{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,27]],"date-time":"2025-09-27T00:05:57Z","timestamp":1758931557782,"version":"3.44.0"},"reference-count":34,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T00:00:00Z","timestamp":1758758400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Henan Academy of Sciences","award":["241819246"],"award-info":[{"award-number":["241819246"]}]}],"content-domain":{"domain":["www.mdpi.com"],"crossmark-restriction":true},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper addresses first- and second-kind Volterra integral equations (VIEs) with discontinuous kernels. A hybrid method combining the Homotopy Analysis Method (HAM) and Physics-Informed Neural Networks (PINNs) is developed. The convergence of the HAM is analyzed. Benchmark examples confirm that the proposed HAM-PINNs approach achieves high accuracy and robustness, demonstrating its effectiveness for complex kernel structures.<\/jats:p>","DOI":"10.3390\/axioms14100726","type":"journal-article","created":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T12:48:28Z","timestamp":1758804508000},"page":"726","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Homotopy Analysis Method and Physics-Informed Neural Networks for Solving Volterra Integral Equations with Discontinuous Kernels"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2307-0891","authenticated-orcid":false,"given":"Samad","family":"Noeiaghdam","sequence":"first","affiliation":[{"name":"Institute of Mathematics, Henan Academy of Sciences, Zhengzhou 450046, China"},{"name":"Department of Mathematical Sciences, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai 602105, India"}]},{"given":"Md Asadujjaman","family":"Miah","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, University of Dhaka, Dhaka 1000, Bangladesh"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6233-8299","authenticated-orcid":false,"given":"Sanda","family":"Micula","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Babes-Bolyai University, 400084 Cluj-Napoca, Romania"},{"name":"Academy of Romanian Scientists, 50044 Bucharest, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/j.matcom.2025.05.011","article-title":"Superconvergent spectral methods for system of modified Volterra Integral Equations","volume":"239","author":"Kumar","year":"2026","journal-title":"Math. 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