","author":"Guo, Ben-Yu","year":"2006","journal-title":"J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0885-7474","issn-type":"print"},{"issue":"5","key":"13","doi-asserted-by":"publisher","first-page":"1011","DOI":"10.1016\/j.apnum.2008.04.003","article-title":"Generalized Jacobi polynomials\/functions and their applications","volume":"59","author":"Guo, Ben-Yu","year":"2009","journal-title":"Appl. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0168-9274","issn-type":"print"},{"issue":"1","key":"14","doi-asserted-by":"publisher","first-page":"445","DOI":"10.1137\/120873984","article-title":"Error estimates for a semidiscrete finite element method for fractional order parabolic equations","volume":"51","author":"Jin, Bangti","year":"2013","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"15","series-title":"North-Holland Mathematics Studies","isbn-type":"print","volume-title":"Theory and applications of fractional differential equations","volume":"204","author":"Kilbas, Anatoly A.","year":"2006","ISBN":"https:\/\/id.crossref.org\/isbn\/9780444518323"},{"issue":"5","key":"16","doi-asserted-by":"publisher","first-page":"795","DOI":"10.1016\/j.camwa.2012.12.011","article-title":"Fractional Sturm-Liouville problem","volume":"66","author":"Klimek, M.","year":"2013","journal-title":"Comput. Math. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0898-1221","issn-type":"print"},{"key":"17","series-title":"Springer Monographs in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-05014-5","volume-title":"Hypergeometric orthogonal polynomials and their $q$-analogues","author":"Koekoek, Roelof","year":"2010","ISBN":"https:\/\/id.crossref.org\/isbn\/9783642050138"},{"key":"18","unstructured":"A. Kuijlaars, A. Mart\u0131nez-Finkelshtein, and R. Orive, Orthogonality of Jacobi polynomials with general parameters, Electronic Transactions on Numerical Analysis, 19(1):1\u201317, 2005."},{"issue":"3","key":"19","doi-asserted-by":"publisher","first-page":"383","DOI":"10.2478\/s13540-012-0028-x","article-title":"Spectral approximations to the fractional integral and derivative","volume":"15","author":"Li, Changpin","year":"2012","journal-title":"Fract. Calc. Appl. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/1311-0454","issn-type":"print"},{"issue":"3","key":"20","doi-asserted-by":"publisher","first-page":"2108","DOI":"10.1137\/080718942","article-title":"A space-time spectral method for the time fractional diffusion equation","volume":"47","author":"Li, Xianjuan","year":"2009","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"5","key":"21","doi-asserted-by":"publisher","first-page":"1016","DOI":"10.4208\/cicp.020709.221209a","article-title":"Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation","volume":"8","author":"Li, Xianjuan","year":"2010","journal-title":"Commun. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/1815-2406","issn-type":"print"},{"issue":"1","key":"22","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1016\/j.cam.2003.09.028","article-title":"Numerical solution of the space fractional Fokker-Planck equation","volume":"166","author":"Liu, F.","year":"2004","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"issue":"1","key":"23","doi-asserted-by":"publisher","first-page":"249","DOI":"10.1016\/j.jcp.2005.05.017","article-title":"Finite difference methods for two-dimensional fractional dispersion equation","volume":"211","author":"Meerschaert, Mark M.","year":"2006","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"1","key":"24","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1016\/j.cam.2004.01.033","article-title":"Finite difference approximations for fractional advection-dispersion flow equations","volume":"172","author":"Meerschaert, Mark M.","year":"2004","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"issue":"1","key":"25","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1016\/S0370-1573(00)00070-3","article-title":"The random walk\u2019s guide to anomalous diffusion: a fractional dynamics approach","volume":"339","author":"Metzler, Ralf","year":"2000","journal-title":"Phys. Rep.","ISSN":"https:\/\/id.crossref.org\/issn\/0370-1573","issn-type":"print"},{"key":"26","series-title":"Mathematics in Science and Engineering","isbn-type":"print","volume-title":"Fractional differential equations","volume":"198","author":"Podlubny, Igor","year":"1999","ISBN":"https:\/\/id.crossref.org\/isbn\/0125588402"},{"key":"27","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-71041-7","volume-title":"Spectral methods","volume":"41","author":"Shen, Jie","year":"2011","ISBN":"https:\/\/id.crossref.org\/isbn\/9783540710400"},{"issue":"2","key":"28","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1016\/j.apnum.2005.03.003","article-title":"A fully discrete difference scheme for a diffusion-wave system","volume":"56","author":"Sun, Zhi-zhong","year":"2006","journal-title":"Appl. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0168-9274","issn-type":"print"},{"key":"29","series-title":"American Mathematical Society Colloquium Publications, Vol. XXIII","volume-title":"Orthogonal polynomials","author":"Szeg\u0151, G\u00e1bor","year":"1975","edition":"4"},{"issue":"2","key":"30","doi-asserted-by":"publisher","first-page":"813","DOI":"10.1016\/j.jcp.2006.05.030","article-title":"A second-order accurate numerical method for the two-dimensional fractional diffusion equation","volume":"220","author":"Tadjeran, Charles","year":"2007","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"5","key":"31","doi-asserted-by":"publisher","first-page":"A2444--A2458","DOI":"10.1137\/12086491X","article-title":"A fast finite difference method for two-dimensional space-fractional diffusion equations","volume":"34","author":"Wang, Hong","year":"2012","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"key":"32","doi-asserted-by":"publisher","first-page":"495","DOI":"10.1016\/j.jcp.2013.06.031","article-title":"Fractional Sturm-Liouville eigen-problems: theory and numerical approximation","volume":"252","author":"Zayernouri, Mohsen","year":"2013","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"3","key":"33","doi-asserted-by":"publisher","first-page":"1443","DOI":"10.1137\/12089421X","article-title":"Sharp error bounds for Jacobi expansions and Gegenbauer-Gauss quadrature of analytic functions","volume":"51","author":"Zhao, Xiaodan","year":"2013","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2016-85-300\/S0025-5718-2015-03035-X\/S0025-5718-2015-03035-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2016-85-300\/S0025-5718-2015-03035-X\/S0025-5718-2015-03035-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:49:11Z","timestamp":1776797351000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2016-85-300\/S0025-5718-2015-03035-X\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,10,1]]},"references-count":33,"journal-issue":{"issue":"300","published-print":{"date-parts":[[2016,7]]}},"alternative-id":["S0025-5718-2015-03035-X"],"URL":"https:\/\/doi.org\/10.1090\/mcom3035","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2015,10,1]]}}}