{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T01:37:58Z","timestamp":1767922678974,"version":"3.49.0"},"reference-count":26,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,20]],"date-time":"2025-01-20T00:00:00Z","timestamp":1737331200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This work aims to provide approximate solutions for singularly perturbed problems with periodic boundary conditions using quintic B-splines and collocation. The well-known Shishkin mesh strategy is applied for mesh construction. Convergence analysis demonstrates that the method achieves parameter-uniform convergence with fourth-order accuracy in the maximum norm. Numerical examples are presented to validate the theoretical estimates. Additionally, the standard hybrid finite difference scheme, a cubic spline scheme, and the proposed method are compared to demonstrate the effectiveness of the present approach.<\/jats:p>","DOI":"10.3390\/axioms14010073","type":"journal-article","created":{"date-parts":[[2025,1,20]],"date-time":"2025-01-20T12:32:52Z","timestamp":1737376372000},"page":"73","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Quintic Spline-Based Computational Method for Solving Singularly Perturbed Periodic Boundary Value Problems"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0882-9249","authenticated-orcid":false,"given":"Puvaneswari","family":"Arumugam","sequence":"first","affiliation":[{"name":"Department of Mathematics, University College of Engineering, Anna University, Tiruchirappalli 620024, Tamilnadu, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0162-2760","authenticated-orcid":false,"given":"Valanarasu","family":"Thynesh","sequence":"additional","affiliation":[{"name":"Department of Mathematics, CDOE, Bharathidasan University, Tiruchirappalli 620024, Tamilnadu, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2228-0969","authenticated-orcid":false,"given":"Chandru","family":"Muthusamy","sequence":"additional","affiliation":[{"name":"Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 632014, Tamil Nadu, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2791-6230","authenticated-orcid":false,"given":"Higinio","family":"Ramos","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Scientific Computing Group, University of Salamanca, Plaza de la Merced, 37008 Salamanca, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Miller, J.J.H., O\u2019Riordan, E., and Shishkin, G.I. (2012). Fitted Numerical Methods for Singular Perturbation Problems: Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions, World Scientific Publishing Co. Pte. Ltd.","DOI":"10.1142\/9789814390743"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Farrell, P.A., Hegarty, A.F., Miller, J.J.H., O\u2019Riordan, E., and Shishkin, G.I. (2000). Robust Computational Techniques for Boundary Layers, Chapman and Hall\/CRC.","DOI":"10.1201\/9781482285727"},{"key":"ref_3","unstructured":"Roos, H.G., Stynes, M., and Tobiska, L. (2008). Robust Numerical Methods for Singularly Perturbed Differential Equations, Computational Mathematics, Springer."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1847","DOI":"10.1080\/10236198.2018.1543417","article-title":"A collocation method for singularly perturbed differential-difference turning point problems exhibiting boundary\/interior layers","volume":"24","author":"Kumar","year":"2018","journal-title":"J. Differ. Equ. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1007\/s40819-020-00842-1","article-title":"Valanarasu, T; Ramesh Babu, A. A System of Singularly Perturbed Periodic Boundary Value Problem: Hybrid Difference Scheme","volume":"6","author":"Puvaneswari","year":"2020","journal-title":"Int. J. Appl. Comput. Math."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Raja, V., Geetha, N., Mahendran, R., and Senthilkumar, L.S. (2024). Numerical solution for third order singularly perturbed turning point problems with integral boundary condition. J. Appl. Math. Comput., 1\u201321.","DOI":"10.1007\/s12190-024-02266-2"},{"key":"ref_7","unstructured":"Chandru, M., and Shanthi, V. (2014). A boundary value technique for singularly perturbed boundary value problem of reaction-diffusion with non-smooth data. J. Eng. Sci. Technol. Spec. Issue ICMTEA2013 Conf., 32\u201345."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"695","DOI":"10.1016\/S0898-1221(03)90135-0","article-title":"A uniformly convergence difference method for the periodical boundary value problem","volume":"46","author":"Amiraliyev","year":"2003","journal-title":"Int. J. Comput. Math. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"196","DOI":"10.1080\/00207160903370172","article-title":"Uniformly convergent second-order difference scheme for a singularly perturbed periodical boundary value problem","volume":"88","author":"Cen","year":"2011","journal-title":"Int. J. Comput. Math."},{"key":"ref_10","first-page":"157","article-title":"Cubic spline scheme on variable mesh for singularly perturbed periodical boundary value problem","volume":"50","author":"Puvaneswari","year":"2020","journal-title":"Novi Sad J. Math."},{"key":"ref_11","first-page":"457","article-title":"A survey of numerical techniques for solving singularly perturbed ordinary differential equations","volume":"130","author":"Kadalbajoo","year":"2002","journal-title":"Appl. Math. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"913","DOI":"10.1016\/j.cpc.2011.12.017","article-title":"Quintic B-spline collocation method for second order mixed boundary value problem","volume":"183","author":"Lang","year":"2012","journal-title":"Comput. Phys. Commun."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"334","DOI":"10.1016\/j.apnum.2023.01.020","article-title":"Uniformly convergent scheme for fourth-order singularly perturbed convection-diffusion ODE","volume":"186","author":"Singh","year":"2023","journal-title":"Appl. Numer. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1872","DOI":"10.1007\/s10910-022-01393-0","article-title":"Spline-based parameter-uniform scheme for fourth-order singularly perturbed differential equations","volume":"60","author":"Singh","year":"2022","journal-title":"J. Math. Chem."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Yousaf, M.Z., Srivastava, H.M., Abbas, M., Nazir, T., Mohammed, P.O., Vivas-Cortez, M., and Chorfi, N. (2023). A Novel quintic B-spline technique for numerical solutions of the fourth-order singular singularly-perturbed problems. Symmetry, 15.","DOI":"10.3390\/sym15101929"},{"key":"ref_16","first-page":"74","article-title":"Quintic B-Spline Galerkin method for fifth order boundary value problems","volume":"5","author":"Viswanadham","year":"2010","journal-title":"ARPN J. Eng. Appl. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"541","DOI":"10.1007\/s40010-021-00759-4","article-title":"Two-parameter singular perturbation boundary value problems via quintic B-spline method","volume":"92","author":"Mishra","year":"2022","journal-title":"Proc. Natl. Acad. Sci. India Sect. A Phys. Sci."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Micula, G. (1999). Handbook of Splines, Kluwer Academic Publishers.","DOI":"10.1007\/978-94-011-5338-6"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"865","DOI":"10.1080\/00207160.2018.1458098","article-title":"A parameter-uniform method for singularly perturbed turning point problems exhibiting interior or twin boundary layers","volume":"96","author":"Kumar","year":"2019","journal-title":"Int. J. Comput. Math."},{"key":"ref_20","first-page":"713","article-title":"Comparative study of singularly perturbed two-point BVPs via: Fitted mesh finite difference method, B-spline collocation method","volume":"204","author":"Kadalbajoo","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Puvaneswari, A., and Valanarasu, T. (2024). Spline approximation methods for second order singularly perturbed convection-diffusion equation with integral boundary condition. Indian J. Pure Appl. Math., 1\u201312.","DOI":"10.1007\/s13226-024-00692-3"},{"key":"ref_22","first-page":"473","article-title":"An asymptotic numerical method for singularly perturbed fourth order ODE of convection-diffusion type turning point problem","volume":"24","author":"Chandru","year":"2016","journal-title":"Neural Parallel Sci. Comput."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"De Boor, C. (1978). A Practical Guide to Splines, Springer.","DOI":"10.1007\/978-1-4612-6333-3"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Hall, C.A. (1968). On error bounds for spline interpolation. J. Approx. Theory I, 209\u2013218.","DOI":"10.1016\/0021-9045(68)90025-7"},{"key":"ref_25","first-page":"248","article-title":"\u03b5-Uniform fitted mesh finite difference methods for general singular perturbation problems","volume":"179","author":"Kadalbajoo","year":"2006","journal-title":"Appl. Math. Comput."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"495","DOI":"10.14317\/jami.2016.495","article-title":"A Schwarz method for fourth-order singularly perturbed reaction-diffusion problem with discontinuous source term","volume":"34","author":"Chandru","year":"2016","journal-title":"J. Appl. Math. Inform."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/1\/73\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:32:06Z","timestamp":1759919526000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/1\/73"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,20]]},"references-count":26,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["axioms14010073"],"URL":"https:\/\/doi.org\/10.3390\/axioms14010073","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1,20]]}}}