{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,2]],"date-time":"2026-04-02T17:44:05Z","timestamp":1775151845849,"version":"3.50.1"},"reference-count":42,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2021,12,27]],"date-time":"2021-12-27T00:00:00Z","timestamp":1640563200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2021,12,27]],"date-time":"2021-12-27T00:00:00Z","timestamp":1640563200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001412","name":"Council of Scientific and Industrial Research, India","doi-asserted-by":"publisher","award":["09\/1217(0035)\/2018-EMR-I"],"award-info":[{"award-number":["09\/1217(0035)\/2018-EMR-I"]}],"id":[{"id":"10.13039\/501100001412","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2022,2]]},"DOI":"10.1007\/s40314-021-01733-x","type":"journal-article","created":{"date-parts":[[2021,12,27]],"date-time":"2021-12-27T19:02:32Z","timestamp":1640631752000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["An efficient numerical method for coupled systems of singularly perturbed parabolic delay problems"],"prefix":"10.1007","volume":"41","author":[{"family":"Aakansha","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sunil","family":"Kumar","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Joginder","family":"Singh","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,12,27]]},"reference":[{"key":"1733_CR1","doi-asserted-by":"publisher","first-page":"1221","DOI":"10.1016\/j.aml.2010.06.002","volume":"23","author":"I Amiraliyeva","year":"2010","unstructured":"Amiraliyeva I, Erdogan F, Amiraliyev G (2010) A uniform numerical method for dealing with a singularly perturbed delay initial value problem. Appl Math Lett 23:1221\u20131225","journal-title":"Appl Math Lett"},{"key":"1733_CR2","doi-asserted-by":"publisher","first-page":"552","DOI":"10.1016\/j.cam.2006.05.032","volume":"205","author":"A Ansari","year":"2007","unstructured":"Ansari A, Bakr S, Shishkin G (2007) A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations. J Comput Appl Math 205:552\u2013566","journal-title":"J Comput Appl Math"},{"key":"1733_CR3","doi-asserted-by":"publisher","first-page":"1475","DOI":"10.1016\/j.camwa.2010.11.010","volume":"61","author":"EB Bashier","year":"2011","unstructured":"Bashier EB, Patidar KC (2011a) A fitted numerical method for a system of partial delay differential equations. Comput Math Appl 61:1475\u20131492","journal-title":"Comput Math Appl"},{"key":"1733_CR4","first-page":"4728","volume":"217","author":"EB Bashier","year":"2011","unstructured":"Bashier EB, Patidar KC (2011b) A novel fitted operator finite difference method for a singularly perturbed delay parabolic partial differential equation. Appl Math Comput 217:4728\u20134739","journal-title":"Appl Math Comput"},{"key":"1733_CR5","doi-asserted-by":"publisher","first-page":"313","DOI":"10.1007\/BF00276489","volume":"22","author":"S Busenberg","year":"1985","unstructured":"Busenberg S, Mahaffy J (1985) Interaction of spatial diffusion and delays in models of genetic control by repression. J Math Biol 22:313\u2013333","journal-title":"J Math Biol"},{"key":"1733_CR6","doi-asserted-by":"publisher","first-page":"173","DOI":"10.1080\/00207160801989875","volume":"87","author":"Z Cen","year":"2010","unstructured":"Cen Z (2010) A second-order finite difference scheme for a class of singularly perturbed delay differential equations. Int J Comput Math 87:173\u2013185","journal-title":"Int J Comput Math"},{"key":"1733_CR7","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1007\/BF01048789","volume":"1","author":"SN Chow","year":"1989","unstructured":"Chow SN, Lin XB, Mallet-Paret J (1989) Transition layers for singularly perturbed delay differential equations with monotone nonlinearities. J Dyn Differ Equ 1:3\u201343","journal-title":"J Dyn Differ Equ"},{"key":"1733_CR8","first-page":"57","volume":"243","author":"C Clavero","year":"2014","unstructured":"Clavero C, Gracia J (2014) An improved uniformly convergent scheme in space for 1D parabolic reaction-diffusion systems. Appl Math Comput 243:57\u201373","journal-title":"Appl Math Comput"},{"key":"1733_CR9","doi-asserted-by":"publisher","first-page":"465","DOI":"10.1007\/s11075-018-0557-4","volume":"81","author":"P Das","year":"2019","unstructured":"Das P (2019) An a posteriori based convergence analysis for a nonlinear singularly perturbed system of delay differential equations on an adaptive mesh. Numer Algorithms 81:465\u2013487","journal-title":"Numer Algorithms"},{"key":"1733_CR10","first-page":"65","volume":"4","author":"J Dyson","year":"1996","unstructured":"Dyson J, Bressan RV, Webb GF (1996) A singular transport equation modelling a proliferating maturity structured cell population. Can Appl Math Q 4:65\u201395","journal-title":"Can Appl Math Q"},{"key":"1733_CR11","doi-asserted-by":"publisher","first-page":"382","DOI":"10.1016\/j.cam.2017.11.017","volume":"333","author":"F Erdogan","year":"2018","unstructured":"Erdogan F, Cen Z (2018) A uniformly almost second order convergent numerical method for singularly perturbed delay differential equations. J Comput Appl Math 333:382\u2013394","journal-title":"J Comput Appl Math"},{"key":"1733_CR12","doi-asserted-by":"publisher","DOI":"10.1201\/9781482285727","volume-title":"Robust computational techniques for boundary layers","author":"PA Farrell","year":"2000","unstructured":"Farrell PA, Hegarty AF, Miller JJH, O\u2019Riordan E, Shishkin GI (2000) Robust computational techniques for boundary layers. Chapman & Hall, Boca Raton"},{"key":"1733_CR13","first-page":"178","volume":"10","author":"V Franklin","year":"2010","unstructured":"Franklin V, Paramasivam M, Valarmathi S, Miller J (2010) Second order parameter-uniform convergence for a finite difference method for a singularly perturbed linear parabolic system. Int J Numer Anal Model 10:178\u2013202","journal-title":"Int J Numer Anal Model"},{"key":"1733_CR14","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1023\/A:1004651430364","volume":"106","author":"V Glizer","year":"2000","unstructured":"Glizer V (2000) Asymptotic solution of a boundary-value problem for linear singularly-perturbed functional differential equations arising in optimal control theory. J Optim Theory Appl 106:309\u2013335","journal-title":"J Optim Theory Appl"},{"key":"1733_CR15","doi-asserted-by":"publisher","first-page":"655","DOI":"10.1016\/j.camwa.2013.12.009","volume":"67","author":"JL Gracia","year":"2014","unstructured":"Gracia JL, Clavero C (2014) \u2018Uniformly convergent additive finite difference schemes for singularly perturbed parabolic reaction\u2013diffusion systems. Comput Math Appl 67:655\u2013670","journal-title":"Comput Math Appl"},{"key":"1733_CR16","first-page":"339","volume":"39","author":"JK Hale","year":"1994","unstructured":"Hale JK (1994) Partial neutral functional differential equations. Rev Roumain Math Pures Appl 39:339\u2013344","journal-title":"Rev Roumain Math Pures Appl"},{"key":"1733_CR17","doi-asserted-by":"publisher","first-page":"118","DOI":"10.1002\/num.22421","volume":"36","author":"AS Hendy","year":"2020","unstructured":"Hendy AS, Pimenov VG, Mac\u00edas-D\u00edaz JE (2020) Convergence and stability estimates in difference setting for time-fractional parabolic equations with functional delay. Numer Methods Partial Differ Equ 36:118\u2013132","journal-title":"Numer Methods Partial Differ Equ"},{"key":"1733_CR18","first-page":"1","volume":"1996","author":"G Hetzer","year":"1996","unstructured":"Hetzer G (1996) Global existence, uniqueness, and continuous dependence for a reaction\u2013diffusion equation with memory. Electron J Differ Equ 1996:1\u201316","journal-title":"Electron J Differ Equ"},{"key":"1733_CR19","first-page":"371","volume":"134","author":"MK Kadalbajoo","year":"2003","unstructured":"Kadalbajoo MK, Patidar KC (2003) Singularly perturbed problems in partial differential equations: a survey. Appl Math Comput 134:371\u2013429","journal-title":"Appl Math Comput"},{"key":"1733_CR20","volume-title":"Delay differential equations with applications in population dynamics","author":"Y Kuang","year":"1993","unstructured":"Kuang Y (1993) Delay differential equations with applications in population dynamics, vol 191. Academic Press, London"},{"key":"1733_CR21","doi-asserted-by":"publisher","first-page":"1355","DOI":"10.1016\/j.camwa.2014.09.004","volume":"68","author":"S Kumar","year":"2014","unstructured":"Kumar S, Kumar M (2014) High order parameter-uniform discretization for singularly perturbed parabolic partial differential equations with time delay. Comput Math Appl 68:1355\u20131367","journal-title":"Comput Math Appl"},{"key":"1733_CR22","first-page":"20","volume":"20","author":"M Kumar","year":"2020","unstructured":"Kumar M, Singh J, Kumar S, Aakansha G (2020) A robust numerical method for a coupled system of singularly perturbed parabolic delay problems. Eng Comput 20:20","journal-title":"Eng Comput"},{"key":"1733_CR23","first-page":"1","volume":"20","author":"K Kumar","year":"2021","unstructured":"Kumar K, Podila P, Das P, Ramos H (2021) A graded mesh refinement approach for boundary layer originated singularly perturbed time-delayed parabolic convection diffusion problems. Math Methods Appl Sci 20:1\u201319","journal-title":"Math Methods Appl Sci"},{"key":"1733_CR24","doi-asserted-by":"publisher","DOI":"10.1090\/mmono\/023","volume-title":"Linear and quasilinear equation of parabolic type: translations of mathematical monographs","author":"OA Ladyzhenskaya","year":"1968","unstructured":"Ladyzhenskaya OA, Solonnikov VA, Uraltseva NN (1968) Linear and quasilinear equation of parabolic type: translations of mathematical monographs. American Mathematical Society, USA"},{"key":"1733_CR25","doi-asserted-by":"publisher","DOI":"10.1142\/3302","volume-title":"Second order parabolic differential equations","author":"GM Lieberman","year":"1996","unstructured":"Lieberman GM (1996) Second order parabolic differential equations. World Scientific, Singapore"},{"key":"1733_CR26","doi-asserted-by":"publisher","first-page":"605","DOI":"10.1007\/BF02459969","volume":"51","author":"A Longtin","year":"1989","unstructured":"Longtin A, Milton JG (1989) Modelling autonomous oscillations in the human pupil light reflex using non-linear delay-differential equations. Bull Math Biol 51:605\u2013624","journal-title":"Bull Math Biol"},{"key":"1733_CR27","doi-asserted-by":"publisher","first-page":"287","DOI":"10.1126\/science.267326","volume":"197","author":"MC Mackey","year":"1977","unstructured":"Mackey MC, Glass L (1977) Oscillation and chaos in physiological control systems. Science 197:287\u2013289","journal-title":"Science"},{"key":"1733_CR28","first-page":"1","volume":"1","author":"MC Mackey","year":"1993","unstructured":"Mackey MC, Rey A (1993) Multistability and boundary layer development in a transport equation with retarded arguments. Can Appl Math Q 1:1\u201321","journal-title":"Can Appl Math Q"},{"key":"1733_CR29","doi-asserted-by":"publisher","first-page":"627","DOI":"10.1093\/imanum\/23.4.627","volume":"23","author":"N Madden","year":"2003","unstructured":"Madden N, Stynes M (2003) A uniformly convergent numerical method for a coupled system of two singularly perturbed linear reaction\u2013diffusion problems. IMA J Numer Anal 23:627\u2013644","journal-title":"IMA J Numer Anal"},{"key":"1733_CR30","doi-asserted-by":"publisher","first-page":"519","DOI":"10.1016\/0025-5564(88)90081-8","volume":"90","author":"JM Mahaffy","year":"1988","unstructured":"Mahaffy JM (1988) Genetic control models with diffusion and delays. Math Biosci 90:519\u2013533","journal-title":"Math Biosci"},{"key":"1733_CR31","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1007\/BF01790539","volume":"145","author":"J Mallet-Paret","year":"1986","unstructured":"Mallet-Paret J, Nussbaum RD (1986) Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation. Ann Mat 145:33\u2013128","journal-title":"Ann Mat"},{"key":"1733_CR32","doi-asserted-by":"publisher","DOI":"10.1142\/2933","volume-title":"Fitted numerical methods for singular perturbation problems","author":"JJH Miller","year":"1996","unstructured":"Miller JJH, O\u2019Riordan E, Shishkin GI (1996) Fitted numerical methods for singular perturbation problems. World Scientific, Singapore"},{"key":"1733_CR33","doi-asserted-by":"publisher","first-page":"266","DOI":"10.1016\/j.apnum.2017.03.013","volume":"118","author":"VG Pimenov","year":"2017","unstructured":"Pimenov VG, Hendy AS (2017) BDF-type shifted Chebyshev approximation scheme for fractional functional differential equations with delay and its error analysis. Appl Numer Math 118:266\u2013276","journal-title":"Appl Numer Math"},{"key":"1733_CR34","doi-asserted-by":"publisher","first-page":"433","DOI":"10.1016\/j.cam.2016.02.039","volume":"318","author":"VG Pimenov","year":"2017","unstructured":"Pimenov VG, Hendy AS, De Staelen RH (2017) On a class of non-linear delay distributed order fractional diffusion equations. J Comput Appl Math 318:433\u2013443","journal-title":"J Comput Appl Math"},{"issue":"3\u20134","key":"1733_CR35","doi-asserted-by":"publisher","first-page":"603","DOI":"10.1080\/00207160701199773","volume":"85","author":"H Ramos","year":"2008","unstructured":"Ramos H, Vigo-Aguiar J (2008) A new algorithm appropriate for solving singular and singularly perturbed autonomous initial-value problems. Int J Comput Math 85(3\u20134):603\u2013611","journal-title":"Int J Comput Math"},{"issue":"1","key":"1733_CR36","doi-asserted-by":"publisher","first-page":"38","DOI":"10.1007\/s10910-009-9625-2","volume":"48","author":"H Ramos","year":"2010","unstructured":"Ramos H, Vigo-Aguiar J, Natesan S, Garcia-Rubio R, Queiruga M (2010) Numerical solution of nonlinear singularly perturbed problems on nonuniform meshes by using a non-standard algorithm. J Math Chem 48(1):38\u201354","journal-title":"J Math Chem"},{"key":"1733_CR37","volume-title":"Robust numerical methods for singularly perturbed differential equations, Springer series in computational mathematics","author":"HG Roos","year":"2008","unstructured":"Roos HG, Stynes M, Tobiska L (2008) Robust numerical methods for singularly perturbed differential equations, Springer series in computational mathematics, 2nd edn. Springer, Berlin","edition":"2"},{"key":"1733_CR38","doi-asserted-by":"publisher","first-page":"1849","DOI":"10.1002\/num.22256","volume":"34","author":"J Singh","year":"2018","unstructured":"Singh J, Kumar S, Kumar M (2018) A domain decomposition method for solving singularly perturbed parabolic reaction\u2013diffusion problems with time delay. Numer Methods Partial Differ Equ 34:1849\u20131866","journal-title":"Numer Methods Partial Differ Equ"},{"key":"1733_CR39","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4419-7646-8","volume-title":"An introduction to delay differential equations with applications to the life sciences","author":"HL Smith","year":"2011","unstructured":"Smith HL (2011) An introduction to delay differential equations with applications to the life sciences, vol 57. Springer, New York"},{"key":"1733_CR40","doi-asserted-by":"publisher","first-page":"2033","DOI":"10.1134\/S0965542510120067","volume":"50","author":"PN Vabishchevich","year":"2010","unstructured":"Vabishchevich PN (2010) Additive schemes for certain operator-differential equations. Comput Math Math Phys 50:2033\u20132043","journal-title":"Comput Math Math Phys"},{"key":"1733_CR41","doi-asserted-by":"publisher","first-page":"2413","DOI":"10.1007\/s10915-019-01091-1","volume":"81","author":"Q Zhang","year":"2019","unstructured":"Zhang Q, Li T (2019) Asymptotic stability of compact and linear $$\\theta $$-methods for space fractional delay generalized diffusion equation. J Sci Comput 81:2413\u20132446","journal-title":"J Sci Comput"},{"key":"1733_CR42","first-page":"1","volume":"14","author":"Q Zhang","year":"2017","unstructured":"Zhang Q, Mei M, Zhang C (2017) Higher-order linearized multistep finite difference methods for non-Fickian delay reaction\u2013diffusion equations. Int J Numer Anal Model 14:1\u201319","journal-title":"Int J Numer Anal Model"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-021-01733-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-021-01733-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-021-01733-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,2,11]],"date-time":"2023-02-11T12:25:30Z","timestamp":1676118330000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-021-01733-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,27]]},"references-count":42,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2022,2]]}},"alternative-id":["1733"],"URL":"https:\/\/doi.org\/10.1007\/s40314-021-01733-x","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,12,27]]},"assertion":[{"value":"12 May 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 September 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 November 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 December 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"29"}}