{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,17]],"date-time":"2026-01-17T21:56:17Z","timestamp":1768686977464,"version":"3.49.0"},"reference-count":38,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2021,12,9]],"date-time":"2021-12-09T00:00:00Z","timestamp":1639008000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100007446","name":"King Khalid University","doi-asserted-by":"publisher","award":["2\/172\/42"],"award-info":[{"award-number":["2\/172\/42"]}],"id":[{"id":"10.13039\/501100007446","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The main focus of this paper was to find the approximate solution of a class of second-order multi-pantograph delay differential equations with singularity. We used the shifted version of Vieta\u2013Lucas polynomials with some symmetries as the main base to develop a collocation approach for solving the aforementioned differential equations. Moreover, an error bound of the present approach by using the maximum norm was computed and an error estimation technique based on the residual function is presented. Finally, the validity and applicability of the presented collocation scheme are shown via four numerical test examples.<\/jats:p>","DOI":"10.3390\/sym13122370","type":"journal-article","created":{"date-parts":[[2021,12,10]],"date-time":"2021-12-10T02:07:18Z","timestamp":1639102038000},"page":"2370","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":25,"title":["Application of Vieta\u2013Lucas Series to Solve a Class of Multi-Pantograph Delay Differential Equations with Singularity"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6116-4928","authenticated-orcid":false,"given":"Mohammad","family":"Izadi","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman 76169-14111, Iran"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5838-7063","authenticated-orcid":false,"given":"\u015euayip","family":"Y\u00fczba\u015f\u0131","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Akdeniz University, Antalya TR 07058, Turkey"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4564-6211","authenticated-orcid":false,"given":"Khursheed J.","family":"Ansari","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"409","DOI":"10.2298\/FIL1802409Y","article-title":"A Galerkin-like approach to solve multi-pantograph type delay differential equations","volume":"32","year":"2018","journal-title":"Filomat"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1140\/epjp\/s13360-020-00449-x","article-title":"Solving a new design of nonlinear second-order Lane-Emden pantograph delay differential model via Bernoulli collocation method","volume":"135","author":"Adel","year":"2020","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_3","first-page":"126123","article-title":"An efficient approximation technique applied to a non-linear Lane-Emden pantograph delay differential model","volume":"401","author":"Izadi","year":"2021","journal-title":"Appl. Math. Comput."},{"key":"ref_4","unstructured":"Kuang, Y. (1993). Delay Differential Equations with Applications in Population Dynamics, Academic Press."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"574","DOI":"10.1016\/j.apnum.2005.04.025","article-title":"Delay-differential-algebraic equations in control theory","volume":"56","author":"Shampine","year":"2006","journal-title":"Appl. Numer. Math."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Campbell, S.A. (2007). Time delays in neural systems. Handbook of Brain Connectivity, Springer.","DOI":"10.1007\/978-3-540-71512-2_2"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/S0025-5564(00)00006-7","article-title":"A delay-differential model of HIV infection of CD4+ T-cells","volume":"165","author":"Culshaw","year":"2000","journal-title":"Math. Biosci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1103\/PhysRevLett.78.203","article-title":"Mechanism of time-delayed feedback control","volume":"78","author":"Just","year":"1997","journal-title":"Phys. Rev. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1109\/JQE.2010.2104135","article-title":"Modeling passive mode-locking in quantum dot lasers: A comparison between a finite-difference traveling-wave model and a delayed differential equation approach","volume":"47","author":"Rossetti","year":"2011","journal-title":"IEEE J. Quant. Electron."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"551","DOI":"10.1175\/1520-0485(1989)019<0551:OTROOE>2.0.CO;2","article-title":"On the role of off-equatorial oceanic Rossby waves during ENSO","volume":"19","author":"Battisti","year":"1989","journal-title":"J. Phys. Oceanogr."},{"key":"ref_11","unstructured":"Forde, J.E. (2005). Delay Differential Equation Models in Mathematical Biology. [Ph.D. Dissertation, University of Michigan]."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Shampine, L.F., and Thompson, L. (2009). Numerical Solution of Delay Differential Equations. Delay Differential Equations, Springer.","DOI":"10.1007\/978-0-387-85595-0_9"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"855","DOI":"10.1137\/0152048","article-title":"Analysis of a model representing stage-structured population growth with state-dependent time delay","volume":"52","author":"Aiello","year":"1992","journal-title":"SIAM J. Appl. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1098\/rspa.1971.0078","article-title":"The dynamics of a current collection system for an electric locomotive","volume":"322","author":"Ockendon","year":"1971","journal-title":"Proc. R. Soc. Lond. A"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1909","DOI":"10.1103\/PhysRevA.52.1909","article-title":"Universal superpositions of coherent states and self-similar potentials","volume":"52","author":"Spiridonov","year":"1995","journal-title":"Phys. Rev. A"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1017\/S0956792510000367","article-title":"A Mellin transform solution to a second-order pantograph equation with linear dispersion arising in a cell growth model","volume":"22","author":"Brunt","year":"2011","journal-title":"Eur. J. Appl. Math."},{"key":"ref_17","first-page":"561","article-title":"Functional differential equations for cell-growth models with dispersion","volume":"4","author":"Wake","year":"2000","journal-title":"Commun. Appl. Anal."},{"key":"ref_18","first-page":"1299","article-title":"A new method of solving singular multi-pantograph delay differential equation in reproducing kernel space","volume":"2","author":"Du","year":"2008","journal-title":"Appl. Math. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2140022-49","DOI":"10.1142\/S0218348X21400223","article-title":"Design of neuro-swarming heuristic solver for multi-pantograph singular delay differential equation","volume":"29","author":"Sabir","year":"2021","journal-title":"Fractals"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Mosavi, A., Shokri, M., Mansor, Z., Qasem, S.N., Band, S.S., and Mohammadzadeh, A. (2020). Machine learning for modeling the singular multi-pantograph equations. Entropy, 22.","DOI":"10.3390\/e22091041"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"6529","DOI":"10.1016\/j.apm.2015.02.006","article-title":"Shifted Legendre approximation with the residual correction to solve pantograph-delay type differential equations","volume":"39","author":"Sezer","year":"2015","journal-title":"Appl. Math. Model."},{"key":"ref_22","first-page":"2021031","article-title":"A novel matrix technique for multi-order pantograph differential equations of fractional order","volume":"477","author":"Izadi","year":"2021","journal-title":"Proc. Roy. Soc. Lond. Ser. A Math. Phys. Engrg. Sci."},{"key":"ref_23","first-page":"395","article-title":"A Taylor operation method for solutions of generalized pantograph type delay differential equations","volume":"42","author":"Ismailov","year":"2018","journal-title":"Turk. J. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"055219","DOI":"10.1088\/1402-4896\/ab6ba2","article-title":"Approximate analytical solution of the linear and nonlinear multi-pantograph delay differential equations","volume":"95","author":"Bahgat","year":"2020","journal-title":"Phys. Script."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/j.apnum.2013.11.003","article-title":"A new Jacobi rational-Gauss collocation method for numerical solution of generalized pantograph equations","volume":"77","author":"Doha","year":"2014","journal-title":"Appl. Numer. Math."},{"key":"ref_26","first-page":"383","article-title":"Runge-Kutta methods for the multi-pantograph delay equation","volume":"163","author":"Li","year":"2005","journal-title":"Appl. Math. Comput."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.cam.2016.02.025","article-title":"Solving generalized pantograph equations by shifted orthonormal Bernstein polynomials","volume":"303","author":"Javadi","year":"2016","journal-title":"J. Comput. Appl. Math."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"431","DOI":"10.5937\/KgJMath1803431Y","article-title":"Convergence analysis of least squares-Epsilon-Ritz algorithm for solving a general class of pantograph equations","volume":"42","author":"Yousefi","year":"2018","journal-title":"Kragujevac J. Math."},{"key":"ref_29","first-page":"61","article-title":"Solving the Basset equation via Chebyshev collocation and LDG methods","volume":"9","author":"Izadi","year":"2021","journal-title":"J. Math. Model."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Babaei, A., Jafari, H., and Banihashemi, S. (2020). A collocation approach for solving time-fractional stochastic heat equation driven by an additive noise. Symmetry, 12.","DOI":"10.3390\/sym12060904"},{"key":"ref_31","first-page":"123","article-title":"Fractional polynomial approximations to the solution of fractional Riccati equation","volume":"51","author":"Izadi","year":"2019","journal-title":"Punjab Univ. J. Math."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Izadi, M., and Srivastava, H.M. (2021). Generalized Bessel quasilinearlization technique applied to Bratu and Lane-Emden type equations of arbitrary order. Fractal Fract., 5.","DOI":"10.3390\/fractalfract5040179"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Izadi, M., Y\u00fczbas\u0131, \u015e., and Cattani, C. (2021). Approximating solutions to fractional-order Bagley-Torvik equation via generalized Bessel polynomial on large domains. Ric. Mat.","DOI":"10.1007\/s11587-021-00650-9"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1016\/j.apnum.2021.07.006","article-title":"Two spectral Legendre\u2019s derivative algorithms for Lane-Emden, Bratu equations, and singular perturbed problems","volume":"169","author":"Abdelhakem","year":"2021","journal-title":"Appl. Numer. Math."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Izadi, M., Y\u00fczbas\u0131, \u015e., and Noeiaghdam, S. (2021). Approximating solutions of non-linear Troesch\u2019s problem via an efficient quasi-linearization Bessel approach. Mathematics, 9.","DOI":"10.3390\/math9161841"},{"key":"ref_36","first-page":"223","article-title":"Vieta polynomials","volume":"40","author":"Horadam","year":"2002","journal-title":"Fibonacci Quart."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"626","DOI":"10.1186\/s13662-020-03085-y","article-title":"Vieta-Lucas polynomials for solving a fractional-order mathematical physics model","volume":"2020","author":"Agarwal","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_38","first-page":"118","article-title":"Residual correction of the Hermite polynomial solutions of the generalized pantograph equations","volume":"3","author":"Gok","year":"2015","journal-title":"New Trends Math. Sci."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/12\/2370\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:43:47Z","timestamp":1760168627000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/12\/2370"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,9]]},"references-count":38,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2021,12]]}},"alternative-id":["sym13122370"],"URL":"https:\/\/doi.org\/10.3390\/sym13122370","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,12,9]]}}}