{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T14:53:00Z","timestamp":1753887180943,"version":"3.41.2"},"reference-count":23,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2019,6,16]],"date-time":"2019-06-16T00:00:00Z","timestamp":1560643200000},"content-version":"vor","delay-in-days":166,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11371079"],"award-info":[{"award-number":["11371079"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Complexity"],"published-print":{"date-parts":[[2019,1]]},"abstract":"<jats:p>In this work, a mechanical quadrature method based on modified trapezoid formula is used for solving weakly singular Volterra integral equation with proportional delays. An improved Gronwall inequality is testified and adopted to prove the existence and uniqueness of the solution of the original equation. Then, we study the convergence and the error estimation of the mechanical quadrature method. Moreover, Richardson extrapolation based on the asymptotic expansion of error not only possesses a high accuracy but also has the posterior error estimate which can be used to design self\u2010adaptive algorithm. Numerical experiments demonstrate the efficiency and applicability of the proposed method.<\/jats:p>","DOI":"10.1155\/2019\/4813802","type":"journal-article","created":{"date-parts":[[2019,6,16]],"date-time":"2019-06-16T23:31:23Z","timestamp":1560727883000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Mechanical Quadrature Method for Solving Delay Volterra Integral Equation with Weakly Singular Kernels"],"prefix":"10.1155","volume":"2019","author":[{"given":"Li","family":"Zhang","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2191-4739","authenticated-orcid":false,"given":"Jin","family":"Huang","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7331-1630","authenticated-orcid":false,"given":"Yubin","family":"Pan","sequence":"additional","affiliation":[]},{"given":"Xiaoxia","family":"Wen","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2019,6,16]]},"reference":[{"key":"e_1_2_12_1_2","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(76)90004-4"},{"key":"e_1_2_12_2_2","doi-asserted-by":"publisher","DOI":"10.1016\/0025-5564(76)90042-0"},{"key":"e_1_2_12_3_2","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/16\/2\/308"},{"volume-title":"Mathematics of Computation","year":"2009","author":"Brunner H.","key":"e_1_2_12_4_2"},{"key":"e_1_2_12_5_2","doi-asserted-by":"publisher","DOI":"10.1007\/s40065-017-0162-8"},{"key":"e_1_2_12_6_2","doi-asserted-by":"publisher","DOI":"10.1140\/epjp\/i2017-11344-9"},{"key":"e_1_2_12_7_2","doi-asserted-by":"publisher","DOI":"10.1007\/s40314-017-0550-x"},{"key":"e_1_2_12_8_2","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/330\/05886"},{"key":"e_1_2_12_9_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.apnum.2015.12.005"},{"key":"e_1_2_12_10_2","doi-asserted-by":"publisher","DOI":"10.1016\/s0377-0427(01)00596-9"},{"key":"e_1_2_12_11_2","first-page":"1","article-title":"A Legendre\u2013Petrov\u2013Galerkin method for solving Volterra integro-differential equations with proportional delays","author":"Cai H.","year":"2018","journal-title":"International Journal of Computer Mathematics"},{"key":"e_1_2_12_12_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2015.01.100"},{"key":"e_1_2_12_13_2","doi-asserted-by":"publisher","DOI":"10.1007\/s11464-009-0010-z"},{"key":"e_1_2_12_14_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.fss.2016.04.004"},{"key":"e_1_2_12_15_2","doi-asserted-by":"publisher","DOI":"10.4208\/aamm.10-m11147"},{"key":"e_1_2_12_16_2","doi-asserted-by":"publisher","DOI":"10.1137\/100818595"},{"key":"e_1_2_12_17_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2017.05.026"},{"key":"e_1_2_12_18_2","doi-asserted-by":"publisher","DOI":"10.1090\/mcom\/3023"},{"key":"e_1_2_12_19_2","doi-asserted-by":"publisher","DOI":"10.1515\/jnma-2014-0014"},{"key":"e_1_2_12_20_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01061258"},{"key":"e_1_2_12_21_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2008.03.024"},{"volume-title":"High Precision Algorithm for Multidimensional Singular Integral","year":"2017","author":"Huang J.","key":"e_1_2_12_22_2"},{"key":"e_1_2_12_23_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-247X(02)00369-4"}],"container-title":["Complexity"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/complexity\/2019\/4813802.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/complexity\/2019\/4813802.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/2019\/4813802","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,7]],"date-time":"2024-08-07T11:39:59Z","timestamp":1723030799000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/2019\/4813802"}},"subtitle":[],"editor":[{"given":"Danilo","family":"Comminiello","sequence":"additional","affiliation":[]}],"short-title":[],"issued":{"date-parts":[[2019,1]]},"references-count":23,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["10.1155\/2019\/4813802"],"URL":"https:\/\/doi.org\/10.1155\/2019\/4813802","archive":["Portico"],"relation":{},"ISSN":["1076-2787","1099-0526"],"issn-type":[{"type":"print","value":"1076-2787"},{"type":"electronic","value":"1099-0526"}],"subject":[],"published":{"date-parts":[[2019,1]]},"assertion":[{"value":"2019-02-15","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2019-05-21","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2019-06-16","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"4813802"}}