{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T06:08:08Z","timestamp":1778738888591,"version":"3.51.4"},"reference-count":16,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T00:00:00Z","timestamp":1778630400000},"content-version":"vor","delay-in-days":132,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2026,1,1]],"date-time":"2026-01-01T00:00:00Z","timestamp":1767225600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2026,1]]},"abstract":"<jats:p>\n                    In this paper, we investigate a set of nonlinear (\n                    <jats:italic>q<\/jats:italic>\n                    ,\n                    <jats:italic>\u03c4<\/jats:italic>\n                    )\u2010fractional Fredholm integrodifferential equations that involve memory\u2010type integral kernels and generalized fractional derivatives. Using a Galerkin technique based on (\n                    <jats:italic>q<\/jats:italic>\n                    ,\n                    <jats:italic>\u03c4<\/jats:italic>\n                    )\u2010Legendre polynomials, we designed an approximation solution and provided a numerical scheme for calculating the integral terms and fractional derivative. Through comparison with benchmark functions and residual analysis, the approximation\u2019s convergence is confirmed. In addition, we propose adequate conditions under which perturbed solutions stay close to the real solution in order to study the Ulam\u2013Hyers stability of the generalized equation. The existence, uniqueness, and stability of the solution under Lipschitz\u2010type conditions on the kernel function and the nonlinearity are demonstrated using a fixed\u2010point theorem. The numerical tests show the method\u2019s robustness and validate the theoretical results. These findings offer a solid foundation for the study and modeling of nonlinear systems in the (\n                    <jats:italic>q<\/jats:italic>\n                    ,\n                    <jats:italic>\u03c4<\/jats:italic>\n                    )\u2010setting that are controlled by nonlocal fractional models.\n                  <\/jats:p>","DOI":"10.1155\/jama\/5594933","type":"journal-article","created":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T05:13:46Z","timestamp":1778735626000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Solvability and Stability of Solutions of (\n                    <i>q<\/i>\n                    ,\n                    <i>\u03c4<\/i>\n                    )\u2010Fractional Integro\u2010Differential Models"],"prefix":"10.1155","volume":"2026","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6326-8456","authenticated-orcid":false,"given":"Shaher","family":"Momani","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9341-025X","authenticated-orcid":false,"given":"Rabha W.","family":"Ibrahim","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0286-7244","authenticated-orcid":false,"given":"Dumitru","family":"Baleanu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2026,5,13]]},"reference":[{"key":"e_1_2_12_1_2","doi-asserted-by":"publisher","DOI":"10.1017\/s0080456800002751"},{"key":"e_1_2_12_2_2","doi-asserted-by":"crossref","unstructured":"KacV. G.andCheungP. 2002 113 Springer New York.","DOI":"10.2752\/9781847888945\/UNRAVRAG0010"},{"key":"e_1_2_12_3_2","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-0348-0431-8","volume-title":"A Comprehensive Treatment of q-Calculus","author":"Ernst T.","year":"2012"},{"key":"e_1_2_12_4_2","article-title":"Quantum Calculus-Based Error Estimations in Hermite-Hadamard Type Inequalities","author":"Samraiz M.","year":"2025","journal-title":"Sahand Communications in Mathematical Analysis"},{"key":"e_1_2_12_5_2","volume-title":"Generalized Fredholm Equations","author":"Hernandez R. T. P.","year":"2019"},{"key":"e_1_2_12_6_2","doi-asserted-by":"publisher","DOI":"10.3390\/app12178498"},{"key":"e_1_2_12_7_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.apnum.2021.04.008"},{"key":"e_1_2_12_8_2","doi-asserted-by":"publisher","DOI":"10.1002\/mma.10943"},{"key":"e_1_2_12_9_2","doi-asserted-by":"publisher","DOI":"10.1007\/s12190-025-02412-4"},{"key":"e_1_2_12_10_2","doi-asserted-by":"publisher","DOI":"10.1007\/s12190-025-02365-8"},{"key":"e_1_2_12_11_2","article-title":"Analytical Solutions of Heat Transfer Model in Two-Dimensional Case of Neutrosophic Fredholm Integro-Di Erential Equations","volume":"79","author":"Darweesh A.","year":"2025","journal-title":"Neutrosophic Sets and Systems"},{"key":"e_1_2_12_12_2","doi-asserted-by":"publisher","DOI":"10.1007\/s11075-024-02005-5"},{"key":"e_1_2_12_13_2","doi-asserted-by":"publisher","DOI":"10.1186\/s13662-020-02811-w"},{"key":"e_1_2_12_14_2","doi-asserted-by":"publisher","DOI":"10.1088\/1402-4896\/ad0c3c"},{"key":"e_1_2_12_15_2","doi-asserted-by":"publisher","DOI":"10.31489\/2025m3\/214-230"},{"key":"e_1_2_12_16_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.padiff.2025.101305"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/jama\/5594933","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/full-xml\/10.1155\/jama\/5594933","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/jama\/5594933","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T05:13:51Z","timestamp":1778735631000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/jama\/5594933"}},"subtitle":[],"editor":[{"given":"Serbay","family":"Duran","sequence":"additional","affiliation":[],"role":[{"role":"editor","vocabulary":"crossref"}]}],"short-title":[],"issued":{"date-parts":[[2026,1]]},"references-count":16,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,1]]}},"alternative-id":["10.1155\/jama\/5594933"],"URL":"https:\/\/doi.org\/10.1155\/jama\/5594933","archive":["Portico"],"relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"value":"1110-757X","type":"print"},{"value":"1687-0042","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1]]},"assertion":[{"value":"2025-06-04","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2026-04-01","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2026-05-13","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"5594933"}}