{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,20]],"date-time":"2026-02-20T18:48:37Z","timestamp":1771613317828,"version":"3.50.1"},"reference-count":22,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,12,28]],"date-time":"2024-12-28T00:00:00Z","timestamp":1735344000000},"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>It is common knowledge that studying integral equations accompanied by and related to phase delay is significant, and that significance grows when considering the problem\u2019s time factor. Through this study, one may predict the material\u2019s state for a short time or infer its state before beginning the investigation. In this work, a phase-lag mixed integral equation (P-MIE) with a continuous kernel in time and a singular kernel in position is studied in (2 + 1) dimensions in the space L2([a,b]\u00d7[c,d])\u00d7C[0,T],T&lt;1. The properties of fractional integrals are used to generate the mixed integral equation (MIE). Certain assumptions are considered in order to examine convergence, uniqueness of solution, and estimation error. We achieve a class of two-dimensional Fredholm integral equations (FIEs) with time-dependent coefficients after applying the separation technique. After that, we will get a linear algebraic system (LAS) in 2Ds applying the product Nystr\u04e7m method (PNM). The convergence of the LAS\u2019s unique solution is covered. Two applications on the MIE with a logarithmic kernel and a Carleman function are discussed to illustrate the viability and efficiency of the applied techniques. At the end, a valuable conclusion is established.<\/jats:p>","DOI":"10.3390\/sym17010036","type":"journal-article","created":{"date-parts":[[2024,12,31]],"date-time":"2024-12-31T13:26:25Z","timestamp":1735651585000},"page":"36","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["The Effect of Fractional Order of Time Phase Delay via a Mixed Integral Equation in (2 + 1) Dimensions with an Extended Discontinuous Kernel"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1696-283X","authenticated-orcid":false,"given":"Sameeha A.","family":"Raad","sequence":"first","affiliation":[{"name":"Mathematics Department, Faculty of Sciences, Umm Al-Qura University, Makkah 21955, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3312-6510","authenticated-orcid":false,"given":"Mohammed A.","family":"Abdou","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Education, Alexandria University, Alexandria 21526, Egypt"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"24379","DOI":"10.3934\/math.20231243","article-title":"Physical phenomena of spectral relationships via quadratic third kind mixed integral equation with discontinuous kernel","volume":"8","author":"Alhazmi","year":"2023","journal-title":"AIMS Math."},{"key":"ref_2","first-page":"7856207","article-title":"On the solution of quadratic nonlinear integral equation with different singular kernels","volume":"1","author":"Basseem","year":"2020","journal-title":"Math. 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