{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,25]],"date-time":"2026-07-25T14:16:11Z","timestamp":1784988971241,"version":"3.55.0"},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2020,10,30]],"date-time":"2020-10-30T00:00:00Z","timestamp":1604016000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,10,30]],"date-time":"2020-10-30T00:00:00Z","timestamp":1604016000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"name":"Natural Science Foundation of Guangdong Province of China","award":["2018A030313236"],"award-info":[{"award-number":["2018A030313236"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Fundation of China","doi-asserted-by":"crossref","award":["11971123"],"award-info":[{"award-number":["11971123"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2021,9]]},"DOI":"10.1007\/s11075-020-01031-3","type":"journal-article","created":{"date-parts":[[2020,10,30]],"date-time":"2020-10-30T18:02:36Z","timestamp":1604080956000},"page":"93-111","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Spectral collocation method for Caputo fractional terminal value problems"],"prefix":"10.1007","volume":"88","author":[{"given":"Zhendong","family":"Gu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yinying","family":"Kong","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2020,10,30]]},"reference":[{"issue":"4","key":"1031_CR1","doi-asserted-by":"publisher","first-page":"1367","DOI":"10.1016\/j.chaos.2006.01.098","volume":"33","author":"WM Ahmad","year":"2007","unstructured":"Ahmad, W.M., El-Khazali, R.: Fractional-order dynamical models of love. Chaos Solitons Fract. 33(4), 1367\u20131375 (2007)","journal-title":"Chaos Solitons Fract."},{"issue":"20-21","key":"1031_CR2","doi-asserted-by":"publisher","first-page":"8903","DOI":"10.1016\/j.apm.2013.04.019","volume":"37","author":"M Atabakzadeh","year":"2013","unstructured":"Atabakzadeh, M., Akrami, M., Erjaee, G.: Chebyshev operational matrix method for solving multi-order fractional ordinary differential equations. Appl. Math. Model. 37(20-21), 8903\u20138911 (2013)","journal-title":"Appl. Math. Model."},{"issue":"1","key":"1031_CR3","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1051\/mmnp\/2018010","volume":"13","author":"A Atangana","year":"2018","unstructured":"Atangana, A., Owolabi, K. M., Atangana, A., Mophou, G., Hristov, J., Hammouch, Z.: New numerical approach for fractional differential equations. Math. Modell. Nat. Phenom. 13(1), 3 (2018)","journal-title":"Math. Modell. Nat. Phenom."},{"issue":"10","key":"1031_CR4","doi-asserted-by":"publisher","first-page":"2492","DOI":"10.1109\/TIP.2007.904971","volume":"16","author":"J Bai","year":"2007","unstructured":"Bai, J., Feng, X.C.: Fractional-order anisotropic diffusion for image denoising. IEEE Trans. Image Process. 16(10), 2492\u20132502 (2007)","journal-title":"IEEE Trans. Image Process."},{"key":"1031_CR5","doi-asserted-by":"crossref","unstructured":"Brunner, H.: Collocation methods for Volterra integral and related functional differential equations, vol. 15. Cambridge University Press (2004)","DOI":"10.1017\/CBO9780511543234"},{"issue":"1","key":"1031_CR6","doi-asserted-by":"publisher","first-page":"134","DOI":"10.1007\/BF00879562","volume":"91","author":"M Caputo","year":"1971","unstructured":"Caputo, M., Mainardi, F.: A new dissipation model based on memory mechanism. Pure Appl. Geophys. 91(1), 134\u2013147 (1971)","journal-title":"Pure Appl. Geophys."},{"key":"1031_CR7","first-page":"603","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, 603\u20131638 (2016)","journal-title":"Math. Comput."},{"issue":"4","key":"1031_CR8","first-page":"1042","volume":"41","author":"Y Cokun","year":"2017","unstructured":"Cokun, Y., Arslan, M.: Terminal value problem for causal differential equations with a C aputo fractional derivative. Turk. J. Math. 41(4), 1042\u20131052 (2017)","journal-title":"Turk. J. Math."},{"issue":"APR.","key":"1031_CR9","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(APR.), 424\u2013448 (2018)","journal-title":"Appl. Math. Model."},{"key":"1031_CR10","doi-asserted-by":"crossref","unstructured":"Diethelm, K.: The analysis of fractional differential equations. Springer. Berlin (2010)","DOI":"10.1007\/978-3-642-14574-2"},{"key":"1031_CR11","unstructured":"Drici, Z., Devi, J.V., Mcrae, F.: On the comparison principle and existence results for terminal value problems. Nonlinear Stud. 21(2), 269\u2013282"},{"key":"1031_CR12","unstructured":"Faires, J.D., Burden, R.L.: Numerical methods. Brooks\/Cole Pub Co. (1998)"},{"issue":"1","key":"1031_CR13","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1515\/cmam-2013-0022","volume":"14","author":"NJ Ford","year":"2014","unstructured":"Ford, N.J., Morgado, M.L., Rebelo, M.: High order numerical methods for fractional terminal value problems. Comput. Methods Appl. Math. 14(1), 55\u201370 (2014)","journal-title":"Comput. Methods Appl. Math."},{"key":"1031_CR14","doi-asserted-by":"publisher","first-page":"392","DOI":"10.1016\/j.cam.2014.06.013","volume":"275","author":"NJ Ford","year":"2015","unstructured":"Ford, N.J., Morgado, M.L., Rebelo, M.: A nonpolynomial collocation method for fractional terminal value problems. J. Comput. Appl. Math. 275, 392\u2013402 (2015)","journal-title":"J. Comput. Appl. Math."},{"key":"1031_CR15","doi-asserted-by":"crossref","first-page":"140","DOI":"10.1016\/j.amc.2014.05.012","volume":"241","author":"JR Graef","year":"2014","unstructured":"Graef, J.R., Kong, L., Wang, M.: A C hebyshev spectral method for solving R iemann-L iouville fractional boundary value problems. Appl. Math. Comput. 241, 140\u2013150 (2014)","journal-title":"Appl. Math. Comput."},{"key":"1031_CR16","doi-asserted-by":"publisher","first-page":"2677","DOI":"10.1007\/s10444-019-09703-y","volume":"45","author":"Z Gu","year":"2019","unstructured":"Gu, Z.: Spectral collocation method for system of weakly singular V olterra integral equations. Adv. Comput. Math. 45, 2677\u20132699 (2019)","journal-title":"Adv. Comput. Math."},{"issue":"4","key":"1031_CR17","doi-asserted-by":"publisher","first-page":"311","DOI":"10.1515\/jnma-2014-0014","volume":"22","author":"Z Gu","year":"2014","unstructured":"Gu, Z., Chen, Y.: Chebyshev spectral-collocation method for a class of weakly singular V olterra integral equations with proportional delay. J. Numer. Math. 22(4), 311\u2013342 (2014)","journal-title":"J. Numer. Math."},{"issue":"3","key":"1031_CR18","doi-asserted-by":"publisher","first-page":"1554","DOI":"10.1007\/s10915-017-0506-8","volume":"74","author":"C Huang","year":"2018","unstructured":"Huang, C., Zhang, Z., Song, Q.: Spectral methods for substantial fractional differential equations. J. Sci. Comput. 74(3), 1554\u20131574 (2018)","journal-title":"J. Sci. Comput."},{"key":"1031_CR19","doi-asserted-by":"publisher","first-page":"94","DOI":"10.1016\/j.cam.2015.02.058","volume":"293","author":"B Jin","year":"2016","unstructured":"Jin, B., Lazarov, R., Lu, X., Zhou, Z.: A simple finite element method for boundary value problems with a R iemann-L iouville derivative. J. Comput. Appl. Math. 293, 94\u2013111 (2016)","journal-title":"J. Comput. Appl. Math."},{"issue":"1","key":"1031_CR20","doi-asserted-by":"publisher","first-page":"445","DOI":"10.1137\/120873984","volume":"51","author":"B Jin","year":"2012","unstructured":"Jin, B., Lazarov, R., Zhou, Z.: Error estimates for a semidiscrete finite element method for fractional order parabolic equations. SIAM J. Numer. Anal. 51(1), 445\u2013466 (2012)","journal-title":"SIAM J. Numer. Anal."},{"issue":"5","key":"1031_CR21","doi-asserted-by":"publisher","first-page":"118","DOI":"10.20852\/ntmsci.2017.190","volume":"3","author":"HD Kasmaei","year":"2017","unstructured":"Kasmaei, H.D., Senol, M.: On the numerical solution of nonlinear fractional-integro differential equations. Trends Math. Sci. 3(5), 118\u2013127 (2017)","journal-title":"Trends Math. Sci."},{"key":"1031_CR22","doi-asserted-by":"crossref","unstructured":"Kopteva, N., Stynes, M.: Analysis and numerical solution of a R iemann-L iouville fractional derivarive two-point boundary value problem 43, 77\u201399 (2017)","DOI":"10.1007\/s10444-016-9476-x"},{"issue":"3","key":"1031_CR23","doi-asserted-by":"publisher","first-page":"2108","DOI":"10.1137\/080718942","volume":"47","author":"X Li","year":"2009","unstructured":"Li, X., Xu, C.: A space-time spectral method for the time fractional diffusion equation. Siam J. Numer. Anal. 47(3), 2108\u20132131 (2009)","journal-title":"Siam J. Numer. Anal."},{"issue":"1","key":"1031_CR24","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 C aputo two-point boundary value problems. J. Sci. Comput. 76(1), 390\u2013425 (2018)","journal-title":"J. Sci. Comput."},{"issue":"2","key":"1031_CR25","doi-asserted-by":"publisher","first-page":"897","DOI":"10.1007\/s10444-018-9645-1","volume":"45","author":"H Liang","year":"2019","unstructured":"Liang, H., Stynes, M.: Collocation methods for general R iemann-L iouville two-point boundary value problems. Adv. Comput. Math. 45(2), 897\u2013928 (2019)","journal-title":"Adv. Comput. Math."},{"issue":"4","key":"1031_CR26","doi-asserted-by":"publisher","first-page":"872","DOI":"10.1109\/TASSP.1981.1163636","volume":"29","author":"R Marks","year":"1981","unstructured":"Marks, R., Hall, M.M.W.: Differintegral interpolation from a bandlimited signal\u2019s samples. IEEE Trans. Acoust. Speech Signal Process. 29(4), 872\u2013877 (1981)","journal-title":"IEEE Trans. Acoust. Speech Signal Process."},{"issue":"1","key":"1031_CR27","doi-asserted-by":"publisher","first-page":"249","DOI":"10.1016\/j.jcp.2005.05.017","volume":"211","author":"MM Meerschaert","year":"2006","unstructured":"Meerschaert, M.M., Scheffler, H.-P., Tadjeran, C.: Finite difference methods for two-dimensional fractional dispersion equation. J. Comput. Phys. 211(1), 249\u2013261 (2006)","journal-title":"J. Comput. Phys."},{"issue":"5","key":"1031_CR28","doi-asserted-by":"publisher","first-page":"1239","DOI":"10.1515\/fca-2017-0065","volume":"20","author":"ML Morgado","year":"2017","unstructured":"Morgado, M.L., Rebelo, M.: Well-posedness and numerical approximation of tempered fractional terminal value problems. Fract. Calc. Appl. Anal. 20(5), 1239\u20131262 (2017)","journal-title":"Fract. Calc. Appl. Anal."},{"issue":"2","key":"1031_CR29","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(2), 167\u2013176 (2011)","journal-title":"J. Comput. Appl. Math."},{"issue":"13","key":"1031_CR30","doi-asserted-by":"publisher","first-page":"3349","DOI":"10.1016\/j.cam.2012.03.002","volume":"236","author":"A Pedas","year":"2012","unstructured":"Pedas, A., Tamme, E.: Piecewise polynomial collocation for linear boundary value problems of fractional differential equations. J. Comput. Appl. Math. 236(13), 3349\u20133359 (2012)","journal-title":"J. Comput. Appl. Math."},{"issue":"2","key":"1031_CR31","doi-asserted-by":"publisher","first-page":"437","DOI":"10.4208\/nmtma.2017.s11","volume":"10","author":"C Sheng","year":"2017","unstructured":"Sheng, C., Jie, S.: A hybrid spectral element method for fractional two-point boundary value problems. Numer. Math. Theory Methods Appl. 10(2), 437\u2013464 (2017)","journal-title":"Numer. Math. Theory Methods Appl."},{"issue":"3","key":"1031_CR32","doi-asserted-by":"publisher","first-page":"616","DOI":"10.1016\/j.cnsns.2009.04.029","volume":"15","author":"L Song","year":"2010","unstructured":"Song, L., Xu, S., Yang, J.: Dynamical models of happiness with fractional order. Commun. Nonlinear Sci. Numer. Simul. 15(3), 616\u2013628 (2010)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"issue":"2","key":"1031_CR33","doi-asserted-by":"publisher","first-page":"698","DOI":"10.1093\/imanum\/dru011","volume":"35","author":"M Stynes","year":"2015","unstructured":"Stynes, M., Gracia, J.L.: A finite difference method for a two-point boundary value problem with a C aputo fractional derivative. Ima J. Numer. Anal. 35(2), 698\u2013721 (2015)","journal-title":"Ima J. Numer. Anal."},{"issue":"2","key":"1031_CR34","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1016\/j.apnum.2005.03.003","volume":"56","author":"ZZ Sun","year":"2006","unstructured":"Sun, Z.Z., Wu, X.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56(2), 193\u2013209 (2006)","journal-title":"Appl. Numer. Math."},{"issue":"1","key":"1031_CR35","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1007\/s11075-015-0041-3","volume":"72","author":"S Vong","year":"2016","unstructured":"Vong, S., Lyu, P., Chen, X., Lei, S.L.: High order finite difference method for time-space fractional differential equations with C aputo and R iemann-L iouville derivatives. Numer. Algorithm. 72(1), 195\u2013210 (2016)","journal-title":"Numer. Algorithm."},{"issue":"6","key":"1031_CR36","doi-asserted-by":"publisher","first-page":"1285","DOI":"10.1007\/s10955-007-9294-0","volume":"126","author":"BJ West","year":"2007","unstructured":"West, B.J.: Fractional calculus in bioengineering. J. Stat. Phys. 126(6), 1285\u20131286 (2007)","journal-title":"J. Stat. Phys."},{"issue":"1","key":"1031_CR37","first-page":"65","volume":"172","author":"X Yin","year":"2013","unstructured":"Yin, X., Zhou, J., Bing, H.: Finite difference approximations for fractional advection-dispersion equations. J. Comput. Appl. Math. 172(1), 65\u201377 (2013)","journal-title":"J. Comput. Appl. Math."},{"issue":"1","key":"1031_CR38","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1016\/j.cam.2019.01.046","volume":"357","author":"M Zaky","year":"2019","unstructured":"Zaky, M.: Recovery of high order accuracy in J acobi spectral collocation methods for fractional terminal value problems with non-smooth solutions. J. Comput. Appl. Math. 357(1), 103\u2013122 (2019)","journal-title":"J. Comput. Appl. Math."},{"issue":"1","key":"1031_CR39","doi-asserted-by":"publisher","first-page":"A40","DOI":"10.1137\/130933216","volume":"36","author":"M Zayernouri","year":"2014","unstructured":"Zayernouri, M., Karniadakis, G.E.: Fractional spectral collocation method. SIAM J. Sci. Comput. 36(1), A40\u2013A62 (2014)","journal-title":"SIAM J. Sci. Comput."},{"issue":"6","key":"1031_CR40","doi-asserted-by":"publisher","first-page":"A2976","DOI":"10.1137\/130910865","volume":"35","author":"F Zeng","year":"2013","unstructured":"Zeng, F., Li, C., Liu, F., Turner, I.: The use of finite difference\/element approaches for solving the time-fractional subdiffusion equation. SIAM J. Sci. Comput. 35(6), A2976\u2013A3000 (2013)","journal-title":"SIAM J. Sci. Comput."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-020-01031-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-020-01031-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-020-01031-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,26]],"date-time":"2023-10-26T07:34:14Z","timestamp":1698305654000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-020-01031-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,10,30]]},"references-count":40,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2021,9]]}},"alternative-id":["1031"],"URL":"https:\/\/doi.org\/10.1007\/s11075-020-01031-3","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,10,30]]},"assertion":[{"value":"16 May 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 October 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 October 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}