{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:36:10Z","timestamp":1776785770450,"version":"3.51.2"},"reference-count":17,"publisher":"American Mathematical Society (AMS)","issue":"255","license":[{"start":{"date-parts":[[2007,4,3]],"date-time":"2007-04-03T00:00:00Z","timestamp":1175558400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>In this paper, parameter-uniform numerical methods for a class of singularly perturbed parabolic partial differential equations with two small parameters on a rectangular domain are studied. Parameter-explicit theoretical bounds on the derivatives of the solutions are derived. The solution is decomposed into a sum of regular and singular components. A numerical algorithm based on an upwind finite difference operator and an appropriate piecewise uniform mesh is constructed. Parameter-uniform error bounds for the numerical approximations are established. Numerical results are given to illustrate the parameter-uniform convergence of the numerical approximations.<\/p>","DOI":"10.1090\/s0025-5718-06-01846-1","type":"journal-article","created":{"date-parts":[[2006,5,24]],"date-time":"2006-05-24T14:43:01Z","timestamp":1148481781000},"page":"1135-1154","source":"Crossref","is-referenced-by-count":71,"title":["Parameter-uniform finite difference schemes for singularly perturbed parabolic diffusion-convection-reaction problems"],"prefix":"10.1090","volume":"75","author":[{"given":"E.","family":"O\u2019Riordan","sequence":"first","affiliation":[]},{"given":"M.","family":"Pickett","sequence":"additional","affiliation":[]},{"given":"G.","family":"Shishkin","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2006,4,3]]},"reference":[{"key":"1","series-title":"Applied Mathematics (Boca Raton)","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1201\/9781482285727","volume-title":"Robust computational techniques for boundary layers","volume":"16","author":"Farrell, P. 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Shishkin, Discrete approximation of singularly perturbed elliptic and parabolic equations, Russian Academy of Sciences,Ural Section, Ekaterinburg.(1992)"},{"key":"13","unstructured":"G. I. Shishkin and V. A. Titov, A difference scheme for a differential equation with two small parameters at the derivatives (Russian), Chisl. Metody Meh. Sploshn. Sredy, (1976), 7 (2), 145\u2013155."},{"issue":"5","key":"14","first-page":"1066","article-title":"A difference scheme for a singularly perturbed equation of parabolic type with a discontinuous initial condition","volume":"300","author":"Shishkin, G. I.","year":"1988","journal-title":"Dokl. Akad. 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