{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,8]],"date-time":"2025-11-08T13:01:02Z","timestamp":1762606862138},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2016,10,20]],"date-time":"2016-10-20T00:00:00Z","timestamp":1476921600000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2017,4]]},"DOI":"10.1007\/s10444-016-9490-z","type":"journal-article","created":{"date-parts":[[2016,10,20]],"date-time":"2016-10-20T04:36:04Z","timestamp":1476938164000},"page":"385-409","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Piecewise spectral collocation method for system of Volterra integral equations"],"prefix":"10.1007","volume":"43","author":[{"given":"Zhendong","family":"Gu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2016,10,20]]},"reference":[{"key":"9490_CR1","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511543234","volume-title":"Collocation Methods for Volterra Integral and Related Functional Differential Equations","author":"H Brunner","year":"2004","unstructured":"Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Differential Equations. Cambridge University Press, London (2004)"},{"key":"9490_CR2","first-page":"63","volume":"608\u2013616","author":"J Biazar","year":"2012","unstructured":"Biazar, J., Ebrahimi, H.: Chebyshev wavelets approach for nonlinear systems of Volterra integral equations. Comput. Math. Appl. 608\u2013616, 63 (2012)","journal-title":"Comput. Math. Appl."},{"key":"9490_CR3","doi-asserted-by":"crossref","first-page":"811","DOI":"10.1016\/j.amc.2013.10.006","volume":"225","author":"F Calio","year":"2013","unstructured":"Calio, F., Garralda-Guillem, A.I., Marchetti, E., Ruiz Galn, M.: Numerical approaches for systems of VolterraCFredholm integral equations. Appl. Math. Comput. 225, 811\u2013821 (2013)","journal-title":"Appl. Math. Comput."},{"key":"9490_CR4","doi-asserted-by":"crossref","unstructured":"Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral method fundamentals in single domains Spring-Verlag (2006)","DOI":"10.1007\/978-3-540-30726-6"},{"key":"9490_CR5","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-540-30728-0","volume-title":"Spectral method evolution to complex geometries and application to fluid dynamics","author":"C Canuto","year":"2007","unstructured":"Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral method evolution to complex geometries and application to fluid dynamics. Springer, Berlin (2007)"},{"key":"9490_CR6","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1007\/s10543-007-0120-5","volume":"47","author":"G Capobianco","year":"2007","unstructured":"Capobianco, G., Conte, D., Del Prete, I., Russo, E.: Fast Runge-Kutta methods for nonlinear convolution systems of Volterra integral euquations. BIT Numer. Math. 47, 259\u2013275 (2007)","journal-title":"BIT Numer. Math."},{"key":"9490_CR7","doi-asserted-by":"crossref","first-page":"67","DOI":"10.2140\/camcos.2013.8.67","volume":"8","author":"Y Chen","year":"2013","unstructured":"Chen, Y., Gu, Z.: Legendre spectral-collocation method for VIDEs with non-vanishing delay. Commun. Appl. Math. Comput. Sci. 8, 67\u201398 (2013)","journal-title":"Commun. Appl. Math. Comput. Sci."},{"key":"9490_CR8","doi-asserted-by":"crossref","first-page":"47","DOI":"10.4208\/jcm.1208-m3497","volume":"31","author":"Y Chen","year":"2013","unstructured":"Chen, Y., Li, X., Tang, T.: A note on Jacobi spectral-collocation methods for weakly singular Volterra integral equations with smooth solutions. J. Comput. Math. 31, 47\u201356 (2013)","journal-title":"J. Comput. Math."},{"key":"9490_CR9","doi-asserted-by":"crossref","first-page":"938","DOI":"10.1016\/j.cam.2009.08.057","volume":"233","author":"Y Chen","year":"2009","unstructured":"Chen, Y., Tang, Y.: Spectral methods for weakly singular Volterra integral equations with smooth solutions. J. Comput. Appl. Math. 233, 938\u2013950 (2009)","journal-title":"J. Comput. Appl. Math."},{"key":"9490_CR10","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1090\/S0025-5718-09-02269-8","volume":"79","author":"Y Chen","year":"2010","unstructured":"Chen, Y., Tang, T.: Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equation with a weakly singular kernel. Math. Comput. 79, 147\u2013167 (2010)","journal-title":"Math. Comput."},{"key":"9490_CR11","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1090\/conm\/586\/11637","volume":"586","author":"Z Gu","year":"2013","unstructured":"Gu, Z., Chen, Y.: Chebyshev spectral-collocation method for Volterra integral equations. Contemp. Math. 586, 163\u2013170 (2013)","journal-title":"Contemp. Math."},{"key":"9490_CR12","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1007\/s10092-013-0083-7","volume":"51","author":"Z Gu","year":"2014","unstructured":"Gu, Z., Chen, Y.: Legendre spectral collocation method for Volterra integral equations with non-vanishing delay. Calcolo 51, 151\u2013174 (2014)","journal-title":"Calcolo"},{"key":"9490_CR13","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1023\/A:1016681018268","volume":"14","author":"B Guo","year":"2001","unstructured":"Guo, B., Wang, L.: Jacobi interpolation approximations and their applications to singular differential equations. Adv. Comput. Math. 14, 227\u2013276 (2001)","journal-title":"Adv. Comput. Math."},{"key":"9490_CR14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jat.2004.03.008","volume":"128","author":"B Guo","year":"2004","unstructured":"Guo, B., Wang, L.: Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces. J. Approximation Theory 128, 1\u201341 (2004)","journal-title":"J. Approximation Theory"},{"key":"9490_CR15","doi-asserted-by":"crossref","first-page":"10225","DOI":"10.1016\/j.amc.2013.03.123","volume":"219","author":"W Jiang","year":"2013","unstructured":"Jiang, W., Chen, Z.: Solving a system of linear Volterra integral equations using the new reproducing kernel method. Appl. Math. Comput. 219, 10225\u201310230 (2013)","journal-title":"Appl. Math. Comput."},{"key":"9490_CR16","doi-asserted-by":"crossref","first-page":"400","DOI":"10.1016\/j.apm.2009.04.013","volume":"34","author":"R Katani","year":"2010","unstructured":"Katani, R., Shahmorad, S.: Block by block method for the systems of nonlinear Volterra integral equations. Appl. Math. Model. 34, 400\u2013406 (2010)","journal-title":"Appl. Math. Model."},{"key":"9490_CR17","doi-asserted-by":"crossref","first-page":"1735","DOI":"10.1137\/120876241","volume":"51","author":"X Li","year":"2013","unstructured":"Li, X., Tang, T., Xu, C.: Parallel in time algorithm with spectral-subdomain enhancement for volterra integral equations. SIAM J. Numer. Anal. 51, 1735\u20131756 (2013)","journal-title":"SIAM J. Numer. Anal."},{"key":"9490_CR18","doi-asserted-by":"crossref","first-page":"500","DOI":"10.1016\/j.amc.2007.05.001","volume":"195","author":"K Maleknejad","year":"2008","unstructured":"Maleknejad, K., Shamloo, A.S.: Numerical solution of singular Volterra integral equations system of convolution type by using operational matrices. Appl. Math. Comput. 195, 500\u2013505 (2008)","journal-title":"Appl. Math. Comput."},{"key":"9490_CR19","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1016\/j.jksus.2010.05.010","volume":"22","author":"F Mirzaee","year":"2010","unstructured":"Mirzaee, F.: Numerical computational solution of the linear Volterra integral equations system via rationalized Haar functions. J. King Saud Univ. (Science) 22, 265\u2013268 (2010)","journal-title":"J. King Saud Univ. (Science)"},{"key":"9490_CR20","doi-asserted-by":"publisher","unstructured":"Mirzaee, F., Bimesl, S.: A new Euler matrix method for solving systems of linear Volterra integral equations with variable coefficients. Journal of the Egyptian Mathematical Society. doi: 10.1016\/j.joems.2013.06.016 (2013)","DOI":"10.1016\/j.joems.2013.06.016"},{"key":"9490_CR21","volume-title":"Numerical Analysis: a Second Course","author":"JM Ortega","year":"1972","unstructured":"Ortega, J.M.: Numerical Analysis: a Second Course. Academic Press, New York (1972)"},{"key":"9490_CR22","doi-asserted-by":"crossref","first-page":"1143","DOI":"10.1016\/j.amc.2006.09.012","volume":"187","author":"M Rabbani","year":"2007","unstructured":"Rabbani, M., Maleknejad, K., Aghazadeh, N.: Numerical computational solution of the Volterra integral equations system of the second kind by using an expansion method. Appl. Math. Comput. 187, 1143\u20131146 (2007)","journal-title":"Appl. Math. Comput."},{"key":"9490_CR23","doi-asserted-by":"crossref","first-page":"755","DOI":"10.1016\/j.camwa.2011.05.057","volume":"62","author":"N Sahin","year":"2011","unstructured":"Sahin, N., Yuzbas, S., Gulsu, M.: A collocation approach for solving systems of linear Volterra integral equations with variable coefficients. Comput. Math. Appl. 62, 755\u2013769 (2011)","journal-title":"Comput. Math. Appl."},{"key":"9490_CR24","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1007\/s12190-012-0582-8","volume":"40","author":"ORN Samadi","year":"2012","unstructured":"Samadi, O.R.N., Tohidi, E.: The spectral method for solving systems of Volterra integral equations. J. Appl. Math. Comput. 40, 477\u2013497 (2012)","journal-title":"J. Appl. Math. Comput."},{"key":"9490_CR25","volume-title":"Spectral and High-Order Methods with Applications","author":"J Shen","year":"2006","unstructured":"Shen, J., Tang, T.: Spectral and High-Order Methods with Applications. Science Press, Beijing (2006)"},{"key":"9490_CR26","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-540-71041-7","volume-title":"Spectral Method Algorithms, Analysis and Applications","author":"J Shen","year":"2011","unstructured":"Shen, J., Tang, T., Wang, L.: Spectral Method Algorithms, Analysis and Applications. Springer, Berlin (2011)"},{"issue":"4","key":"9490_CR27","doi-asserted-by":"crossref","first-page":"1953","DOI":"10.1137\/130915200","volume":"52","author":"CT Sheng","year":"2014","unstructured":"Sheng, C.T., Wang, Z.Q., Guo, B.Y.: A multistep Legendre\u2013Gauss spectral collocation method for nonlinear volterra integral equations[J]. SIAM J. Numer. Anal. 52(4), 1953\u20131980 (2014)","journal-title":"SIAM J. Numer. Anal."},{"key":"9490_CR28","doi-asserted-by":"crossref","first-page":"3451","DOI":"10.1016\/j.apm.2010.02.034","volume":"34","author":"HH Sorkun","year":"2010","unstructured":"Sorkun, H.H., Yalcinbas, S.: Approximate solutions of linear Volterra integral equation systems with variable coefficients. Appl. Math. Model. 34, 3451\u20133464 (2010)","journal-title":"Appl. Math. Model."},{"key":"9490_CR29","doi-asserted-by":"crossref","first-page":"3486","DOI":"10.1016\/j.cnsns.2010.12.021","volume":"16","author":"H Taghvafard","year":"2011","unstructured":"Taghvafard, H., Erjaee, G.H.: On solving a system of singular Volterra integral equations of convolution type. Commun. Nonlinear Sci. Numer. Simulat. 16, 3486\u20133492 (2011)","journal-title":"Commun. Nonlinear Sci. Numer. Simulat."},{"key":"9490_CR30","doi-asserted-by":"crossref","first-page":"547","DOI":"10.1016\/j.amc.2007.12.041","volume":"201","author":"A Tahmasbi","year":"2008","unstructured":"Tahmasbi, A., Fard, O.S.: Numerical solution of linear Volterra integral equations system of the second kind. Appl. Math. Comput. 201, 547\u2013552 (2008)","journal-title":"Appl. Math. Comput."},{"key":"9490_CR31","first-page":"825","volume":"26","author":"T Tang","year":"2008","unstructured":"Tang, T., Xu, X., Cheng, J.: On Spectral methods for Volterra integral equation and the convergence analysis. J. Comput. Math. 26, 825\u2013837 (2008)","journal-title":"J. Comput. Math."},{"key":"9490_CR32","unstructured":"Volterra, V., Variazioni, E.: fluttuazioni del numero d\u2019indinvidui in specie animali conviventi. Memorie del R. Comitato talassografico italiano, Men. CXXXI (1927)"},{"key":"9490_CR33","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1007\/s11464-009-0002-z","volume":"4","author":"Z Wan","year":"2009","unstructured":"Wan, Z., Chen, Y., Huang, Y.: Legendre spectral Galerkin method for second-kind Volterra integral equations. Front. Math. China 4, 181\u2013193 (2009)","journal-title":"Front. Math. China"},{"key":"9490_CR34","doi-asserted-by":"crossref","first-page":"419","DOI":"10.4208\/nmtma.2011.m1028","volume":"4","author":"Y Wei","year":"2011","unstructured":"Wei, Y., Chen, Y.: Convergence analysis of the Legendre spectral collocation methods for second order Volterra integro-differential equations. Numer. Math. Theory Methods Appl. 4, 419\u2013438 (2011)","journal-title":"Numer. Math. Theory Methods Appl."},{"key":"9490_CR35","doi-asserted-by":"crossref","first-page":"1","DOI":"10.4208\/aamm.10-m1055","volume":"4","author":"Y Wei","year":"2012","unstructured":"Wei, Y., Chen, Y.: Convergence analysis of the spectral methods for weakly singular Volterra integro-differential equations with smooth solutions. Adv. Appl. Math. Mech. 4, 1\u201320 (2012)","journal-title":"Adv. Appl. Math. Mech."},{"key":"9490_CR36","doi-asserted-by":"crossref","first-page":"672","DOI":"10.1007\/s10915-012-9595-6","volume":"53","author":"Y Wei","year":"2012","unstructured":"Wei, Y., Chen, Y.: Legendre spectral collocation methods for pantograph volterra Delay-Integro-Differential equations. J. Sci. Comput. 53, 672\u2013688 (2012)","journal-title":"J. Sci. Comput."},{"key":"9490_CR37","doi-asserted-by":"crossref","first-page":"424","DOI":"10.4208\/nmtma.2013.1125nm","volume":"6","author":"Y Wei","year":"2013","unstructured":"Wei, Y., Chen, Y.: A spectral method for neutral volterra Integro-Differential equation with weakly singular kernel. Numer. Math. Theor. Meth. Appl. 6, 424\u2013446 (2013)","journal-title":"Numer. Math. Theor. Meth. Appl."},{"key":"9490_CR38","first-page":"81","volume":"15-29","author":"Y Wei","year":"2014","unstructured":"Wei, Y., Chen, Y.: Legendrespectralcollocationmethodforneutralandhigh-ordervolterraintegro-differentialequation. Appl. Numer. Math. 15-29, 81 (2014)","journal-title":"Appl. Numer. Math."},{"key":"9490_CR39","doi-asserted-by":"crossref","first-page":"414","DOI":"10.1007\/s10915-012-9577-8","volume":"53","author":"Z Xie","year":"2012","unstructured":"Xie, Z., Li, X., Tang, T.: Convergence analysis of spectral galerkin methods for volterra type integral equations. J. Sci. Comput. 53, 414\u2013434 (2012)","journal-title":"J. Sci. Comput."},{"key":"9490_CR40","doi-asserted-by":"crossref","first-page":"2398","DOI":"10.1016\/j.cam.2011.11.026","volume":"236","author":"L Yang","year":"2012","unstructured":"Yang, L., Shen, J., Wang, Y.: The reproducing kernel method for solving the system of the linear Volterra integral equations with variable coefficients. J. Comput. Appl. Math. 236, 2398\u20132405 (2012)","journal-title":"J. Comput. Appl. Math."}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-016-9490-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10444-016-9490-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-016-9490-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,8,20]],"date-time":"2023-08-20T14:41:58Z","timestamp":1692542518000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10444-016-9490-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,10,20]]},"references-count":40,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2017,4]]}},"alternative-id":["9490"],"URL":"https:\/\/doi.org\/10.1007\/s10444-016-9490-z","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"value":"1019-7168","type":"print"},{"value":"1572-9044","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,10,20]]}}}