{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,8,2]],"date-time":"2025-08-02T18:17:28Z","timestamp":1754158648999,"version":"3.41.2"},"reference-count":43,"publisher":"Emerald","issue":"2","license":[{"start":{"date-parts":[[2020,8,6]],"date-time":"2020-08-06T00:00:00Z","timestamp":1596672000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.emerald.com\/insight\/site-policies"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["EC"],"published-print":{"date-parts":[[2021,2,8]]},"abstract":"<jats:sec>\n<jats:title content-type=\"abstract-subheading\">Purpose<\/jats:title>\n<jats:p>The purpose of this paper is to design and analyze a robust numerical method for a coupled system of singularly perturbed parabolic delay partial differential equations (PDEs).<\/jats:p>\n<\/jats:sec>\n<jats:sec>\n<jats:title content-type=\"abstract-subheading\">Design\/methodology\/approach<\/jats:title>\n<jats:p>Some a priori bounds on the regular and layer parts of the solution and their derivatives are derived. Based on these a priori bounds, appropriate layer adapted meshes of Shishkin and generalized Shishkin types are defined in the spatial direction. After that, the problem is discretized using an implicit Euler scheme on a uniform mesh in the time direction and the central difference scheme on layer adapted meshes of Shishkin and generalized Shishkin types in the spatial direction.<\/jats:p>\n<\/jats:sec>\n<jats:sec>\n<jats:title content-type=\"abstract-subheading\">Findings<\/jats:title>\n<jats:p>The method is proved to be robust convergent of almost second-order in space and first-order in time. Numerical results are presented to support the theoretical error bounds.<\/jats:p>\n<\/jats:sec>\n<jats:sec>\n<jats:title content-type=\"abstract-subheading\">Originality\/value<\/jats:title>\n<jats:p>A coupled system of singularly perturbed parabolic delay PDEs is considered and some a priori bounds are derived. A numerical method is developed for the problem, where appropriate layer adapted Shishkin and generalized Shishkin meshes are considered. Error analysis of the method is given for both Shishkin and generalized Shishkin meshes.<\/jats:p>\n<\/jats:sec>","DOI":"10.1108\/ec-04-2020-0191","type":"journal-article","created":{"date-parts":[[2020,8,6]],"date-time":"2020-08-06T11:45:27Z","timestamp":1596714327000},"page":"964-988","source":"Crossref","is-referenced-by-count":2,"title":["A robust numerical method for a coupled system of singularly perturbed parabolic delay problems"],"prefix":"10.1108","volume":"38","author":[{"given":"Mukesh","family":"Kumar","sequence":"first","affiliation":[]},{"given":"Joginder","family":"Singh","sequence":"additional","affiliation":[]},{"given":"Sunil","family":"Kumar","sequence":"additional","affiliation":[]},{"given":"Aakansha","family":"Aakansha","sequence":"additional","affiliation":[]}],"member":"140","published-online":{"date-parts":[[2020,8,6]]},"reference":[{"issue":"4","key":"key2021043009431623600_ref001","doi-asserted-by":"crossref","first-page":"663","DOI":"10.1007\/s11075-009-9306-z","article-title":"A finite difference scheme for a class of singularly perturbed initial value problems for delay differential equations","volume":"52","year":"2009","journal-title":"Numerical Algorithms"},{"issue":"1","key":"key2021043009431623600_ref002","doi-asserted-by":"crossref","first-page":"552","DOI":"10.1016\/j.cam.2006.05.032","article-title":"A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations","volume":"205","year":"2007","journal-title":"Journal of Computational and Applied Mathematics"},{"issue":"6","key":"key2021043009431623600_ref003","doi-asserted-by":"crossref","first-page":"1475","DOI":"10.1016\/j.camwa.2010.11.010","article-title":"A fitted numerical method for a system of partial delay differential equations","volume":"61","year":"2011","journal-title":"Computers and Mathematics with Applications"},{"issue":"9","key":"key2021043009431623600_ref004","doi-asserted-by":"crossref","first-page":"4728","DOI":"10.1016\/j.amc.2010.11.028","article-title":"A novel fitter operator finite difference method for a singularly perturbed delay parabolic partial differential equation","volume":"217","year":"2011","journal-title":"Applied Mathematics and Computation"},{"issue":"3","key":"key2021043009431623600_ref005","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1007\/BF00276489","article-title":"Interaction of spatial diffusion and delays in models of genetic control by repression","volume":"22","year":"1985","journal-title":"Journal of Mathematical Biology"},{"issue":"1","key":"key2021043009431623600_ref006","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1080\/00207160801989875","article-title":"A second-order finite difference scheme for a class of singularly perturbed delay differential equations","volume":"87","year":"2010","journal-title":"International Journal of Computer Mathematics"},{"key":"key2021043009431623600_ref007","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/j.amc.2014.05.081","article-title":"An improved uniformly convergent scheme in space for 1d parabolic reaction\u2013diffusion systems","volume":"243","year":"2014","journal-title":"Applied Mathematics and Computation"},{"issue":"3","key":"key2021043009431623600_ref008","doi-asserted-by":"crossref","first-page":"490","DOI":"10.1080\/00207160.2017.1290439","article-title":"Second-order uniformly convergent numerical method for singularly perturbed delay parabolic partial differential equations","volume":"95","year":"2018","journal-title":"International Journal of Computer Mathematics"},{"key":"key2021043009431623600_ref009","first-page":"65","article-title":"A singular transport equation modelling a proliferating maturity structured cell population","volume":"4","year":"1996","journal-title":"Canadian Applied Mathematics Quarterly"},{"issue":"1","key":"key2021043009431623600_ref010","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1007\/s11075-011-9480-7","article-title":"Fitted finite difference method for singularly perturbed delay differential equations","volume":"59","year":"2012","journal-title":"Numerical Algorithms"},{"volume-title":"Robust Computational Techniques for Boundary Layers","year":"2000","key":"key2021043009431623600_ref011"},{"key":"key2021043009431623600_ref012","first-page":"178","article-title":"Second order parameter-uniform convergence for a finite difference method for a singularly perturbed linear parabolic system","volume":"10","year":"2013","journal-title":"International Journal of Numerical Analysis and Modeling"},{"volume-title":"Introduction to Time-Delay Systems: Analysis and Control","year":"2014","key":"key2021043009431623600_ref013"},{"issue":"1","key":"key2021043009431623600_ref014","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1108\/EC-03-2019-0115","article-title":"Numerical analysis and simulation of delay parabolic partial differential equation involving a small parameter","volume":"37","year":"2019","journal-title":"Engineering Computations"},{"issue":"1","key":"key2021043009431623600_ref015","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.cam.2006.06.005","article-title":"A uniformly convergent scheme for a system of reaction-diffusion equations","volume":"206","year":"2007","journal-title":"Journal of Computational and Applied Mathematics"},{"issue":"3","key":"key2021043009431623600_ref016","doi-asserted-by":"crossref","first-page":"474","DOI":"10.1080\/00207160.2018.1432856","article-title":"A parameter-uniform higher order finite difference scheme for singularly perturbed time-dependent parabolic problem with two small parameters","volume":"96","year":"2019","journal-title":"International Journal of Computer Mathematics"},{"issue":"1","key":"key2021043009431623600_ref017","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1002\/num.22203","article-title":"Higher order numerical approximation for time dependent singularly perturbed differential-difference convection-diffusion equations","volume":"34","year":"2018","journal-title":"Numerical Methods for Partial Differential Equations"},{"key":"key2021043009431623600_ref018","first-page":"339","article-title":"Partial neutral functional differential equations","volume":"39","year":"1994","journal-title":"Revue Roumaine de Math\u00e9matiques Pures et Appliqu\u00e9es"},{"issue":"1","key":"key2021043009431623600_ref019","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1002\/num.22421","article-title":"Convergence and stability estimates in difference setting for time-fractional parabolic equations with functional delay","volume":"36","year":"2020","journal-title":"Numerical Methods for Partial Differential Equations"},{"key":"key2021043009431623600_ref020","first-page":"1047","article-title":"A functional reaction-diffusion equation from climate modeling: S-shapedness of the principal branch of fixed points of the time 1 map","volume":"8","year":"1995","journal-title":"Differential Integral Equations"},{"issue":"8","key":"key2021043009431623600_ref021","doi-asserted-by":"crossref","first-page":"3641","DOI":"10.1016\/j.amc.2010.09.059","article-title":"A brief survey on numerical methods for solving singularly perturbed problems","volume":"217","year":"2010","journal-title":"Applied Mathematics and Computation"},{"volume-title":"Delay Differential Equations with Applications in Population Dynamics","year":"1993","key":"key2021043009431623600_ref022"},{"issue":"10","key":"key2021043009431623600_ref023","doi-asserted-by":"crossref","first-page":"1355","DOI":"10.1016\/j.camwa.2014.09.004","article-title":"High order parameter-uniform discretization for singularly perturbed parabolic partial differential equations with time delay","volume":"68","year":"2014","journal-title":"Computers and Mathematics with Applications"},{"issue":"2","key":"key2021043009431623600_ref024","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1007\/s11075-016-0258-9","article-title":"A second order uniformly convergent numerical scheme for parameterized singularly perturbed delay differential problems","volume":"76","year":"2017","journal-title":"Numerical Algorithms"},{"volume-title":"Linear and Quasilinear Equation of Parabolic Type: Translations of Mathematical Monographs","year":"1968","key":"key2021043009431623600_ref025"},{"issue":"1","key":"key2021043009431623600_ref026","first-page":"121","article-title":"Accurate solution of a system of coupled singularly perturbed reaction-diffusion equations","volume":"1","year":"2004","journal-title":"Computing"},{"issue":"4300","key":"key2021043009431623600_ref027","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1126\/science.267326","article-title":"Oscillation and chaos in physiological control systems","volume":"197","year":"1977","journal-title":"Science"},{"key":"key2021043009431623600_ref028","first-page":"1","article-title":"Multistability and boundary layer development in a transport equation with retarded arguments","volume":"1","year":"1993","journal-title":"Canadian Applied Mathematics Quarterly"},{"issue":"4","key":"key2021043009431623600_ref029","doi-asserted-by":"crossref","first-page":"627","DOI":"10.1093\/imanum\/23.4.627","article-title":"A uniformly convergent numerical method for a coupled system of two singularly perturbed linear reaction-diffusion problems","volume":"23","year":"2003","journal-title":"IMA Journal of Numerical Analysis"},{"issue":"1","key":"key2021043009431623600_ref030","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1007\/BF00275860","article-title":"Models of genetic control by repression with time delays and spatial effects","volume":"20","year":"1984","journal-title":"Journal of Mathematical Biology"},{"volume-title":"Fitted Numerical Methods for Singular Perturbation Problems","year":"1996","key":"key2021043009431623600_ref031"},{"issue":"6","key":"key2021043009431623600_ref032","doi-asserted-by":"crossref","first-page":"2407","DOI":"10.1002\/num.22420","article-title":"A fitted numerical method for parabolic turning point singularly perturbed problems with an interior layer","volume":"35","year":"2019","journal-title":"Numerical Methods for Partial Differential Equations"},{"issue":"176","key":"key2021043009431623600_ref033","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1090\/S0025-5718-1986-0856702-7","article-title":"A uniformly accurate finite-element method for a singularly perturbed one-dimensional reaction-diffusion problem","volume":"47","year":"1986","journal-title":"Mathematics of Computation"},{"key":"key2021043009431623600_ref034","doi-asserted-by":"crossref","first-page":"266","DOI":"10.1016\/j.apnum.2017.03.013","article-title":"BDF-type shifted Chebyshev approximation scheme for fractional functional differential equations with delay and its error analysis","volume":"118","year":"2017","journal-title":"Applied Numerical Mathematics"},{"issue":"3","key":"key2021043009431623600_ref035","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1515\/amcs-2017-0033","article-title":"A numerical solution for a class of time fractional diffusion equations with delay","volume":"27","year":"2017","journal-title":"International Journal of Applied Mathematics and Computer Science"},{"key":"key2021043009431623600_ref036","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1016\/j.cam.2016.02.039","article-title":"On a class of non-linear delay distributed order fractional diffusion equations","volume":"318","year":"2017","journal-title":"Journal of Computational and Applied Mathematics"},{"key":"key2021043009431623600_ref037","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s11075-019-00815-6","article-title":"Numerical approximation for a class of singularly perturbed delay differential equations with boundary and interior layer(s)","year":"2019","journal-title":"Numerical Algorithms"},{"key":"key2021043009431623600_ref038","article-title":"Robust numerical methods for singularly perturbed differential equations","volume-title":"Springer Series in Computational Mathematics","year":"2008","edition":"2nd ed."},{"issue":"5","key":"key2021043009431623600_ref039","doi-asserted-by":"crossref","first-page":"1849","DOI":"10.1002\/num.22256","article-title":"A domain decomposition method for solving singularly perturbed parabolic reaction-diffusion problems with time delay","volume":"34","year":"2018","journal-title":"Numerical Methods for Partial Differential Equations"},{"key":"key2021043009431623600_ref040","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1007\/s006070170002","article-title":"A high-order scheme for quasilinear boundary value problems with two small parameters","volume":"67","year":"2001","journal-title":"Computing"},{"issue":"3","key":"key2021043009431623600_ref041","doi-asserted-by":"crossref","first-page":"2413","DOI":"10.1007\/s10915-019-01091-1","article-title":"Asymptotic stability of compact and linear \u03b8-methods for space fractional delay generalized diffusion equation","volume":"81","year":"2019","journal-title":"Journal of Scientific Computing"},{"key":"key2021043009431623600_ref042","first-page":"1","article-title":"Higher-order linearized multistep finite difference methods for non-Fickian delay reaction-diffusion equations","volume":"14","year":"2017","journal-title":"International Journal of Numerical Analysis and Modeling"},{"key":"key2021043009431623600_ref043","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1016\/j.cam.2016.04.016","article-title":"The compact and crank\u2013Nicolson ADI schemes for two-dimensional semilinear multidelay parabolic equations","volume":"306","year":"2016","journal-title":"Journal of Computational and Applied Mathematics"}],"container-title":["Engineering Computations"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.emerald.com\/insight\/content\/doi\/10.1108\/EC-04-2020-0191\/full\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.emerald.com\/insight\/content\/doi\/10.1108\/EC-04-2020-0191\/full\/html","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,7,25]],"date-time":"2025-07-25T01:19:24Z","timestamp":1753406364000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.emerald.com\/ec\/article\/38\/2\/964-988\/446183"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,6]]},"references-count":43,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,8,6]]},"published-print":{"date-parts":[[2021,2,8]]}},"alternative-id":["10.1108\/EC-04-2020-0191"],"URL":"https:\/\/doi.org\/10.1108\/ec-04-2020-0191","relation":{},"ISSN":["0264-4401","0264-4401"],"issn-type":[{"type":"print","value":"0264-4401"},{"type":"print","value":"0264-4401"}],"subject":[],"published":{"date-parts":[[2020,8,6]]}}}