{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,11]],"date-time":"2025-11-11T13:39:43Z","timestamp":1762868383937,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2020,12,17]],"date-time":"2020-12-17T00:00:00Z","timestamp":1608163200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this study, we present a spectral method for solving nonlinear Volterra integral equations with weakly singular kernels based on the Genocchi polynomials. Many other interesting results concerning nonlinear equations with discontinuous symmetric kernels with application of group symmetry have remained beyond this paper. In the proposed approach, relying on the useful properties of Genocchi polynomials, we produce an operational matrix and a related coefficient matrix to convert nonlinear Volterra integral equations with weakly singular kernels into a system of algebraic equations. This method is very fast and gives high-precision answers with good accuracy in a low number of repetitions compared to other methods that are available. The error boundaries for this method are also presented. Some illustrative examples are provided to demonstrate the capability of the proposed method. Also, the results derived from the new method are compared to Euler\u2019s method to show the superiority of the proposed method.<\/jats:p>","DOI":"10.3390\/sym12122105","type":"journal-article","created":{"date-parts":[[2020,12,17]],"date-time":"2020-12-17T21:21:49Z","timestamp":1608240109000},"page":"2105","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Matrix Method by Genocchi Polynomials for Solving Nonlinear Volterra Integral Equations with Weakly Singular Kernels"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5902-4830","authenticated-orcid":false,"given":"Elham","family":"Hashemizadeh","sequence":"first","affiliation":[{"name":"Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj 3149968111, Iran"}]},{"given":"Mohammad Ali","family":"Ebadi","sequence":"additional","affiliation":[{"name":"Young Researchers and Elite Club, Karaj Branch, Islamic Azad University, Karaj 3149968111, Iran"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2307-0891","authenticated-orcid":false,"given":"Samad","family":"Noeiaghdam","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Programming, South Ural State University, Lenin prospect 76, 454080 Chelyabinsk, Russia"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1016\/S0377-0427(02)00862-2","article-title":"On a symptotic methods for Fredholm\u2013Volterra integral equation of the second kind in contact problems","volume":"154","author":"Abdou","year":"2003","journal-title":"J. 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