{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T18:53:59Z","timestamp":1778698439627,"version":"3.51.4"},"reference-count":32,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,1,23]],"date-time":"2025-01-23T00:00:00Z","timestamp":1737590400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,1,23]],"date-time":"2025-01-23T00:00:00Z","timestamp":1737590400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100002386","name":"Cairo University","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100002386","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2025,6]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>In this paper, a new approach to time-fractional partial integro-differential equations with weakly singular kernels (<jats:bold>\n              <jats:italic>TFPIDE<\/jats:italic>\n            <\/jats:bold>) is presented. The suggested method produces a spectral semi-analytic solution by using shifted first-kind Chebyshev polynomials (<jats:bold>\n              <jats:italic>SCP1K<\/jats:italic>\n            <\/jats:bold>) as basis functions. To satisfy homogeneous beginning and boundary requirements, a suitable collection of basis functions should be chosen. The unknown expansion coefficients are then found using the Petrov\u2013Galerkin technique. Interestingly, we obtain precise equations for each of the related matrices\u2019 elements. These matrices follow a clear pattern that facilitates the inversion procedure and allows the algebraic problem generated by the Petrov\u2013Galerkin technique to be solved. The work contributes to a better knowledge of the dependability of the approach by thoroughly examining convergence and error analysis. Numerical examples demonstrate the applicability, accuracy, and efficiency of the suggested technique, supplemented by comparisons with previous research. The outcomes demonstrate how well this method works for solving time fractional partial integro-differential equations, highlighting its importance as a useful contribution to the body of knowledge in the area.<\/jats:p>","DOI":"10.1007\/s12190-025-02371-w","type":"journal-article","created":{"date-parts":[[2025,1,23]],"date-time":"2025-01-23T04:47:08Z","timestamp":1737607628000},"page":"3891-3911","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["Chebyshev Petrov\u2013Galerkin method for nonlinear time-fractional integro-differential equations with a mildly singular kernel"],"prefix":"10.1007","volume":"71","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0403-8797","authenticated-orcid":false,"given":"Y. H.","family":"Youssri","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1467-640X","authenticated-orcid":false,"given":"A. G.","family":"Atta","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,1,23]]},"reference":[{"key":"2371_CR1","doi-asserted-by":"publisher","first-page":"5456","DOI":"10.1007\/s40314-018-0645-z","volume":"37","author":"F Ghanbari","year":"2018","unstructured":"Ghanbari, F., Ghanbari, K., Mokhtary, P.: Generalized Jacobi\u2013Galerkin method for nonlinear fractional differential algebraic equations. Comput. Appl. Math. 37, 5456\u20135475 (2018)","journal-title":"Comput. Appl. Math."},{"key":"2371_CR2","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s11075-021-01165-y","volume":"89","author":"MA Zaky","year":"2022","unstructured":"Zaky, M.A., Abo-Gabal, H., Hafez, R.M., Doha, E.H.: Computational and theoretical aspects of Romanovski-Bessel polynomials and their applications in spectral approximations. Numer. Algor. 89, 1\u201335 (2022)","journal-title":"Numer. Algor."},{"issue":"1","key":"2371_CR3","doi-asserted-by":"publisher","DOI":"10.1088\/1402-4896\/ad1700","volume":"99","author":"YH Youssri","year":"2024","unstructured":"Youssri, Y.H., Ismail, M., Atta, A.G.: Chebyshev Petrov\u2013Galerkin procedure for the time-fractional heat equation with nonlocal conditions. Phys. Scr. 99(1), 015251 (2024)","journal-title":"Phys. Scr."},{"issue":"1","key":"2371_CR4","first-page":"1","volume":"14","author":"YH Youssri","year":"2024","unstructured":"Youssri, Y.H., Atta, A.G.: Modal spectral Tchebyshev Petrov\u2013Galerkin stratagem for the time-fractional nonlinear Burgers\u2019 equation. Iran. J. Numer. Anal. Optim. 14(1), 1 (2024)","journal-title":"Iran. J. Numer. Anal. Optim."},{"issue":"1","key":"2371_CR5","doi-asserted-by":"publisher","first-page":"2450002","DOI":"10.1142\/S0129183124500025","volume":"35","author":"M Moustafa","year":"2024","unstructured":"Moustafa, M., Youssri, Y.H., Atta, A.G.: Explicit Chebyshev\u2013Galerkin scheme for the time-fractional diffusion equation. Int. J. Mod. Phys. C 35(1), 2450002 (2024)","journal-title":"Int. J. Mod. Phys. C"},{"issue":"4","key":"2371_CR6","first-page":"207","volume":"14","author":"AG Atta","year":"2023","unstructured":"Atta, A.G., Youssri, Y.H.: Shifted second-kind Chebyshev spectral collocation-based technique for time-fractional KdV\u2013Burgers\u2019 equation. Iran. J. Math. Chem. 14(4), 207\u2013224 (2023)","journal-title":"Iran. J. Math. Chem."},{"issue":"4","key":"2371_CR7","doi-asserted-by":"publisher","first-page":"853","DOI":"10.37256\/cm.4420233594","volume":"4","author":"RM Hafez","year":"2023","unstructured":"Hafez, R.M., Youssri, Y.H., Atta, A.G.: Jacobi rational operational approach for time-fractional sub-diffusion equation on a semi-infinite domain. Contemp. Math. 4(4), 853\u2013876 (2023)","journal-title":"Contemp. Math."},{"issue":"1","key":"2371_CR8","doi-asserted-by":"publisher","first-page":"296","DOI":"10.37256\/cm.5120244054","volume":"5","author":"YH Youssri","year":"2024","unstructured":"Youssri, Y.H., Atta, A.G.: Fej\u00e9r-Quadrature collocation algorithm for solving fractional integro-differential equations via Fibonacci polynomials. Contemp. Math. 5(1), 296\u2013308 (2024)","journal-title":"Contemp. Math."},{"key":"2371_CR9","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2023.115302","volume":"436","author":"W Zheng","year":"2024","unstructured":"Zheng, W., Chen, Y., Zhou, J.: A Legendre spectral method for multidimensional partial Volterra integro-differential equations. J. Comput. Appl. Math. 436, 115302 (2024)","journal-title":"J. Comput. Appl. Math."},{"issue":"1","key":"2371_CR10","doi-asserted-by":"publisher","first-page":"017119","DOI":"10.1063\/5.0131771","volume":"35","author":"H Tu","year":"2023","unstructured":"Tu, H., Wang, Y., Zhang, Y., Liao, H., Liu, W.: Parallel numerical simulation of weakly range-dependent ocean acoustic waveguides by adiabatic modes based on a spectral method. Phys. Fluids 35(1), 017119 (2023)","journal-title":"Phys. Fluids"},{"issue":"3","key":"2371_CR11","doi-asserted-by":"publisher","first-page":"037113","DOI":"10.1063\/5.0138012","volume":"35","author":"H Tu","year":"2023","unstructured":"Tu, H., Wang, Y., Yang, C., Liu, W., Wang, X.: A Chebyshev\u2013Tau spectral method for coupled modes of underwater sound propagation in range-dependent ocean environments. Phys. Fluids 35(3), 037113 (2023)","journal-title":"Phys. Fluids"},{"issue":"4","key":"2371_CR12","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevD.107.044009","volume":"107","author":"RA Konoplya","year":"2023","unstructured":"Konoplya, R.A., Zhidenko, A.: Bernstein spectral method for quasinormal modes of a generic black hole spacetime and application to instability of dilaton-de Sitter solution. Phys. Rev. D. 107(4), 044009 (2023)","journal-title":"Phys. Rev. D."},{"issue":"6","key":"2371_CR13","first-page":"107","volume":"29","author":"YH Youssri","year":"2024","unstructured":"Youssri, Y.H., Atta, A.G.: Radical Petrov\u2013Galerkin approach for the time-fractional KdV\u2013Burgers\u2019 equation. Math. Comput. Appl. 29(6), 107 (2024)","journal-title":"Math. Comput. Appl."},{"issue":"1","key":"2371_CR14","doi-asserted-by":"publisher","first-page":"20240022","DOI":"10.1515\/nleng-2024-0022","volume":"13","author":"YH Youssri","year":"2024","unstructured":"Youssri, Y.H., Atta, A.G., Abu Waar, Z.Y., Moustafa, M.O.: Petrov\u2013Galerkin method for small deflections in fourth-order beam equations in civil engineering. Nonlinear Eng. 13(1), 20240022 (2024)","journal-title":"Nonlinear Eng."},{"key":"2371_CR15","doi-asserted-by":"publisher","first-page":"512","DOI":"10.1016\/j.cma.2017.06.006","volume":"324","author":"E Kharazmi","year":"2017","unstructured":"Kharazmi, E., Zayernouri, M., Em Karniadakis, G.: A Petrov\u2013Galerkin spectral element method for fractional elliptic problems. Comput. Methods Appl. Mech. Eng. 324, 512\u2013536 (2017)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"2371_CR16","doi-asserted-by":"publisher","first-page":"678","DOI":"10.1016\/j.matcom.2021.06.004","volume":"190","author":"RM Hafez","year":"2021","unstructured":"Hafez, R.M., Zaky, M.A., Hendy, A.S.: A novel spectral Galerkin\/Petrov\u2013Galerkin algorithm for the multi-dimensional space-time fractional advection-diffusion\u2013reaction equations with nonsmooth solutions. Math. Comput. Simul. 190, 678\u2013690 (2021)","journal-title":"Math. Comput. Simul."},{"issue":"2","key":"2371_CR17","doi-asserted-by":"publisher","first-page":"207","DOI":"10.1007\/s11803-008-0837-4","volume":"7","author":"M Wang","year":"2008","unstructured":"Wang, M., Au, F.T.K.: Precise integration methods based on the Chebyshev polynomial of the first kind. Earthq. Eng. Eng. Vib. 7(2), 207\u2013216 (2008)","journal-title":"Earthq. Eng. Eng. Vib."},{"key":"2371_CR18","doi-asserted-by":"publisher","first-page":"132","DOI":"10.37256\/cm.4120232254","volume":"4","author":"HM Ahmed","year":"2023","unstructured":"Ahmed, H.M.: Numerical Solutions for Singular Lane-Emden Equations Using Shifted Chebyshev Polynomials of the First Kind. Contemp. Math. 4, 132\u2013149 (2023)","journal-title":"Contemp. Math."},{"key":"2371_CR19","first-page":"277","volume":"12","author":"WM Abd-Elhameed","year":"2024","unstructured":"Abd-Elhameed, W.M., Youssri, Y.H., Atta, A.G.: Tau algorithm for fractional delay differential equations utilizing seventh-kind Chebyshev polynomials. J. Math. Model. 12, 277\u2013299 (2024)","journal-title":"J. Math. Model."},{"issue":"4","key":"2371_CR20","doi-asserted-by":"publisher","first-page":"353","DOI":"10.1515\/156939805775122280","volume":"20","author":"VI Lebedev","year":"2005","unstructured":"Lebedev, V.I., Finogenov, S.A.: Some algorithms for computing Chebyshev normalized first-kind polynomials by roots. Russ. J. Numer. Anal. Math. Model. 20(4), 353\u2013363 (2005)","journal-title":"Russ. J. Numer. Anal. Math. Model."},{"issue":"1","key":"2371_CR21","first-page":"1","volume":"51","author":"ES Shoukralla","year":"2021","unstructured":"Shoukralla, E.S.: Application of Chebyshev polynomials of the second kind to the numerical solution of weakly singular Fredholm integral equations of the first kind. IAENG Int. J. Appl. Math. 51(1), 1\u201316 (2021)","journal-title":"IAENG Int. J. Appl. Math."},{"issue":"3","key":"2371_CR22","doi-asserted-by":"publisher","first-page":"85","DOI":"10.3390\/fractalfract5030085","volume":"5","author":"T Akram","year":"2021","unstructured":"Akram, T., Ali, Z., Rabiei, F., Shah, K., Kumam, P.: A numerical study of nonlinear fractional order partial integro-differential equation with a weakly singular kernel. Fract. Fract. 5(3), 85 (2021)","journal-title":"Fract. Fract."},{"key":"2371_CR23","doi-asserted-by":"publisher","first-page":"666","DOI":"10.1016\/j.cjph.2020.08.017","volume":"67","author":"JR Loh","year":"2020","unstructured":"Loh, J.R., Phang, C., Tay, K.G.: New method for solving fractional partial integro-differential equations by combination of Laplace transform and resolvent kernel method. Chin. J. Phys. 67, 666\u2013680 (2020)","journal-title":"Chin. J. Phys."},{"issue":"9","key":"2371_CR24","doi-asserted-by":"publisher","first-page":"e19307","DOI":"10.1016\/j.heliyon.2023.e19307","volume":"9","author":"M Abbas","year":"2023","unstructured":"Abbas, M., Aslam, S., Abdullah, F.A., Riaz, M.B., Gepreel, K.A.: An efficient spline technique for solving time-fractional integro-differential equations. Heliyon 9(9), e19307 (2023)","journal-title":"Heliyon"},{"issue":"11","key":"2371_CR25","doi-asserted-by":"publisher","first-page":"619","DOI":"10.3390\/fractalfract6110619","volume":"6","author":"AG Atta","year":"2022","unstructured":"Atta, A.G., Abd-Elhameed, W.M., Moatimid, G.M., Youssri, Y.H.: Modal shifted fifth-kind Chebyshev Tau integral approach for solving heat conduction equation. Fract. Fract. 6(11), 619 (2022)","journal-title":"Fract. Fract."},{"issue":"3","key":"2371_CR26","doi-asserted-by":"publisher","first-page":"594","DOI":"10.3390\/sym15030594","volume":"15","author":"EM Abdelghany","year":"2023","unstructured":"Abdelghany, E.M., Abd-Elhameed, W.M., Moatimid, G.M., Youssri, Y.H., Atta, A.G.: A tau approach for solving time-fractional heat equation based on the shifted sixth-kind Chebyshev polynomials. Symmetry 15(3), 594 (2023)","journal-title":"Symmetry"},{"issue":"13","key":"2371_CR27","doi-asserted-by":"publisher","first-page":"14745","DOI":"10.1002\/mma.9343","volume":"46","author":"AG Atta","year":"2023","unstructured":"Atta, A.G., Abd-Elhameed, W.M., Moatimid, G.M., Youssri, Y.H.: Novel spectral schemes to fractional problems with nonsmooth solutions. Math. Methods Appl. Sci. 46(13), 14745\u201314764 (2023)","journal-title":"Math. Methods Appl. Sci."},{"key":"2371_CR28","volume-title":"Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications","author":"I Podlubny","year":"1998","unstructured":"Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Elsevier, San Diego (1998)"},{"issue":"8","key":"2371_CR29","doi-asserted-by":"publisher","first-page":"381","DOI":"10.1007\/s40314-022-02096-7","volume":"41","author":"AG Atta","year":"2022","unstructured":"Atta, A.G., Youssri, Y.H.: Advanced shifted first-kind Chebyshev collocation approach for solving the nonlinear time-fractional partial integro-differential equation with a weakly singular kernel. Comput. Appl. Math. 41(8), 381 (2022)","journal-title":"Comput. Appl. Math."},{"key":"2371_CR30","doi-asserted-by":"publisher","DOI":"10.1201\/9781420036114","volume-title":"Chebyshev Polynomials","author":"JC Mason","year":"2002","unstructured":"Mason, J.C., Handscomb, D.C.: Chebyshev Polynomials. Chapman and Hall, New York (2002)"},{"key":"2371_CR31","doi-asserted-by":"crossref","unstructured":"Youssri, Y.H., Atta, A.G.: Double Tchebyshev spectral Tau algorithm for solving KdV equation, with soliton application. In: Solitons, pp. 451\u2013467. Springer (2022)","DOI":"10.1007\/978-1-0716-2457-9_771"},{"issue":"8","key":"2371_CR32","doi-asserted-by":"publisher","first-page":"350","DOI":"10.1007\/s40314-023-02491-8","volume":"42","author":"J Alavi","year":"2023","unstructured":"Alavi, J., Aminikhah, H.: An efficient parametric finite difference and orthogonal spline approximation for solving the weakly singular nonlinear time-fractional partial integro-differential equation. Comput. Appl. Math. 42(8), 350 (2023)","journal-title":"Comput. Appl. Math."}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-025-02371-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-025-02371-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-025-02371-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,23]],"date-time":"2025-05-23T16:45:39Z","timestamp":1748018739000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-025-02371-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,23]]},"references-count":32,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,6]]}},"alternative-id":["2371"],"URL":"https:\/\/doi.org\/10.1007\/s12190-025-02371-w","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"value":"1598-5865","type":"print"},{"value":"1865-2085","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1,23]]},"assertion":[{"value":"29 October 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 January 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 January 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 January 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"Not applicable.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethics approval and consent to participate"}}]}}