{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T15:13:54Z","timestamp":1787238834371,"version":"build-2736575974"},"reference-count":28,"publisher":"Springer Science and Business Media LLC","issue":"7","license":[{"start":{"date-parts":[[2025,6,28]],"date-time":"2025-06-28T00:00:00Z","timestamp":1751068800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,6,28]],"date-time":"2025-06-28T00:00:00Z","timestamp":1751068800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2025,10]]},"DOI":"10.1007\/s40314-025-03299-4","type":"journal-article","created":{"date-parts":[[2025,6,28]],"date-time":"2025-06-28T02:01:12Z","timestamp":1751076072000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A four-block scheme for nonlinear third-kind Volterra integral equations"],"prefix":"10.1007","volume":"44","author":[{"given":"F.","family":"Afiatdoust","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"M. H.","family":"Heydari","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"M.","family":"Bayram","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"D.","family":"Baleanu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,6,28]]},"reference":[{"key":"3299_CR1","unstructured":"Abramowitz M, Stegun IA (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables. National Bureau of Standards Applied Mathematics Series 55, United States"},{"key":"3299_CR2","doi-asserted-by":"crossref","unstructured":"Afiatdoust F, Heydari MH, Hosseini MM (2023a) A block-by-block approach for nonlinear fractional integro-differential equations. Int J Comput Math 100(11):2140\u20132155","DOI":"10.1080\/00207160.2023.2265500"},{"key":"3299_CR3","doi-asserted-by":"crossref","unstructured":"Afiatdoust F, Heydari MH, Hosseini MM (2023b) A block-by-block strategy for fractional systems of nonlinear weakly singular integro-differential equations. Comput Appl Math 42(6):252","DOI":"10.1007\/s40314-023-02380-0"},{"issue":"1","key":"3299_CR4","first-page":"214","volume":"14","author":"F Afiatdoust","year":"2024","unstructured":"Afiatdoust F, Heydari MH, Hosseini MM (2024) Block-by-block technique for a class of nonlinear systems of fractional integro-differential equations. J Appl Anal Comput 14(1):214\u2013234","journal-title":"J Appl Anal Comput"},{"issue":"3","key":"3299_CR5","doi-asserted-by":"publisher","first-page":"325","DOI":"10.1216\/JIE-2015-27-3-325","volume":"27","author":"SS Allaei","year":"2015","unstructured":"Allaei SS, Yang Z, Brunner H (2015) Existence, uniqueness and regularity of solutions to a class of third-kind Volterra integral equations. J Integr Equ Appl 27(3):325\u2013342","journal-title":"J Integr Equ Appl"},{"issue":"3","key":"3299_CR6","first-page":"1104","volume":"37","author":"SS Allaei","year":"2017","unstructured":"Allaei SS, Yang ZW, Brunner H (2017) Collocation methods for third-kind VIEs. IMA J Numer Anal 37(3):1104\u20131124","journal-title":"IMA J Numer Anal"},{"key":"3299_CR7","doi-asserted-by":"crossref","unstructured":"Aourir E, Izem N, Dastjerdi HL (2024a) A computational approach for solving third kind VIEs by collocation method based on radial basis functions. J Comput Appl Math 440:115636","DOI":"10.1016\/j.cam.2023.115636"},{"key":"3299_CR8","doi-asserted-by":"crossref","unstructured":"Aourir E, Izem N, Dastjerdi HL (2024b) Numerical solutions of a class of linear and nonlinear Volterra integral equations of the third kind using collocation method based on radial basis functions. Comput Appl Math 43(3):117","DOI":"10.1007\/s40314-024-02630-9"},{"key":"3299_CR9","doi-asserted-by":"publisher","DOI":"10.1017\/9781316162491","volume-title":"Volterra integral equations: an introduction to theory and applications","author":"H Brunner","year":"2017","unstructured":"Brunner H (2017) Volterra integral equations: an introduction to theory and applications, vol 30. Cambridge University Press, Cambridge"},{"key":"3299_CR10","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-30728-0","volume-title":"Spectral methods: fundamentals in single domains","author":"C Canuto","year":"2007","unstructured":"Canuto C, Hussaini MY, Quarteroni A, Zang TA (2007) Spectral methods: fundamentals in single domains. Springer, Berlin"},{"key":"3299_CR11","doi-asserted-by":"crossref","first-page":"126685","DOI":"10.1016\/j.cam.2022.114405","volume":"414","author":"H Chen","year":"2022","unstructured":"Chen H, Ma J (2022) Solving the third-kind Volterra integral equation via the boundary value technique: Lagrange polynomial versus fractional interpolation. Appl Math Comput 414:126685","journal-title":"Appl Math Comput"},{"key":"3299_CR12","doi-asserted-by":"publisher","first-page":"909","DOI":"10.1007\/s12190-021-01547-4","volume":"68","author":"N Karamollahi","year":"2022","unstructured":"Karamollahi N, Heydari M, Loghmani GB (2022) An interpolation-based method for solving Volterra integral equations. J Appl Math Comput 68:909\u2013940","journal-title":"J Appl Math Comput"},{"issue":"2","key":"3299_CR13","doi-asserted-by":"publisher","first-page":"400","DOI":"10.1016\/j.apm.2009.04.013","volume":"34","author":"R Katani","year":"2010","unstructured":"Katani R, Shahmorad S (2010) Block by block method for the systems of nonlinear Volterra integral equations. Appl Math Model 34(2):400\u2013406","journal-title":"Appl Math Model"},{"key":"3299_CR14","doi-asserted-by":"crossref","unstructured":"Katani R, Shahmorad S (2012a) A block by block method with Romberg quadrature for the system of Urysohn type Volterra integral equations. Comput Appl Math 31:191\u2013203","DOI":"10.1590\/S1807-03022012000100010"},{"key":"3299_CR15","doi-asserted-by":"crossref","unstructured":"Katani R, Shahmorad S (2012b) The block-by-block method with Romberg quadrature for the solution of nonlinear Volterra integral equations on large intervals. Ukr Math J 64(7):1050\u20131063","DOI":"10.1007\/s11253-012-0698-x"},{"issue":"7","key":"3299_CR16","doi-asserted-by":"publisher","first-page":"1050","DOI":"10.1007\/s11253-012-0698-x","volume":"64","author":"R Katani","year":"2012","unstructured":"Katani R, Shahmorad S (2012) The block-by-block method with Romberg quadrature for the solution of nonlinear Volterra integral equations on large intervals. Ukr Math J 64(7):1050\u20131063","journal-title":"Ukr Math J"},{"issue":"6","key":"3299_CR17","doi-asserted-by":"publisher","first-page":"737","DOI":"10.3846\/13926292.2015.1111266","volume":"20","author":"R Katani","year":"2015","unstructured":"Katani R, Shahmorad S (2015) A block by block method for solving system of Volterra integral equations with continuous and Abel kernels. Math Model Anal 20(6):737\u2013753","journal-title":"Math Model Anal"},{"key":"3299_CR18","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"},{"key":"3299_CR19","doi-asserted-by":"publisher","first-page":"113458","DOI":"10.1016\/j.cam.2021.113458","volume":"392","author":"X Ma","year":"2021","unstructured":"Ma X, Huang C (2021) Recovery of high order accuracy in spectral collocation method for linear Volterra integral equations of the third-kind with non-smooth solutions. J Comput Appl Math 392:113458","journal-title":"J Comput Appl Math"},{"key":"3299_CR20","doi-asserted-by":"crossref","unstructured":"Nemati S, Lima PM (2018) Numerical solution of a third-kind Volterra integral equation using an operational matrix technique. 2018 European control conference (ECC), pp 3215\u20133220","DOI":"10.23919\/ECC.2018.8550223"},{"key":"3299_CR21","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s40314-019-0947-9","volume":"38","author":"F Shayanfard","year":"2019","unstructured":"Shayanfard F, Dastjerdi HL, Ghaini F (2019) A numerical method for solving Volterra integral equations of the third kind by multistep collocation method. Comput Appl Math 38:1\u201313","journal-title":"Comput Appl Math"},{"key":"3299_CR22","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10092-019-0304-9","volume":"56","author":"H Song","year":"2019","unstructured":"Song H, Yang Z, Brunner H (2019) Analysis of collocation methods for nonlinear Volterra integral equations of the third kind. Calcolo 56:1\u201329","journal-title":"Calcolo"},{"key":"3299_CR23","first-page":"125509","volume":"388","author":"H Song","year":"2021","unstructured":"Song H, Xiao Y, Chen M (2021) Collocation methods for third-kind Volterra integral equations with proportional delays. Appl Math Comput 388:125509","journal-title":"Appl Math Comput"},{"issue":"4","key":"3299_CR24","doi-asserted-by":"publisher","first-page":"191","DOI":"10.1007\/s40314-022-01775-9","volume":"41","author":"H Song","year":"2022","unstructured":"Song H, Yang Z, Xiao Y (2022) Iterated collocation methods for nonlinear third-kind Volterra integral equations with proportional delays. Comput Appl Math 41(4):191","journal-title":"Comput Appl Math"},{"issue":"1","key":"3299_CR25","first-page":"012003","volume":"2279","author":"A Toigonbaeva","year":"2022","unstructured":"Toigonbaeva A, Obodoeva G, Kambarova A (2022) Regularization and uniqueness of solutions of Volterra linear integral equations of the third kind. J Phys: Conf Ser 2279(1):012003","journal-title":"J Phys: Conf Ser"},{"issue":"5","key":"3299_CR26","doi-asserted-by":"publisher","first-page":"161","DOI":"10.1007\/s40314-021-01555-x","volume":"40","author":"F Usta","year":"2021","unstructured":"Usta F (2021) Bernstein approximation technique for numerical solution of Volterra integral equations of the third kind. Comput Appl Math 40(5):161","journal-title":"Comput Appl Math"},{"issue":"9\u201310","key":"3299_CR27","doi-asserted-by":"publisher","first-page":"1145","DOI":"10.1080\/01630560903393188","volume":"30","author":"G Vainikko","year":"2009","unstructured":"Vainikko G (2009) Cordial Volterra integral equations 1. Numer Funct Anal Optim 30(9\u201310):1145\u20131172","journal-title":"Numer Funct Anal Optim"},{"issue":"1","key":"3299_CR28","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1080\/01630563.2010.526412","volume":"32","author":"G Vainikko","year":"2010","unstructured":"Vainikko G (2010) Spline collocation-interpolation method for linear and nonlinear cordial Volterra integral equations. Numer Funct Anal Optim 32(1):83\u2013109","journal-title":"Numer Funct Anal Optim"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03299-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-025-03299-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03299-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,9]],"date-time":"2025-09-09T05:04:23Z","timestamp":1757394263000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-025-03299-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,28]]},"references-count":28,"journal-issue":{"issue":"7","published-print":{"date-parts":[[2025,10]]}},"alternative-id":["3299"],"URL":"https:\/\/doi.org\/10.1007\/s40314-025-03299-4","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,6,28]]},"assertion":[{"value":"6 March 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 May 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 June 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 June 2025","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 that there is not any conflict of interest regarding this paper.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflicts of Interest"}}],"article-number":"334"}}