{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,7]],"date-time":"2026-01-07T07:53:31Z","timestamp":1767772411263,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,9,18]],"date-time":"2023-09-18T00:00:00Z","timestamp":1694995200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Zhejiang Normal University Postdoctoral Research Fund","award":["ZC304022938","61976196","LZ22F030003"],"award-info":[{"award-number":["ZC304022938","61976196","LZ22F030003"]}]},{"name":"Natural Science Foundation of China","award":["ZC304022938","61976196","LZ22F030003"],"award-info":[{"award-number":["ZC304022938","61976196","LZ22F030003"]}]},{"name":"Zhejiang Provincial Natural Science Foundation of China","award":["ZC304022938","61976196","LZ22F030003"],"award-info":[{"award-number":["ZC304022938","61976196","LZ22F030003"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This work uses the collocation approximation method to solve a specific type of backward stochastic Volterra integral equations (BSVIEs). Using Newton\u2019s method, BSVIEs can be solved using block pulse functions and the corresponding stochastic operational matrix of integration. We present examples to illustrate the estimate analysis and to demonstrate the convergence of the two approximating sequences separately. To measure their accuracy, we compare the solutions with values of exact and approximative solutions at a few selected locations using a specified absolute error. We also propose an efficient method for solving a triangular linear algebraic problem using a single integral equation. To confirm the effectiveness of our method, we conduct numerical experiments with issues from real-world applications.<\/jats:p>","DOI":"10.3390\/axioms12090888","type":"journal-article","created":{"date-parts":[[2023,9,18]],"date-time":"2023-09-18T02:17:30Z","timestamp":1695003450000},"page":"888","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Numerical Solution of Nonlinear Backward Stochastic Volterra Integral Equations"],"prefix":"10.3390","volume":"12","author":[{"given":"Mahvish","family":"Samar","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, China"},{"name":"School of Computer Science and Technology, Zhejiang Normal University, Jinhua 321004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9728-2504","authenticated-orcid":false,"given":"Kutorzi","family":"Yao","sequence":"additional","affiliation":[{"name":"School of Mathematics and Institute for Financial Studies, Shandong University, Jinan 250100, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xinzhong","family":"Zhu","sequence":"additional","affiliation":[{"name":"School of Computer Science and Technology, Zhejiang Normal University, Jinhua 321004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/0167-6911(90)90082-6","article-title":"Adapted solution of a backward stochastic differential equation","volume":"14","author":"Pardoux","year":"1990","journal-title":"Syst. 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