{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,2]],"date-time":"2026-02-02T19:46:47Z","timestamp":1770061607752,"version":"3.49.0"},"reference-count":32,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2021,4,22]],"date-time":"2021-04-22T00:00:00Z","timestamp":1619049600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,4,22]],"date-time":"2021-04-22T00:00:00Z","timestamp":1619049600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12071294"],"award-info":[{"award-number":["12071294"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"publisher","award":["2020M681345"],"award-info":[{"award-number":["2020M681345"]}],"id":[{"id":"10.13039\/501100002858","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":[[2021,6]]},"DOI":"10.1007\/s10444-021-09858-7","type":"journal-article","created":{"date-parts":[[2021,4,22]],"date-time":"2021-04-22T08:05:46Z","timestamp":1619078746000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["An hp-version Legendre spectral collocation method for multi-order fractional differential equations"],"prefix":"10.1007","volume":"47","author":[{"given":"Yuling","family":"Guo","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhongqing","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,4,22]]},"reference":[{"key":"9858_CR1","doi-asserted-by":"publisher","first-page":"876","DOI":"10.1016\/j.jcp.2014.10.060","volume":"281","author":"AH Bhrawy","year":"2015","unstructured":"Bhrawy, A.H., Zaky, M.A.: A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations. J. Comput. Phys. 281, 876\u2013895 (2015)","journal-title":"J. Comput. Phys."},{"key":"9858_CR2","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511543234","volume-title":"Collocation methods for volterra integral and related functional equations","author":"H Brunner","year":"2004","unstructured":"Brunner, H.: Collocation methods for volterra integral and related functional equations. Cambridge University Press, Cambridge (2004)"},{"key":"9858_CR3","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-30726-6","volume-title":"Spectral methods: Fundamentals in single domains","author":"C Canuto","year":"2006","unstructured":"Canuto, C., Hussaini, M., Quarteroni, A., Zang, T.: Spectral methods: Fundamentals in single domains. Springer, Berlin (2006)"},{"key":"9858_CR4","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.: Generalized Jacobi functions and their applications to fractional differential equations. Math. Comput. 85, 1603\u20131638 (2016)","journal-title":"Math. Comput."},{"key":"9858_CR5","doi-asserted-by":"publisher","first-page":"284","DOI":"10.1016\/j.cnsns.2017.02.009","volume":"50","author":"A Dabiri","year":"2017","unstructured":"Dabiri, A., Butcher, E.A.: Efficient modified Chebyshev differentiation matrices for fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 50, 284\u2013310 (2017)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"9858_CR6","doi-asserted-by":"publisher","first-page":"424","DOI":"10.1016\/j.apm.2017.12.012","volume":"56","author":"A Dabiri","year":"2018","unstructured":"Dabiri, A., Butcher, E.A.: Numerical solution of multi-order fractional differential equations with multiple delays via spectral collocation methods. Appl. Math. Model. 56, 424\u2013448 (2018)","journal-title":"Appl. Math. Model."},{"key":"9858_CR7","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-14574-2","volume-title":"The analysis of fractional differential equations: An application-oriented exposition using differential operators of caputo type, Lect. Notes Math.","author":"K Diethelm","year":"2010","unstructured":"Diethelm, K.: The analysis of fractional differential equations: An application-oriented exposition using differential operators of caputo type, Lect. Notes Math. Springer, Berlin (2010)"},{"key":"9858_CR8","doi-asserted-by":"publisher","first-page":"194","DOI":"10.1016\/j.apnum.2020.08.003","volume":"158","author":"Y Guo","year":"2020","unstructured":"Guo, Y., Wang, Z.: An hp-version Chebyshev collocation method for nonlinear fractional differential equations. Appl. Numer. Math. 158, 194\u2013211 (2020)","journal-title":"Appl. Numer. Math."},{"key":"9858_CR9","volume-title":"Applications of fractional calculus in physics","author":"R Hilfer","year":"1999","unstructured":"Hilfer, R.: Applications of fractional calculus in physics. World Scientific, Singapore (1999)"},{"key":"9858_CR10","doi-asserted-by":"publisher","first-page":"2665","DOI":"10.1090\/mcom\/2960","volume":"84","author":"B Jin","year":"2015","unstructured":"Jin, B., Lazarov, R., Pasciak, J., Rundell, W.: Variational formulation of problems involving fractional order differential operators. Math. Comput. 84, 2665\u20132700 (2015)","journal-title":"Math. Comput."},{"key":"9858_CR11","doi-asserted-by":"publisher","first-page":"390","DOI":"10.1007\/s10915-017-0622-5","volume":"76","author":"H Liang","year":"2018","unstructured":"Liang, H., Stynes, M.: Collocation methods for general Caputo two-point boundary value problems. J. Sci. Comput. 76, 390\u2013425 (2018)","journal-title":"J. Sci. Comput."},{"key":"9858_CR12","doi-asserted-by":"publisher","first-page":"922","DOI":"10.1137\/17M1113060","volume":"39","author":"A Lischke","year":"2017","unstructured":"Lischke, A., Zayernouri, M., Karniadakis, G.E.: A Petrov-Galerkin spectral method of linear complexity for fractional multiterm ODEs on the half line. SIAM J. Sci. Comput. 39, 922\u2013946 (2017)","journal-title":"SIAM J. Sci. Comput."},{"key":"9858_CR13","doi-asserted-by":"publisher","first-page":"825","DOI":"10.1016\/j.jcp.2014.10.051","volume":"281","author":"Y Liu","year":"2015","unstructured":"Liu, Y., Zhou, Z., Jin, B., Lazarov, R.: The Galerkin finite element method for a multi-term time-fractional diffusion equation. J. Comput. Phys. 281, 825\u2013843 (2015)","journal-title":"J. Comput. Phys."},{"key":"9858_CR14","doi-asserted-by":"publisher","first-page":"538","DOI":"10.1016\/j.jmaa.2010.08.048","volume":"374","author":"Y Luchko","year":"2011","unstructured":"Luchko, Y.: Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation. J. Math. Anal. Appl. 374, 538\u2013548 (2011)","journal-title":"J. Math. Anal. Appl."},{"key":"9858_CR15","doi-asserted-by":"publisher","first-page":"671","DOI":"10.1016\/j.apm.2015.06.014","volume":"40","author":"P Mokhtary","year":"2016","unstructured":"Mokhtary, P., Ghoreishi, F., Srivastava, H.M.: The m\u00fcntz-Legendre tau method for fractional differential equations. Appl. Math. Model 40, 671\u2013684 (2016)","journal-title":"Appl. Math. Model"},{"key":"9858_CR16","doi-asserted-by":"publisher","first-page":"619","DOI":"10.4208\/nmtma.2016.m1429","volume":"9","author":"J Mu","year":"2016","unstructured":"Mu, J., Wang, Z.: A multiple interval Chebyshev-Gauss-Lobatto collocation method for ordinary differential equations. Numer. Math. Theor. Meth. Appl. 9, 619\u2013639 (2016)","journal-title":"Numer. Math. Theor. Meth. Appl."},{"key":"9858_CR17","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1016\/j.cam.2011.06.015","volume":"236","author":"A Pedas","year":"2011","unstructured":"Pedas, A., Tamme, E.: Spline collocation methods for linear multi-term fractional differential equations. J. Comput. Appl. Math. 236, 167\u2013176 (2011)","journal-title":"J. Comput. Appl. Math."},{"key":"9858_CR18","volume-title":"Fractional Differential Equations","author":"I Podlubny","year":"1999","unstructured":"Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)"},{"key":"9858_CR19","doi-asserted-by":"publisher","first-page":"1326","DOI":"10.1016\/j.camwa.2009.07.006","volume":"59","author":"A Saadatmandi","year":"2010","unstructured":"Saadatmandi, A., Dehghan, M.: A new operational matrix for solving fractional order differential equations. Comput. Math. Appl. 59, 1326\u20131336 (2010)","journal-title":"Comput. Math. Appl."},{"key":"9858_CR20","doi-asserted-by":"publisher","first-page":"421","DOI":"10.1007\/s40324-017-0141-1","volume":"75","author":"H Saeedi","year":"2018","unstructured":"Saeedi, H.: A fractional-order operational method for numerical treatment of multi-order fractional partial differential equation with variable coefficients. SeMA J. 75, 421\u2013433 (2018)","journal-title":"SeMA J."},{"key":"9858_CR21","doi-asserted-by":"publisher","first-page":"1953","DOI":"10.1137\/130915200","volume":"52","author":"C Sheng","year":"2014","unstructured":"Sheng, C., Wang, Z., Guo, B.: A multistep Legendre-Gauss spectral collocation method for nonlinear Volterra integral equations. SIAM J. Numer. Anal. 52, 1953\u20131980 (2014)","journal-title":"SIAM J. Numer. Anal."},{"key":"9858_CR22","doi-asserted-by":"publisher","first-page":"616","DOI":"10.1016\/j.mcm.2009.11.002","volume":"51","author":"V Srivastava","year":"2010","unstructured":"Srivastava, V., Rai, K.N.: A multi-term fractional diffusion equation for oxygen delivery through a capillary to tissues. Math. Comput. Model. 51, 616\u2013624 (2010)","journal-title":"Math. Comput. Model."},{"key":"9858_CR23","doi-asserted-by":"publisher","first-page":"467","DOI":"10.1007\/s10915-018-0820-9","volume":"78","author":"H Sun","year":"2019","unstructured":"Sun, H., Zhao, X., Sun, Z.: The temporal second order difference schemes based on the interpolation approximation for the time multi-term fractional wave equation. J. Sci. Comput. 78, 467\u2013498 (2019)","journal-title":"J. Sci. Comput."},{"key":"9858_CR24","volume-title":"Orthogonal polynomials","author":"G Szeg\u00f6","year":"1978","unstructured":"Szeg\u00f6, G.: Orthogonal polynomials. AMS Coll. Publ. 23, Providence (1978)"},{"key":"9858_CR25","volume-title":"Theory and applications of fractional differential equations","author":"JJ Trujillo","year":"2006","unstructured":"Trujillo, J.J., Kilbas, A.A., Srivastava, H.M.: Theory and applications of fractional differential equations. Elsevier, Amsterdam (2006)"},{"key":"9858_CR26","doi-asserted-by":"publisher","first-page":"647","DOI":"10.1007\/s10915-017-0373-3","volume":"72","author":"C Wang","year":"2017","unstructured":"Wang, C., Wang, Z., Jia, H.: An hp-version spectral collocation method for nonlinear Volterra integro-differential equation with weakly singular kernels. J. Sci. Comput. 72, 647\u2013678 (2017)","journal-title":"J. Sci. Comput."},{"key":"9858_CR27","doi-asserted-by":"publisher","first-page":"166","DOI":"10.1007\/s10915-017-0616-3","volume":"76","author":"C Wang","year":"2018","unstructured":"Wang, C., Wang, Z., Wang, L.: A spectral collocation method for nonlinear fractional boundary value problems with a Caputo derivative. J. Sci. Comput. 76, 166\u2013188 (2018)","journal-title":"J. Sci. Comput."},{"key":"9858_CR28","doi-asserted-by":"publisher","first-page":"2285","DOI":"10.1090\/mcom\/3183","volume":"86","author":"Z Wang","year":"2017","unstructured":"Wang, Z., Guo, Y., Yi, L.: An hp-version Legendre-Jacobi spectral collocation method for Volterra integro-differential equations with smooth and weakly singular kernels. Math. Comp. 86, 2285\u20132324 (2017)","journal-title":"Math. Comp."},{"key":"9858_CR29","first-page":"635","volume":"85","author":"Z Wang","year":"2016","unstructured":"Wang, Z., Sheng, C.: An hp-spectral collocation method for nonlinear Volterra integral equations with vanishing variable delays. Math. Comp. 85, 635\u2013666 (2016)","journal-title":"Math. Comp."},{"key":"9858_CR30","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.cam.2019.01.001","volume":"38","author":"R Yan","year":"2019","unstructured":"Yan, R., Sun, Y., Ma, Q., Ding, X.: A spectral collocation method for nonlinear fractional initial value problems with a variable-order fractional derivative. Comput. Appl. Math. 38, 1\u201324 (2019)","journal-title":"Comput. Appl. Math."},{"key":"9858_CR31","doi-asserted-by":"publisher","first-page":"3525","DOI":"10.1007\/s40314-017-0530-1","volume":"37","author":"MA Zaky","year":"2018","unstructured":"Zaky, M. A.: A Legendre spectral quadrature tau method for the multi-term time-fractional diffusion equations. Comput. Appl. Math. 37, 3525\u20133538 (2018)","journal-title":"Comput. Appl. Math."},{"key":"9858_CR32","doi-asserted-by":"publisher","first-page":"144","DOI":"10.1007\/s40314-019-0922-5","volume":"38","author":"MA Zaky","year":"2019","unstructured":"Zaky, M.A., Ameen, I.G.: On the rate of convergence of spectral collocation methods for nonlinear multi-order fractional initial value problems. Comput. Appl. Math. 38, 144\u2013170 (2019)","journal-title":"Comput. Appl. Math."}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-021-09858-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10444-021-09858-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-021-09858-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,6,21]],"date-time":"2021-06-21T08:14:19Z","timestamp":1624263259000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10444-021-09858-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,22]]},"references-count":32,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2021,6]]}},"alternative-id":["9858"],"URL":"https:\/\/doi.org\/10.1007\/s10444-021-09858-7","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"value":"1019-7168","type":"print"},{"value":"1572-9044","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,4,22]]},"assertion":[{"value":"29 July 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 March 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 April 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"37"}}