{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,5]],"date-time":"2026-05-05T01:37:45Z","timestamp":1777945065529,"version":"3.51.4"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"9","license":[{"start":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T00:00:00Z","timestamp":1777680000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T00:00:00Z","timestamp":1777680000000},"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":["12301468"],"award-info":[{"award-number":["12301468"]}],"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":["12571391, 12171322, and 12271366"],"award-info":[{"award-number":["12571391, 12171322, and 12271366"]}],"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":[[2026,10]]},"DOI":"10.1007\/s40314-026-03769-3","type":"journal-article","created":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T08:58:06Z","timestamp":1777712286000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["hp-version $$C^1$$-CPG method with exponential convergence for weakly singular Volterra integro-differential equations"],"prefix":"10.1007","volume":"45","author":[{"given":"Fujun","family":"Liu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lina","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2922-5508","authenticated-orcid":false,"given":"Lijun","family":"Yi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2026,5,2]]},"reference":[{"key":"3769_CR1","volume-title":"Sobolev spaces, pure and applied mathematics","author":"RA Adams","year":"1975","unstructured":"Adams RA (1975) Sobolev spaces, pure and applied mathematics, vol 65. Academic Press, New York"},{"key":"3769_CR2","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1016\/S0045-7825(97)00100-X","volume":"150","author":"I Babu\u0161ka","year":"1997","unstructured":"Babu\u0161ka I, Strouboulis T, Copps K (1997) $$hp$$ optimization of finite element approximations: analysis of the optimal mesh sequences in one dimension. Comput Methods Appl Mech Engrg 150:89\u2013108","journal-title":"Comput Methods Appl Mech Engrg"},{"key":"3769_CR3","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 (2004) Collocation methods for Volterra integral and related functional equations. Cambridge University Press, Cambridge"},{"key":"3769_CR4","doi-asserted-by":"publisher","first-page":"224","DOI":"10.1137\/040619314","volume":"44","author":"H Brunner","year":"2006","unstructured":"Brunner H, Sch\u00f6tzau D (2006) $$hp$$-discontinuous Galerkin time-stepping for Volterra integrodifferential equations. SIAM J Numer Anal 44:224\u2013245","journal-title":"SIAM J Numer Anal"},{"key":"3769_CR5","volume-title":"Volterra integral and differential equations","author":"TA Burton","year":"2005","unstructured":"Burton TA (2005) Volterra integral and differential equations, 2nd edn. Elsevier B. V, Amsterdam","edition":"2"},{"key":"3769_CR6","doi-asserted-by":"publisher","first-page":"455","DOI":"10.1090\/S0025-5718-1981-0606506-0","volume":"36","author":"M Delfour","year":"1981","unstructured":"Delfour M, Hager W, Trochu F (1981) Discontinuous Galerkin methods for ordinary differential equations. Math Comp 36:455\u2013473","journal-title":"Math Comp"},{"key":"3769_CR7","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1016\/S0168-9274(97)00019-6","volume":"24","author":"W Enright","year":"1997","unstructured":"Enright W, Hu M (1997) Continuous Runge-Kutta methods for neutral Volterra integro-differential equations with delay. Appl Numer Math 24:175\u2013190","journal-title":"Appl Numer Math"},{"key":"3769_CR8","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1007\/s11464-009-0008-6","volume":"4","author":"CM Huang","year":"2009","unstructured":"Huang CM, Vandewalle S (2009) Stability of Runge-Kutta-Pouzet methods for Volterra integro-differential equations with delays. Front Math China 4:63\u201387","journal-title":"Front Math China"},{"key":"3769_CR9","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1007\/s40314-021-01435-4","volume":"40","author":"QM Huang","year":"2021","unstructured":"Huang QM, Wang M (2021) Superconvergence of interpolated collocation solutions for weakly singular Volterra integral equations of the second kind. Comput Appl Math 40:71","journal-title":"Comput Appl Math"},{"key":"3769_CR10","doi-asserted-by":"publisher","first-page":"415","DOI":"10.1090\/S0025-5718-1972-0321301-2","volume":"26","author":"BL Hulme","year":"1972","unstructured":"Hulme BL (1972) One-step piecewise polynomial Galerkin methods for initial value problems. Math Comp 26:415\u2013426","journal-title":"Math Comp"},{"key":"3769_CR11","doi-asserted-by":"publisher","first-page":"881","DOI":"10.1090\/S0025-5718-1972-0315899-8","volume":"26","author":"BL Hulme","year":"1972","unstructured":"Hulme BL (1972) Discrete Galerkin and related one-step methods for ordinary differential equations. Math Comp 26:881\u2013891","journal-title":"Math Comp"},{"key":"3769_CR12","doi-asserted-by":"publisher","first-page":"84","DOI":"10.1016\/j.apnum.2020.01.017","volume":"152","author":"SS Li","year":"2020","unstructured":"Li SS, Wang LN, Yi LJ (2020) An $$hp$$-version of $$C^0$$-continuous Petrov-Galerkin method for second-order Volterra integro-differential equations. Appl Numer Math 152:84\u2013104","journal-title":"Appl Numer Math"},{"key":"3769_CR13","first-page":"20","volume":"11","author":"SS Li","year":"2021","unstructured":"Li SS, Wang LN, Yi LJ (2021) An $$hp$$-version of $$C^0$$-continuous Petrov-Galerkin time-stepping method for second-order Volterra integro-differential equations with weakly singular kernels, East Asian. J Appl Math 11:20\u201342","journal-title":"J Appl Math"},{"key":"3769_CR14","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1007\/s11075-024-01874-0","volume":"99","author":"YP Li","year":"2025","unstructured":"Li YP, Liang H, Yuan HF (2025) On the convergence of Galerkin methods for auto-convolution Volterra integro-differential equations. Numer Algorithms 99:183\u2013205","journal-title":"Numer Algorithms"},{"key":"3769_CR15","doi-asserted-by":"publisher","first-page":"2725","DOI":"10.1093\/imanum\/drad072","volume":"44","author":"H Liang","year":"2024","unstructured":"Liang H, Stynes M (2024) A general collocation analysis for weakly singular Volterra integral equations with variable exponent. IMA J Numer Anal 44:2725\u20132751","journal-title":"IMA J Numer Anal"},{"key":"3769_CR16","doi-asserted-by":"publisher","first-page":"937","DOI":"10.1137\/S0036142999336145","volume":"38","author":"T Lin","year":"2000","unstructured":"Lin T, Lin YP, Rao M, Zhang SH (2000) Petrov-Galerkin methods for linear Volterra integro-differential equations. SIAM J Numer Anal 38:937\u2013963","journal-title":"SIAM J Numer Anal"},{"key":"3769_CR17","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970852","volume-title":"Analytical and numerical methods for Volterra equations","author":"P Linz","year":"1985","unstructured":"Linz P (1985) Analytical and numerical methods for Volterra equations. SIAM, Philadelphia, PA"},{"key":"3769_CR18","doi-asserted-by":"publisher","first-page":"1987","DOI":"10.1090\/S0025-5718-2013-02689-0","volume":"82","author":"K Mustapha","year":"2013","unstructured":"Mustapha K (2013) A superconvergent discontinuous Galerkin method for Volterra integro-differential equations, smooth and nonsmooth kernels. Math Comput 82:1987\u20132005","journal-title":"Math Comput"},{"key":"3769_CR19","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1016\/j.cam.2013.03.047","volume":"253","author":"K Mustapha","year":"2013","unstructured":"Mustapha K, Ryan JK (2013) Post-processing discontinuous Galerkin solutions to Volterra integro-differential equations: analysis and simulations. J Comput Appl Math 253:89\u2013103","journal-title":"J Comput Appl Math"},{"key":"3769_CR20","doi-asserted-by":"publisher","first-page":"207","DOI":"10.1007\/s100920070002","volume":"37","author":"D Sch\u00f6tzau","year":"2000","unstructured":"Sch\u00f6tzau D, Schwab C (2000) An $$hp$$ a priori error analysis of the DG time-stepping method for initial value problems. Calcolo 37:207\u2013232","journal-title":"Calcolo"},{"key":"3769_CR21","volume-title":"$$p$$- and $$hp$$- Finite element methods","author":"C Schwab","year":"1998","unstructured":"Schwab C (1998) $$p$$- and $$hp$$- Finite element methods. Oxford University Press, New York"},{"key":"3769_CR22","doi-asserted-by":"publisher","first-page":"725","DOI":"10.1007\/s10543-012-0373-5","volume":"52","author":"M Shakourifar","year":"2012","unstructured":"Shakourifar M, Enright W (2012) Superconvergent interpolants for collocation methods applied to Volterra integro-differential equations with delay. BIT 52:725\u2013740","journal-title":"BIT"},{"key":"3769_CR23","doi-asserted-by":"publisher","first-page":"480","DOI":"10.4208\/aamm.OA-2019-0056","volume":"12","author":"XL Shi","year":"2020","unstructured":"Shi XL, Chen YP, Huang YQ, Huang FL (2020) Spectral collocation methods for second-order Volterra integro-differential equations with weakly singular kernels. Adv Appl Math Mech 12:480\u2013502","journal-title":"Adv Appl Math Mech"},{"key":"3769_CR24","doi-asserted-by":"publisher","first-page":"517","DOI":"10.1134\/S001226612104008X","volume":"57","author":"VV Vlasov","year":"2021","unstructured":"Vlasov VV, Rautian NA (2021) On Volterra integro-differential equations with kernels representable by Stieltjes integrals. Differ Equations 57:517\u2013532","journal-title":"Differ Equations"},{"key":"3769_CR25","doi-asserted-by":"publisher","first-page":"2285","DOI":"10.1090\/mcom\/3183","volume":"86","author":"ZQ Wang","year":"2017","unstructured":"Wang ZQ, Guo YL, Yi LJ (2017) An $$hp$$-version Legendre-Jacobi spectral collocation method for Volterra integro-differential equations with smooth and weakly singular kernels. Math Comput 86:2285\u20132324","journal-title":"Math Comput"},{"key":"3769_CR26","doi-asserted-by":"publisher","first-page":"1455","DOI":"10.1093\/imanum\/drae036","volume":"45","author":"LN Wang","year":"2025","unstructured":"Wang LN, Zhang MZ, Tian HJ, Yi LJ (2025) $$hp$$-version $$C^1$$-continuous Petrov-Galerkin method for nonlinear second-order initial value problems with application to wave equations. IMA J Numer Anal 45:1455\u20131500","journal-title":"IMA J Numer Anal"},{"key":"3769_CR27","doi-asserted-by":"publisher","first-page":"523","DOI":"10.1007\/s10915-004-4796-2","volume":"25","author":"TP Wihler","year":"2005","unstructured":"Wihler TP (2005) An a priori error analysis of the $$hp$$-version of the continuous Galerkin FEM for nonlinear initial value problems. J Sci Comput 25:523\u2013549","journal-title":"J Sci Comput"},{"key":"3769_CR28","first-page":"2212","volume":"9","author":"GZ Xiu","year":"2019","unstructured":"Xiu GZ, Yuan J, Shi B, Wang LY (2019) Hereditary effects of exponentially damped oscillators with past histories. J Appl Anal Comput 9:2212\u20132223","journal-title":"J Appl Anal Comput"},{"key":"3769_CR29","doi-asserted-by":"publisher","first-page":"715","DOI":"10.1007\/s10915-015-9983-9","volume":"65","author":"LJ Yi","year":"2015","unstructured":"Yi LJ (2015) An $$h$$-$$p$$ version of the continuous Petrov-Galerkin finite element method for nonlinear Volterra integro-differential equations. J Sci Comput 65:715\u2013734","journal-title":"J Sci Comput"},{"key":"3769_CR30","doi-asserted-by":"publisher","first-page":"2677","DOI":"10.1137\/15M1006489","volume":"53","author":"LJ Yi","year":"2015","unstructured":"Yi LJ, Guo BQ (2015) An $$h$$-$$p$$ version of the continuous Petrov-Galerkin finite element method for Volterra integro-differential equations with smooth and nonsmooth kernels. SIAM J Numer Anal 53:2677\u20132704","journal-title":"SIAM J Numer Anal"},{"key":"3769_CR31","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2023.115140","volume":"427","author":"LJ Yi","year":"2023","unstructured":"Yi LJ, Zhang MZ, Mao XY (2023) Superconvergent postprocessing of the discontinuous Galerkin time stepping method for nonlinear Volterra integro-differential equations. J Comput Appl Math 427:115140","journal-title":"J Comput Appl Math"},{"key":"3769_CR32","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1093\/imanum\/24.2.193","volume":"24","author":"CJ Zhang","year":"2004","unstructured":"Zhang CJ, Vandewalle S (2004) Stability analysis of Runge-Kutta methods for nonlinear Volterra delay-integro-differential equations. IMA J Numer Anal 24:193\u2013214","journal-title":"IMA J Numer Anal"},{"key":"3769_CR33","doi-asserted-by":"publisher","first-page":"241","DOI":"10.1023\/A:1022333811602","volume":"45","author":"SH Zhang","year":"2000","unstructured":"Zhang SH, Lin T, Lin YP, Rao M (2000) Defect correction and a posteriori error estimation of Petrov-Galerkin methods for nonlinear Volterra integro-differential equations. Appl Math 45:241\u2013263","journal-title":"Appl Math"},{"key":"3769_CR34","doi-asserted-by":"publisher","first-page":"340","DOI":"10.1016\/j.apnum.2021.08.003","volume":"170","author":"MZ Zhang","year":"2021","unstructured":"Zhang MZ, Mao XY, Yi LJ (2021) Exponential convergence of the $$hp$$-version of the composite Gauss-Legendre quadrature for integrals with endpoint singularities. Appl Numer Math 170:340\u2013352","journal-title":"Appl Numer Math"},{"key":"3769_CR35","doi-asserted-by":"publisher","first-page":"671","DOI":"10.1051\/m2an\/2022100","volume":"57","author":"MZ Zhang","year":"2023","unstructured":"Zhang MZ, Mao XY, Yi LJ (2023) Superconvergence and postprocessing of the continuous Galerkin method for nonlinear Volterra integro-differential equations. ESAIM Math Model Numer Anal 57:671\u2013691","journal-title":"ESAIM Math Model Numer Anal"},{"key":"3769_CR36","doi-asserted-by":"publisher","first-page":"330","DOI":"10.1137\/19M1245621","volume":"58","author":"XC Zheng","year":"2020","unstructured":"Zheng XC, Wang H (2020) An optimal-order numerical approximation to variable-order space-fractional diffusion equations on uniform or graded meshes. SIAM J Numer Anal 58:330\u2013352","journal-title":"SIAM J Numer Anal"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-026-03769-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-026-03769-3","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-026-03769-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T08:58:07Z","timestamp":1777712287000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-026-03769-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,5,2]]},"references-count":36,"journal-issue":{"issue":"9","published-print":{"date-parts":[[2026,10]]}},"alternative-id":["3769"],"URL":"https:\/\/doi.org\/10.1007\/s40314-026-03769-3","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,5,2]]},"assertion":[{"value":"29 December 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 March 2026","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 April 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 May 2026","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 declare no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"376"}}