{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,21]],"date-time":"2026-05-21T02:54:14Z","timestamp":1779332054941,"version":"3.51.4"},"reference-count":37,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2023,5,24]],"date-time":"2023-05-24T00:00:00Z","timestamp":1684886400000},"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>Symmetry is presented in many works involving differential and integral equations. Whenever a human is involved in the design of an integral equation, they naturally tend to opt for symmetric features. The most common examples are the Green functions and linguistic kernels that are often designed symmetrically and regularly distributed over the universe of discourse. In the current study, the authors report a study on boundary value problem (BVP) for a nonlinear integro Volterra\u2013Fredholm integral equation with variable coefficients and show the existence of solution by applying some fixed-point theorems. The authors employ various numerical common approaches as the homotopy analysis methodology established by Liao and the modified Adomain decomposition technique to produce a numerical approximate solution, then graphical depiction reveals that both methods are most effective and convenient. In this regard, the authors address the requirements that ensure the existence and uniqueness of the solution for various variations of nonlinearity power. The authors also show numerical examples of how to apply our primary theorems and test the convergence and validity of our suggested approach.<\/jats:p>","DOI":"10.3390\/sym15061144","type":"journal-article","created":{"date-parts":[[2023,5,25]],"date-time":"2023-05-25T01:36:28Z","timestamp":1684978588000},"page":"1144","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["New Numerical Results on Existence of Volterra\u2013Fredholm Integral Equation of Nonlinear Boundary Integro-Differential Type"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1493-6518","authenticated-orcid":false,"given":"Hawsar","family":"HamaRashid","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Education, University of Sulaimani, Sulaimani 46001, Kurdistan Region, Iraq"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9277-8092","authenticated-orcid":false,"given":"Hari Mohan","family":"Srivastava","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada"},{"name":"Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, Baku 1007, Azerbaijan"},{"name":"Center for Converging Humanities, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Republic of Korea"},{"name":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9012-0943","authenticated-orcid":false,"given":"Mudhafar","family":"Hama","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, University of Sulaimani, Sulaimani 46001, Kurdistan Region, Iraq"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6837-8075","authenticated-orcid":false,"given":"Pshtiwan Othman","family":"Mohammed","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Education, University of Sulaimani, Sulaimani 46001, Kurdistan Region, Iraq"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0223-4711","authenticated-orcid":false,"given":"Eman","family":"Al-Sarairah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Khalifa University, Abu Dhabi P.O. Box 127788, United Arab Emirates"},{"name":"Department of Mathematics, Al-Hussein Bin Talal University, P.O. Box 20, Ma\u2019an 71111, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8485-7113","authenticated-orcid":false,"given":"Musawa Yahya","family":"Almusawa","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Jazan University, Jazan 45142, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1007\/s11082-021-02782-6","article-title":"Abundant closed-form solitons for time-fractional integro-differential equation in fluid dynamics","volume":"53","author":"AzZobi","year":"2021","journal-title":"Opt. 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