{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,10]],"date-time":"2026-06-10T10:55:03Z","timestamp":1781088903168,"version":"3.54.1"},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2017,8,18]],"date-time":"2017-08-18T00:00:00Z","timestamp":1503014400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2017,8,18]],"date-time":"2017-08-18T00:00:00Z","timestamp":1503014400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/100000086","name":"Directorate for Mathematical and Physical Sciences","doi-asserted-by":"publisher","award":["DMS-1620262"],"award-info":[{"award-number":["DMS-1620262"]}],"id":[{"id":"10.13039\/100000086","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2018,6]]},"DOI":"10.1007\/s10444-017-9561-9","type":"journal-article","created":{"date-parts":[[2017,8,18]],"date-time":"2017-08-18T11:41:47Z","timestamp":1503056507000},"page":"745-771","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":35,"title":["Spectral element method with geometric mesh for two-sided fractional differential equations"],"prefix":"10.1007","volume":"44","author":[{"given":"Zhiping","family":"Mao","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4885-5732","authenticated-orcid":false,"given":"Jie","family":"Shen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2017,8,18]]},"reference":[{"issue":"3","key":"9561_CR1","doi-asserted-by":"publisher","first-page":"1418","DOI":"10.1137\/130933447","volume":"52","author":"M Chen","year":"2014","unstructured":"Chen, M., Deng, W.: Fourth order accurate scheme for the space fractional diffusion equations. SIAM J. Numer. Anal. 52(3), 1418\u20131438 (2014)","journal-title":"SIAM J. Numer. Anal."},{"key":"9561_CR2","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1016\/j.jcp.2014.10.016","volume":"293","author":"F Chen","year":"2015","unstructured":"Chen, F., Xu, Q., Hesthaven, J.S.: A multi-domain spectral method for time-fractional differential equations. J. Comput. Phys. 293, 157\u2013172 (2015)","journal-title":"J. Comput. Phys."},{"issue":"300","key":"9561_CR3","doi-asserted-by":"publisher","first-page":"1603","DOI":"10.1090\/mcom3035","volume":"85","author":"S Chen","year":"2016","unstructured":"Chen, S., Shen, J., Wang, L.-L.: Generalized Jacobi functions and their applications to fractional differential equations. Math. Comp. 85(300), 1603\u20131638 (2016)","journal-title":"Math. Comp."},{"issue":"2","key":"9561_CR4","doi-asserted-by":"publisher","first-page":"256","DOI":"10.1002\/num.20169","volume":"23","author":"VJ Ervin","year":"2007","unstructured":"Ervin, V.J., Roop, J.P.: Variational solution of fractional advection dispersion equations on bounded domains in $\\mathbb {R}^{d}$. Numer. Methods Partial Differ. Equ. 23(2), 256\u2013281 (2007)","journal-title":"Numer. Methods Partial Differ. Equ."},{"issue":"1","key":"9561_CR5","doi-asserted-by":"publisher","first-page":"521","DOI":"10.1016\/S0301-0104(02)00714-0","volume":"284","author":"R Gorenflo","year":"2002","unstructured":"Gorenflo, R., Mainardi, F., Moretti, D., Pagnini, G., Paradisi, P.: Discrete random walk models for space\u2013time fractional diffusion. Chem. Phys. 284(1), 521\u2013541 (2002)","journal-title":"Chem. Phys."},{"issue":"1-4","key":"9561_CR6","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1007\/s11071-004-3749-5","volume":"38","author":"R Gorenflo","year":"2004","unstructured":"Gorenflo, R., Vivoli, A., Mainardi, F.: Discrete and continuous random walk models for space-time fractional diffusion. Nonlinear Dyn. 38(1-4), 101\u2013116 (2004)","journal-title":"Nonlinear Dyn."},{"issue":"6","key":"9561_CR7","doi-asserted-by":"publisher","first-page":"577","DOI":"10.1007\/BF01389733","volume":"49","author":"W-Z Gui","year":"1986","unstructured":"Gui, W.-Z., Babus\u0307ka, I.: The h, p and h-p versions of the finite element method in 1 dimension. I. The error analysis of the p-version. Numer Math. 49(6), 577\u2013612 (1986)","journal-title":"Numer Math."},{"issue":"6","key":"9561_CR8","doi-asserted-by":"publisher","first-page":"613","DOI":"10.1007\/BF01389734","volume":"49","author":"W-Z Gui","year":"1986","unstructured":"Gui, W.-Z., Babus\u0307ka, I.: The h, p and h-p versions of the finite element method in 1 dimension. II. The error analysis of the h- and h-p versions. Numer Math. 49(6), 613\u2013657 (1986)","journal-title":"Numer Math."},{"key":"9561_CR9","doi-asserted-by":"crossref","unstructured":"Hilfer, R., Butzer, P.L., Westphal, U., Douglas, J., Schneider, W.R., Zaslavsky, G., Nonnemacher, T., Blumen, A., West, B.: Applications of fractional calculus in physics, volume 128. World Scientific (2000)","DOI":"10.1142\/9789812817747_0001"},{"key":"9561_CR10","unstructured":"Jin, B., Lazarov, R., Pasciak, J., Rundell, W.: A finite element method for the fractional Sturm-liouville Problem. arXiv:\n                    1307.5114\n                    \n                   (2013)"},{"issue":"5","key":"9561_CR11","doi-asserted-by":"publisher","first-page":"1261","DOI":"10.1051\/m2an\/2015010","volume":"49","author":"B Jin","year":"2015","unstructured":"Jin, B., Zhou, Z.: A finite element method with singularity reconstruction for fractional boundary value problems. ESAIM Math. Model. Numer Anal. 49(5), 1261\u20131283 (2015)","journal-title":"ESAIM Math. Model. Numer Anal."},{"issue":"1","key":"9561_CR12","doi-asserted-by":"publisher","first-page":"481","DOI":"10.1137\/140992278","volume":"54","author":"B Jin","year":"2016","unstructured":"Jin, B., Lazarov, R., Zhi, Z.: Petrov-Galerkin finite element method for fractional convection-diffusion equations. SIAM J. Numer. Anal. 54(1), 481\u2013503 (2016)","journal-title":"SIAM J. Numer. Anal."},{"issue":"5","key":"9561_CR13","doi-asserted-by":"crossref","first-page":"1016","DOI":"10.4208\/cicp.020709.221209a","volume":"8","author":"X Li","year":"2010","unstructured":"Li, X., Xu, C.: Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation. Commun. Comput. Phys. 8(5), 1016\u20131051 (2010)","journal-title":"Commun. Comput. Phys."},{"issue":"1","key":"9561_CR14","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1016\/j.jcp.2006.06.005","volume":"222","author":"Q Liu","year":"2007","unstructured":"Liu, Q., Liu, F., Turner, I.W., Anh, V.V.: Approximation of the L\u00e9vy-Feller advection-dispersion process by random walk and finite difference method. J. Comput. Phys. 222(1), 57\u201370 (2007)","journal-title":"J. Comput. Phys."},{"key":"9561_CR15","doi-asserted-by":"publisher","first-page":"243","DOI":"10.1016\/j.jcp.2015.11.047","volume":"307","author":"Z Mao","year":"2016","unstructured":"Mao, Z., Shen, J.: Efficient spectral-Galerkin methods for fractional partial differential equations with variable coefficients. J. Comput. Phys. 307, 243\u2013261 (2016)","journal-title":"J. Comput. Phys."},{"key":"9561_CR16","doi-asserted-by":"publisher","first-page":"165","DOI":"10.1016\/j.apnum.2016.04.002","volume":"106","author":"Z Mao","year":"2016","unstructured":"Mao, Z., Chen, S., Shen, J.: Efficient and accurate spectral method using generalized Jacobi functions for solving Riesz fractional differential equations. Appl. Numer. Math. 106, 165\u2013181 (2016)","journal-title":"Appl. Numer. Math."},{"issue":"1","key":"9561_CR17","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/S0370-1573(00)00070-3","volume":"339","author":"R Metzler","year":"2000","unstructured":"Metzler, R., Klafter, J.: The random walk\u2019s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339(1), 1\u201377 (2000)","journal-title":"Phys. Rep."},{"key":"9561_CR18","doi-asserted-by":"crossref","unstructured":"Moffatt, H.K., Zaslavsky, G.M., Comte, P., Tabor, M.: Topological aspects of the dvnamics of fluidi and plasmas (1992)","DOI":"10.1007\/978-94-017-3550-6"},{"key":"9561_CR19","volume-title":"Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications","author":"I Podlubny","year":"1999","unstructured":"Podlubny, I.: Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Academic press, San Diego (1999)"},{"key":"9561_CR20","unstructured":"Samko, S.G., Kilbas, A.A., Mari\u010bev, O.I.: Fractional Integrals and Derivatives. Gordon and Breach Science Publication (1993)"},{"key":"9561_CR21","doi-asserted-by":"crossref","unstructured":"Shen, J., Tang, T., Wang, L.-L.: Spectral Methods: Algorithms, Analysis and Applications. Springer (2011)","DOI":"10.1007\/978-3-540-71041-7"},{"issue":"1","key":"9561_CR22","doi-asserted-by":"publisher","first-page":"13","DOI":"10.1016\/S0378-4371(99)00565-8","volume":"282","author":"VV Yanovsky","year":"2000","unstructured":"Yanovsky, V.V., Chechkin, A.V., Schertzer, D., Tur, A.V.: L\u00e9vy anomalous diffusion and fractional fokker\u2013planck equation. Phys. A Stat. Mech. Appl. 282(1), 13\u201334 (2000)","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"9561_CR23","doi-asserted-by":"publisher","first-page":"495","DOI":"10.1016\/j.jcp.2013.06.031","volume":"252","author":"M Zayernouri","year":"2013","unstructured":"Zayernouri, M., Karniadakis, G.E.: Fractional Sturm-Liouville eigen-problems: theory and numerical approximation. J. Comput. Phys. 252, 495\u2013517 (2013)","journal-title":"J. Comput. Phys."},{"issue":"4","key":"9561_CR24","doi-asserted-by":"publisher","first-page":"B684","DOI":"10.1137\/130940967","volume":"36","author":"M Zayernouri","year":"2014","unstructured":"Zayernouri, M., Karniadakis, G.E.: Discontinuous spectral element methods for time- and space-fractional advection equations. SIAM J. Sci. Comput. 36(4), B684\u2013B707 (2014)","journal-title":"SIAM J. Sci. Comput."},{"issue":"6","key":"9561_CR25","doi-asserted-by":"publisher","first-page":"A2710","DOI":"10.1137\/141001299","volume":"37","author":"F Zeng","year":"2015","unstructured":"Zeng, F., Zhang, Z., Karniadakis, G.E.: A generalized spectral collocation method with tunable accuracy for variable-order fractional differential equations. SIAM J. Sci. Comput. 37(6), A2710\u2013A2732 (2015)","journal-title":"SIAM J. Sci. Comput."},{"key":"9561_CR26","doi-asserted-by":"crossref","unstructured":"Zeng, F., Mao, Z., Karniadakis, G.E.: A generalized spectral collocation method with tunable accuracy for fractional differential equations with end-point singularities. To be appeared on SIAM Journal Science Computer (2016)","DOI":"10.1137\/16M1076083"},{"issue":"6","key":"9561_CR27","doi-asserted-by":"publisher","first-page":"A2865","DOI":"10.1137\/140961560","volume":"36","author":"X Zhao","year":"2014","unstructured":"Zhao, X., Sun, Z.-Z., Hao, Z.-P.: A fourth-order compact ADI scheme for two-dimensional nonlinear space fractional Schr\u00f6dinger equation. SIAM J. Sci. Comput. 36(6), A2865\u2013A2886 (2014)","journal-title":"SIAM J. Sci. Comput."}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10444-017-9561-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-017-9561-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-017-9561-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,5,17]],"date-time":"2020-05-17T15:37:32Z","timestamp":1589729852000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10444-017-9561-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,8,18]]},"references-count":27,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2018,6]]}},"alternative-id":["9561"],"URL":"https:\/\/doi.org\/10.1007\/s10444-017-9561-9","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"value":"1019-7168","type":"print"},{"value":"1572-9044","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,8,18]]},"assertion":[{"value":"15 February 2017","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 August 2017","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 August 2017","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}