{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T07:27:52Z","timestamp":1772263672732,"version":"3.50.1"},"reference-count":30,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2021,4,20]],"date-time":"2021-04-20T00:00:00Z","timestamp":1618876800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2021,4,20]],"date-time":"2021-04-20T00:00:00Z","timestamp":1618876800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11671342"],"award-info":[{"award-number":["11671342"]}],"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":["11771369"],"award-info":[{"award-number":["11771369"]}],"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":["11931003"],"award-info":[{"award-number":["11931003"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2021,6]]},"DOI":"10.1007\/s40314-021-01469-8","type":"journal-article","created":{"date-parts":[[2021,4,20]],"date-time":"2021-04-20T09:03:36Z","timestamp":1618909416000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["High accurate convergent spectral Galerkin methods for nonlinear weakly singular Volterra integro-differential equations"],"prefix":"10.1007","volume":"40","author":[{"given":"Yin","family":"Yang","sequence":"first","affiliation":[]},{"given":"Guoting","family":"Deng","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7211-5307","authenticated-orcid":false,"given":"Emran","family":"Tohidi","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,4,20]]},"reference":[{"issue":"2","key":"1469_CR1","doi-asserted-by":"publisher","first-page":"401","DOI":"10.1016\/j.cam.2003.08.047","volume":"163","author":"P Baratella","year":"2004","unstructured":"Baratella P, Orsi AP (2004) A new approach to the numerical solution of weakly singular Volterra integral equations. J Comput Appl Math 163(2):401\u2013418","journal-title":"J Comput Appl Math"},{"key":"1469_CR2","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511543234","volume-title":"Collocation methods for Volterra integral and related functional differential equations","author":"H Brunner","year":"2004","unstructured":"Brunner H (2004) Collocation methods for Volterra integral and related functional differential equations, vol 15. Cambridge University Press, Cambridge"},{"key":"1469_CR3","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-30726-6","volume-title":"Spectral methods fundamentals in single domains","author":"CG Canuto","year":"2006","unstructured":"Canuto CG, Hussaini MY, Quarteroni A, Zang T (2006) Spectral methods fundamentals in single domains. Springer, Berlin"},{"issue":"4","key":"1469_CR4","doi-asserted-by":"publisher","first-page":"938","DOI":"10.1016\/j.cam.2009.08.057","volume":"233","author":"Y Chen","year":"2009","unstructured":"Chen Y, Tang T (2009) Spectral methods for weakly singular Volterra integral equations with smooth solutions. J Comput Appl Math 233(4):938\u2013950","journal-title":"J Comput Appl Math"},{"issue":"269","key":"1469_CR5","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1090\/S0025-5718-09-02269-8","volume":"79","author":"Y Chen","year":"2010","unstructured":"Chen Y, Tang T (2010) Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equation with a weakly singular kernel. Math Comput 79(269):147\u2013167","journal-title":"Math Comput"},{"issue":"3","key":"1469_CR6","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1007\/BF01400302","volume":"23","author":"J Douglas","year":"1974","unstructured":"Douglas J, Dupont T, Wahlbin L (1974) The stability in $$L^{q}$$ of the $$L^{2}$$-projection into finite element function spaces. Numerische Mathematik 23(3):193\u2013197","journal-title":"Numerische Mathematik"},{"key":"1469_CR7","doi-asserted-by":"publisher","first-page":"109696","DOI":"10.1016\/j.chaos.2020.109696","volume":"134","author":"W Gao","year":"2020","unstructured":"Gao W, Veeresha P, Prakasha DG, Baskonus HM, Yel G (2020) New approach for the model describing the deathly disease in pregnant women using Mittag-Leffler function. Chaos Solitons Fractals 134:109696","journal-title":"Chaos Solitons Fractals"},{"key":"1469_CR8","doi-asserted-by":"publisher","first-page":"109929","DOI":"10.1016\/j.chaos.2020.109929","volume":"138","author":"W Gao","year":"2020","unstructured":"Gao W, Veeresha P, Baskonus HM, Prakasha DG, Kumar P (2020) A new study of unreported cases of 2019-nCOV epidemic outbreaks. Chaos Solitons Fractals 138:109929","journal-title":"Chaos Solitons Fractals"},{"key":"1469_CR9","volume-title":"Weighted inequalities of Hardy type","author":"A Kunfer","year":"2003","unstructured":"Kunfer A, Persson LE (2003) Weighted inequalities of Hardy type. World Scientific, New York"},{"key":"1469_CR10","doi-asserted-by":"crossref","unstructured":"L\u00fc T, Huang Y (2006) Extrapolation method for solving weakly singular nonlinear Volterra integral equations of the second kind. J Math Anal Appl 324:225\u2013237","DOI":"10.1016\/j.jmaa.2005.12.013"},{"key":"1469_CR11","doi-asserted-by":"crossref","unstructured":"Mastroianni G, Occorsio D (2001) Optimal systems of nodes of for Lagrange interpolation on bounded interals. a survey. J Comput Appl Math 134(1\u20132):325\u2013341","DOI":"10.1016\/S0377-0427(00)00557-4"},{"key":"1469_CR12","doi-asserted-by":"crossref","unstructured":"Ragozin D (1970) Polynomial approximation on compact manifolds and homogeneous space. Trans Am Math Soc 150:41\u201353","DOI":"10.1090\/S0002-9947-1970-0410210-0"},{"key":"1469_CR13","doi-asserted-by":"crossref","unstructured":"Ragozin D (1971) Constructive polynomial approximation on spheres and projective spaces. Trans Am Math Soc 162:157\u2013170","DOI":"10.2307\/1995746"},{"key":"1469_CR14","doi-asserted-by":"crossref","unstructured":"Sahu PK, Ray SS (2016) A New Bernoulli Wavelet Method for Numerical Solutions of Nonlinear Weakly Singular Volterra Integro-Differential Equations. Int J Comput Methods 1750022","DOI":"10.1142\/S0219876217500220"},{"key":"1469_CR15","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-71041-7","volume-title":"Spectral methods. Algorithms, analysis and applications","author":"J Shen","year":"2011","unstructured":"Shen J, Tang T, Wang LL (2011) Spectral methods. Algorithms, analysis and applications. Springer, New York"},{"key":"1469_CR16","doi-asserted-by":"publisher","first-page":"576","DOI":"10.1002\/num.22314","volume":"35","author":"X Shi","year":"2019","unstructured":"Shi X, Wei Y, Huang F (2019) Spectral collocation methods for nonlinear weakly singular Volterra integro-differential equations. Numer Methods Partial Differ Equ 35:576\u2013596","journal-title":"Numer Methods Partial Differ Equ"},{"key":"1469_CR17","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1016\/j.amc.2016.11.022","volume":"299","author":"S Sohrabi","year":"2017","unstructured":"Sohrabi S, Ranjbar H, Saei M (2017) Convergence analysis of the Jacobi-collocation method for nonlinear weakly singular Volterra integral equations. App Math Comput 299:141\u2013152","journal-title":"App Math Comput"},{"issue":"6","key":"1469_CR18","first-page":"825","volume":"26","author":"T Tang","year":"2008","unstructured":"Tang T, Xiang XU, Cheng J (2008) On spectral methods for Volterra integral equations and the convergence analysis. J Comput Math 26(6):825\u2013837","journal-title":"J Comput Math"},{"issue":"002","key":"1469_CR19","doi-asserted-by":"publisher","first-page":"216","DOI":"10.4208\/nmtma.2011.42s.6","volume":"4","author":"X Tao","year":"2011","unstructured":"Tao X, Xie Z, Zhou X (2011) Spectral Petrov-Galerkin Methods for the Second Kind Volterra Type Integro-Differential Equations. Numer Math Theor Meth Appl 4(002):216\u2013236","journal-title":"Numer Math Theor Meth Appl"},{"issue":"2","key":"1469_CR20","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10915-017-0373-3","volume":"72","author":"CL Wang","year":"2017","unstructured":"Wang CL, Wang ZQ, Jia HL (2017) An hp-version Spectral Collocation Method for Nonlinear Volterra Integro-differential Equation with Weakly Singular Kernels. J Sci Comput 72(2):1\u201332","journal-title":"J Sci Comput"},{"issue":"01","key":"1469_CR21","doi-asserted-by":"publisher","first-page":"1","DOI":"10.4208\/aamm.10-m1055","volume":"4","author":"Y Wei","year":"2012","unstructured":"Wei Y, Chen Y (2012) Convergence analysis of the spectral methods for weakly singular Volterra integro-differential equations with smooth solutions. Adv Appl Math Mech 4(01):1\u201320","journal-title":"Adv Appl Math Mech"},{"issue":"4","key":"1469_CR22","first-page":"774","volume":"16","author":"D Willett","year":"1965","unstructured":"Willett D (1965) A linear generalization of Gronwall\u2019s inequality. Proc Am Math Soc 16(4):774\u2013778","journal-title":"Proc Am Math Soc"},{"key":"1469_CR23","doi-asserted-by":"publisher","first-page":"519","DOI":"10.1007\/s10092-014-0128-6","volume":"52","author":"Y Yang","year":"2015","unstructured":"Yang Y (2015) Jacobi spectral Galerkin methods for fractional integro-differential equations. Calcolo 52:519\u2013542","journal-title":"Calcolo"},{"key":"1469_CR24","doi-asserted-by":"publisher","first-page":"166","DOI":"10.1016\/j.apnum.2020.10.001","volume":"160","author":"Y Yang","year":"2021","unstructured":"Yang Y, Tang Z (2021) Mapped Spectral collocation methods for Volterra integral equations with noncompact kernels. Appl Numer Math 160:166\u2013177","journal-title":"Appl Numer Math"},{"issue":"6","key":"1469_CR25","doi-asserted-by":"publisher","first-page":"1440","DOI":"10.4208\/aamm.OA-2018-0038","volume":"10","author":"X Yang","year":"2018","unstructured":"Yang X, Yang Y, Chen Y, Liu J (2018) Jacobi spectral collocation method based on Lagrange interpolation polynomials for solving nonlinear fractional integro-differential equations. Adv Appl Math Mech 10(6):1440\u20131458","journal-title":"Adv Appl Math Mech"},{"key":"1469_CR26","doi-asserted-by":"crossref","unstructured":"Yang Y, Tang Z, Huang Y (2019) Numerical solutions for Fredholm integral equations of the second kind with weakly singular kernel using spectral collocation method. Appl Math Comput 349:314\u2013324","DOI":"10.1016\/j.amc.2018.12.035"},{"key":"1469_CR27","doi-asserted-by":"crossref","unstructured":"Yang Y, Tohidi E, Ma X, Kang S (2019) Rigorous convergence analysis of Jacobi spectral Galerkin methods for Volterra integral equations with noncompact kernels. J Comput Appl Math 366:112403","DOI":"10.1016\/j.cam.2019.112403"},{"issue":"4","key":"1469_CR28","doi-asserted-by":"publisher","first-page":"4027","DOI":"10.3934\/math.2020259","volume":"5","author":"G Yel","year":"2020","unstructured":"Yel G, Baskonus HM, Gao W (2020) New dark-bright soliton in the shallow water wave model. AIMS Math 5(4):4027\u20134044","journal-title":"AIMS Math"},{"issue":"8","key":"1469_CR29","first-page":"3807","volume":"16","author":"X Zhang","year":"2014","unstructured":"Zhang X, Li J (2014) Jacobi pseudo-spectral Galerkin method for second kind Volterra integro-differential equations with a weakly singular kernel. J Vibroeng 16(8):3807\u20133826","journal-title":"J Vibroeng"},{"issue":"002","key":"1469_CR30","doi-asserted-by":"publisher","first-page":"281","DOI":"10.1007\/s11464-013-0282-1","volume":"8","author":"R Zhang","year":"2013","unstructured":"Zhang R, Zhu B, Xie H (2013) Spectral methods for weakly singular Volterra integral equations with pantograph delays. Front Math China 8(002):281\u2013299","journal-title":"Front Math China"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-021-01469-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-021-01469-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-021-01469-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,8]],"date-time":"2025-04-08T22:50:47Z","timestamp":1744152647000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-021-01469-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,20]]},"references-count":30,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2021,6]]}},"alternative-id":["1469"],"URL":"https:\/\/doi.org\/10.1007\/s40314-021-01469-8","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,4,20]]},"assertion":[{"value":"16 October 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 January 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 February 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 April 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors state that they do not have any conflict of interest in their submitted manuscript.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"118"}}