{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T18:35:32Z","timestamp":1769020532734,"version":"3.49.0"},"reference-count":18,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,9,7]],"date-time":"2025-09-07T00:00:00Z","timestamp":1757203200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Singularly perturbed integro-partial differential equations with reaction\u00e2\u20ac\u201cdiffusion behavior present significant challenges due to boundary layers arising from small perturbation parameters, which complicate the development of accurate and efficient numerical schemes for physical and engineering models. In this study, a uniformly convergent higher-order method is proposed to address these challenges. The approach applies the implicit Euler method for temporal discretization on a uniform mesh and central differences on a Shishkin mesh for spatial approximation, and utilizes the trapezoidal rule for evaluating integral terms; further, extrapolation techniques are incorporated in both time and space to increase accuracy. Numerical analysis demonstrates that the base scheme achieves first-order convergence, while extrapolation enhances convergence rates to second-order in time and fourth-order in space. Theoretical results confirm uniform convergence with respect to the perturbation parameter, and comprehensive numerical experiments validate these analytical claims. Findings indicate that the proposed scheme is reliable, efficient, and particularly effective in attaining fourth-order spatial accuracy when solving singularly perturbed integro-partial differential equations of reaction\u00e2\u20ac\u201cdiffusion type, thus providing a robust numerical tool for complex applications in science and engineering.<\/jats:p>","DOI":"10.3390\/sym17091475","type":"journal-article","created":{"date-parts":[[2025,9,8]],"date-time":"2025-09-08T08:06:32Z","timestamp":1757318792000},"page":"1475","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Stable and Convergent High-Order Numerical Schemes for Parabolic Integro-Differential Equations with Small Coefficients"],"prefix":"10.3390","volume":"17","author":[{"given":"Lolugu","family":"Govindarao","sequence":"first","affiliation":[{"name":"Amrita School of Physical Science, Amrita Vishwa Vidyapeetham, Coimbatore 641112, Tamilnadu, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4392-0742","authenticated-orcid":false,"given":"Khalil S.","family":"Al-Ghafri","sequence":"additional","affiliation":[{"name":"Mathematics and Computing Skills Unit, University of Technology and Applied Sciences, P.O. Box 466, Ibri 516, Oman"}]},{"given":"Jugal","family":"Mohapatra","sequence":"additional","affiliation":[{"name":"Department of Mathematics, National Institute of Technology Rourkela, Rourkela 769008, Odisha, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9779-2371","authenticated-orcid":false,"given":"Th\u0227i Anh","family":"Nhan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Menlo College, 1000 El Camino Real, Atherton, CA 94027, USA"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Miller, J.J.H., O\u2019Riordan, E., and Shishkin, G.I. (1996). Fitted Numerical Methods for Singular Perturbation Problems, World Scientific.","DOI":"10.1142\/2933"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Shishkin, G.I., and Shishkina, L.P. (2009). Difference Methods for Singular Perturbation Problems, Chapman and Hall. 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Notes"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"167","DOI":"10.3846\/mma.2018.011","article-title":"Parameter-uniform improved hybrid numerical scheme for singularly perturbed problems with interior layers","volume":"2","author":"Mukherjee","year":"2018","journal-title":"Math. Model. Anal."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1007\/s12591-015-0265-7","article-title":"Numerical treatment for the class of time dependent singularly perturbed parabolic problems with general shift arguments","volume":"25","author":"Bansal","year":"2015","journal-title":"Differ. Equ. Dyn. Syst."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1002\/num.20030","article-title":"Higher order numerical methods for one-dimensional parabolic singularly perturbed problems with regular layers","volume":"21","author":"Clavero","year":"2005","journal-title":"Numer. Methods Part. Differ. 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