{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,18]],"date-time":"2026-03-18T00:52:39Z","timestamp":1773795159776,"version":"3.50.1"},"reference-count":32,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,9,16]],"date-time":"2022-09-16T00:00:00Z","timestamp":1663286400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China (NSFC)","doi-asserted-by":"publisher","award":["11961009"],"award-info":[{"award-number":["11961009"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China (NSFC)","doi-asserted-by":"publisher","award":["11901135"],"award-info":[{"award-number":["11901135"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China (NSFC)","doi-asserted-by":"publisher","award":["[2020]1Y015"],"award-info":[{"award-number":["[2020]1Y015"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China (NSFC)","doi-asserted-by":"publisher","award":["QJJ2022015"],"award-info":[{"award-number":["QJJ2022015"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China (NSFC)","doi-asserted-by":"publisher","award":["QJJ2022047"],"award-info":[{"award-number":["QJJ2022047"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Guizhou Provincial Science and Technology Projects","award":["11961009"],"award-info":[{"award-number":["11961009"]}]},{"name":"Guizhou Provincial Science and Technology Projects","award":["11901135"],"award-info":[{"award-number":["11901135"]}]},{"name":"Guizhou Provincial Science and Technology Projects","award":["[2020]1Y015"],"award-info":[{"award-number":["[2020]1Y015"]}]},{"name":"Guizhou Provincial Science and Technology Projects","award":["QJJ2022015"],"award-info":[{"award-number":["QJJ2022015"]}]},{"name":"Guizhou Provincial Science and Technology Projects","award":["QJJ2022047"],"award-info":[{"award-number":["QJJ2022047"]}]},{"name":"Department of Education of Guizhou Province","award":["11961009"],"award-info":[{"award-number":["11961009"]}]},{"name":"Department of Education of Guizhou Province","award":["11901135"],"award-info":[{"award-number":["11901135"]}]},{"name":"Department of Education of Guizhou Province","award":["[2020]1Y015"],"award-info":[{"award-number":["[2020]1Y015"]}]},{"name":"Department of Education of Guizhou Province","award":["QJJ2022015"],"award-info":[{"award-number":["QJJ2022015"]}]},{"name":"Department of Education of Guizhou Province","award":["QJJ2022047"],"award-info":[{"award-number":["QJJ2022047"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we present a high-order approximate solution with uniform accuracy for nonlinear 3D Volterra integral equations. This numerical scheme is constructed based on the three-dimensional block cubic Lagrangian interpolation method. At the same time, we give the local truncation error analysis of the numerical scheme based on Taylor\u2019s theorem. Through theoretical analysis, we reach the conclusion that the optimal convergence order of this high-order numerical scheme is 4. Finally, we verify the effectiveness and applicability of the method through four numerical examples.<\/jats:p>","DOI":"10.3390\/axioms11090476","type":"journal-article","created":{"date-parts":[[2022,9,18]],"date-time":"2022-09-18T22:12:43Z","timestamp":1663539163000},"page":"476","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A High-Order Approximate Solution for the Nonlinear 3D Volterra Integral Equations with Uniform Accuracy"],"prefix":"10.3390","volume":"11","author":[{"given":"Zi-Qiang","family":"Wang","sequence":"first","affiliation":[{"name":"School of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, China"}]},{"given":"Ming-Dan","family":"Long","sequence":"additional","affiliation":[{"name":"School of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, China"}]},{"given":"Jun-Ying","family":"Cao","sequence":"additional","affiliation":[{"name":"School of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,16]]},"reference":[{"key":"ref_1","first-page":"56","article-title":"Singular nonlinear integral equation and its application in viscoelastic nonlinear material","volume":"4","author":"Abdou","year":"2008","journal-title":"Int. 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