{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,17]],"date-time":"2025-11-17T14:31:11Z","timestamp":1763389871645,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,9,12]],"date-time":"2024-09-12T00:00:00Z","timestamp":1726099200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Prince Sattam bin Abdulaziz University","award":["PSAU\/2023\/01\/26664"],"award-info":[{"award-number":["PSAU\/2023\/01\/26664"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>A quadratic integral Equation (QIE) of the second kind with continuous kernels is solved in the space C([0,T]\u00d7[0,T]). The existence of at least one solution to the QIE is discussed in this article. Our evidence depends on a suitable combination of the measures of the noncompactness approach and the fixed-point principle of Darbo. The quadratic integral equation can be used to derive a system of integral equations of the second kind using the quadrature method. With the aid of two different polynomials, Laguerre and Hermite, the system of integral equations is solved using the collocation method. In each numerical approach, the estimation of the error is discussed. Finally, using some examples, the accuracy and scalability of the proposed method are demonstrated along with comparisons. Mathematica 11 was used to obtain all of the results from the techniques that were shown.<\/jats:p>","DOI":"10.3390\/axioms13090621","type":"journal-article","created":{"date-parts":[[2024,9,12]],"date-time":"2024-09-12T06:31:40Z","timestamp":1726122700000},"page":"621","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Analytical and Numerical Approaches via Quadratic Integral Equations"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1457-6927","authenticated-orcid":false,"given":"Jihan","family":"Alahmadi","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mohamed A.","family":"Abdou","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Education, Alexandria University, Alexandria 21511, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5865-278X","authenticated-orcid":false,"given":"Mohamed A.","family":"Abdel-Aty","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Faculty of Science, Benha University, Benha 13518, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"18","DOI":"10.5539\/jmr.v9n5p18","article-title":"New Model for Solving Mixed Integral Equation of the First Kind with Generalized Potential Kernel","volume":"9","author":"Alhazmi","year":"2017","journal-title":"J. Math. Res."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1007\/s40314-023-02212-1","article-title":"Block-by-block method for solving non-linear Volterra integral equation of the first kind","volume":"42","author":"Ghiat","year":"2023","journal-title":"Comp. Appl. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1007\/s10915-022-01953-1","article-title":"Optimal Control in a Bounded Domain for Wave Propagating in the Whole Space: Coupling Through Boundary Integral Equations","volume":"92","author":"Gong","year":"2022","journal-title":"J. Sci. Comput."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"735","DOI":"10.1080\/09720502.2020.1861787","article-title":"Solving Volterra integral equation by using a new transformation","volume":"24","author":"Jaabar","year":"2021","journal-title":"J. Interdiscip. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"27488","DOI":"10.3934\/math.20231406","article-title":"New algorithms for solving nonlinear mixed integral equations","volume":"8","author":"Matoog","year":"2023","journal-title":"AIMS Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1007\/s10915-023-02155-z","article-title":"Fractional Collocation Method for Third-Kind Volterra Integral Equations with Nonsmooth Solutions","volume":"95","author":"Ma","year":"2023","journal-title":"J. Sci. Comput."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1815","DOI":"10.1007\/s11784-016-0336-6","article-title":"On some iterative numerical methods for a Volterra functional integral equation of the second kind","volume":"19","author":"Micula","year":"2017","journal-title":"J. Fixed Point Theory Appl."},{"key":"ref_8","first-page":"935","article-title":"An iterative numerical method for Fredholm\u2013Volterra integral equations of the second kind","volume":"270","author":"Micula","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1080\/09720502.2020.1731965","article-title":"Solution of non linear Fredholm integral equation involving constant delay by BEM with piecewise linear approximation","volume":"23","author":"Sarkar","year":"2020","journal-title":"J. Interdiscip. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"196","DOI":"10.1016\/j.cam.2015.08.018","article-title":"A new spectral meshless radial point interpolation(SMRPI) method for the two\u2013dimensional Fredholm integral equations on general domains with error analysis","volume":"294","author":"Fatahi","year":"2016","journal-title":"J. Comput. Appl. Math."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Noeiaghdam, S., Dreglea, A., He, J., Avazzadeh, Z., Suleman, M., Fariborzi Araghi, M.A., Sidorov, D.N., and Sidorov, N. (2020). Error estimation of the homotopy perturbation method to solve second kind Volterra integral equations with piecewise smooth kernels. Application of the CADNA library. Symmetry, 12.","DOI":"10.3390\/sym12101730"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"e11827","DOI":"10.1016\/j.heliyon.2022.e11827","article-title":"Solution of nonlinear mixed integral equation via collocation method basing on orthogonal polynomials","volume":"8","author":"Jaan","year":"2022","journal-title":"Heliyon"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Abusalim, S.A., Abdou, M.A., Nasr, M.E., and Abdel\u2013Aty, M.A. (2023). An Algorithm for the Solution of Nonlinear Volterra\u2013Fredholm Integral Equations with a Singular Kernel. Fractal Fract, 7.","DOI":"10.3390\/fractalfract7100730"},{"key":"ref_14","first-page":"3203","article-title":"Analytical and Numerical Discussion for the Phase-Lag Volterra-Fredholm Integral Equation with Singular Kernel","volume":"13","author":"Abdou","year":"2023","journal-title":"J. Appl. Anal. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Abusalim, S.A., Abdou, M.A., Abdel\u2013Aty, M.A., and Nasr, M.E. (2023). Hybrid Functions Approach via Nonlinear Integral Equations with Symmetric and Nonsymmetrical Kernel in Two Dimensions. Symmetry, 15.","DOI":"10.20944\/preprints202305.0599.v1"},{"key":"ref_16","first-page":"377","article-title":"CAS wavelet method for the numerical solution of boundary integral equations with logarithmic singular kernels","volume":"4","author":"Adibi","year":"2014","journal-title":"Int. J. Math. Model. Comput."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"12","DOI":"10.3103\/S1068362322010022","article-title":"Solvability of quadratic integral equations with singular kernel","volume":"57","author":"Abdou","year":"2022","journal-title":"J. Contemp. Math. Anal."},{"key":"ref_18","first-page":"637","article-title":"Application of Fibonacci collocation method for solving Volterra\u2013Fredholm integral equations","volume":"273","author":"Mirzaee","year":"2016","journal-title":"Appl. Math. Comput."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1016\/j.cam.2016.11.004","article-title":"Solving system of Volterra\u2013Fredholm integral equations with Bernstein polynomials and hybrid Bernstein Block-Pulse functions","volume":"315","author":"Hesameddini","year":"2017","journal-title":"J. Comput. Appl. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1007\/s43994-022-00016-3","article-title":"Certain results associated with mixed integral equations and their comparison via numerical methods","volume":"9","author":"Alhazmi","year":"2023","journal-title":"J. Umm Al-Qura Univ. Appl. Sci."},{"key":"ref_21","first-page":"805","article-title":"Numerical solution of linear Fredholm integral equations via two\u2013dimensional modification of hat functions","volume":"250","author":"Mirzaee","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1016\/j.cam.2014.07.018","article-title":"Bernoulli polynomials for the numerical solution of some classes of linear and nonlinear integral equations","volume":"275","author":"Bazm","year":"2015","journal-title":"J. Comput. Appl. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"4465","DOI":"10.1002\/mma.4907","article-title":"On the numerical solution of stochastic quadratic integral equations via operational matrix method","volume":"41","author":"Mirzaee","year":"2018","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Benkerrouche, A., Souid, M.S., Stamov, G., and Stamova, I. (2022). On the solutions of a quadratic integral equation of the Urysohn type of fractional variable order. Entropy, 24.","DOI":"10.3390\/e24070886"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1016\/j.apnum.2020.02.011","article-title":"A computational method for a class of systems of nonlinear variable-order fractional quadratic integral equations","volume":"153","author":"Heydari","year":"2020","journal-title":"Appl. Numer. Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"6009","DOI":"10.2298\/FIL2217009M","article-title":"On some properties of Riemann\u2013Liouville fractional operator in Orlicz spaces and applications to quadratic integral equations","volume":"36","author":"Metwali","year":"2022","journal-title":"Filomat"},{"key":"ref_27","first-page":"1588","article-title":"Analytical results for quadratic integral equations with phase-lag term","volume":"10","author":"Abdou","year":"2020","journal-title":"J. Appl. Anal. Comput."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"19","DOI":"10.12785\/msl\/020103","article-title":"Positive continuous solution of a quadratic integral equation of fractional orders","volume":"2","author":"Hashem","year":"2013","journal-title":"Math. Sci. Lett."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"8095","DOI":"10.2298\/FIL2324095A","article-title":"Analytical and numerical discussion for the quadratic integral equations","volume":"37","author":"Abdou","year":"2023","journal-title":"Filomat"},{"key":"ref_30","first-page":"65","article-title":"Application of modified hat functions for solving nonlinear quadratic integral equations","volume":"6","author":"Mirzaee","year":"2016","journal-title":"Iran J. Numer. Anal. Opt."},{"key":"ref_31","first-page":"283","article-title":"Shole Haghighi, A. Existence of solutions of infinite systems of integral equations in two variables via measure of noncompactness","volume":"246","author":"Arab","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"7856207","DOI":"10.1155\/2020\/7856207","article-title":"On the solution of quadratic nonlinear integral Equation with different singular kernels","volume":"2020","author":"Basseem","year":"2020","journal-title":"Math. Probl. Eng."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Bana\u015b, J., Jleli, M., Mursaleen, M., Samet, B., and Vetro, C. (2017). Advances in Nonlinear Analysis via the Concept of Measure of Noncompactness, Springer Nature.","DOI":"10.1007\/978-981-10-3722-1"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"343","DOI":"10.24193\/fpt-ro.2023.1.19","article-title":"On the attractivity of the solutions of a problem involving Hilfer fractional derivative via the measure of noncompactness","volume":"24","author":"Pourhadi","year":"2023","journal-title":"Fixed Point Theory"},{"key":"ref_35","unstructured":"Delves, L.M., and Mohamed, J.L. (1988). Computational Methods for Integral Equations, Cambridge University Press."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"17486","DOI":"10.3934\/math.2022964","article-title":"Approximation of solutions for nonlinear functional integral equations","volume":"7","author":"Mishra","year":"2022","journal-title":"AIMS Math."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/9\/621\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:54:33Z","timestamp":1760111673000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/9\/621"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,9,12]]},"references-count":36,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2024,9]]}},"alternative-id":["axioms13090621"],"URL":"https:\/\/doi.org\/10.3390\/axioms13090621","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,9,12]]}}}