{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T17:17:54Z","timestamp":1781284674208,"version":"3.54.1"},"reference-count":20,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2019,12,4]],"date-time":"2019-12-04T00:00:00Z","timestamp":1575417600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2019,12,4]],"date-time":"2019-12-04T00:00:00Z","timestamp":1575417600000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2020,11]]},"DOI":"10.1007\/s11075-019-00850-3","type":"journal-article","created":{"date-parts":[[2019,12,4]],"date-time":"2019-12-04T07:02:31Z","timestamp":1575442951000},"page":"1005-1027","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["An efficient and uniformly convergent scheme for one-dimensional parabolic singularly perturbed semilinear systems of reaction-diffusion type"],"prefix":"10.1007","volume":"85","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1263-1996","authenticated-orcid":false,"given":"C.","family":"Clavero","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"J. C.","family":"Jorge","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2019,12,4]]},"reference":[{"key":"850_CR1","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1007\/978-3-319-67202-1_3","volume":"120","author":"I Boglaev","year":"2017","unstructured":"Boglaev, I.: Uniformly convergent monotone iterates for nonlinear parabolic reaction-diffusion systems. Lecture Notes in Computational Science and Engineering 120, 35\u201348 (2017)","journal-title":"Lecture Notes in Computational Science and Engineering"},{"key":"850_CR2","doi-asserted-by":"publisher","first-page":"188","DOI":"10.1093\/imanum\/drp033","volume":"31","author":"NM Chadha","year":"2011","unstructured":"Chadha, N.M., Kopteva, N.: A robust grid equidistribution method for a one-dimensional singularly perturbed semilinear reaction-diffusion problem. IMA J. Numer. Anal. 31, 188\u2013211 (2011)","journal-title":"IMA J. Numer. Anal."},{"key":"850_CR3","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1007\/s10444-010-9163-2","volume":"35","author":"NM Chadha","year":"2011","unstructured":"Chadha, N.M., Kopteva, N.: Maximum norm a posteriori error estimate for a 3d singularly perturbed semilinear reaction-diffusion problem. Adv. Comput. Math. 35, 33\u201355 (2011)","journal-title":"Adv. Comput. Math."},{"key":"850_CR4","doi-asserted-by":"publisher","first-page":"655","DOI":"10.1016\/j.camwa.2013.12.009","volume":"67","author":"C Clavero","year":"2014","unstructured":"Clavero, C., Gracia, J.L.: Uniformly convergent additive finite difference schemes for singularly perturbed parabolic reaction-diffusion system. Comput. Math. Appl. 67, 655\u2013670 (2014)","journal-title":"Comput. Math. Appl."},{"key":"850_CR5","doi-asserted-by":"publisher","first-page":"1097","DOI":"10.1007\/s11075-018-0518-y","volume":"80","author":"C Clavero","year":"2019","unstructured":"Clavero, C., Gracia, J.L.: Uniformly convergent additive schemes for 2D singularly perturbed parabolic systems of reaction-diffusion type. Numer. Algorithms 80, 1097\u20131120 (2019)","journal-title":"Numer. Algorithms"},{"key":"850_CR6","first-page":"428","volume":"7","author":"C Clavero","year":"2010","unstructured":"Clavero, C., Gracia, J.L., Lisbona, F.: Second order uniform approximations for the solution of time dependent singularly perturbed reaction-diffusion systems. Int. J. Numer. Anal. Mod. 7, 428\u2013443 (2010)","journal-title":"Int. J. Numer. Anal. Mod."},{"key":"850_CR7","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.cam.2018.05.019","volume":"344","author":"C Clavero","year":"2018","unstructured":"Clavero, C., Jorge, J.C.: Solving efficiently one dimensional parabolic singularly perturbed reaction-diffusion systems: a splitting by components. J. Comp. Appl. Math. 344, 1\u201314 (2018)","journal-title":"J. Comp. Appl. Math."},{"key":"850_CR8","doi-asserted-by":"publisher","first-page":"265","DOI":"10.1016\/j.amc.2014.10.023","volume":"249","author":"P Das","year":"2014","unstructured":"Das, P., Natesan, S.: Optimal error estimate using mesh equidistribution technique for singularly perturbed system of reaction-diffusion boundary-value problems. Appl. Math. Comp. 249, 265\u2013277 (2014)","journal-title":"Appl. Math. Comp."},{"key":"850_CR9","doi-asserted-by":"crossref","unstructured":"Farrell, P.A., Hegarty, A.F., Miller, J.J.H., O\u2019Riordan, E., Shishkin, G.I.: Robust Computational Techniques for Boundary Layers, Applied Mathematics, vol. 16 Chapman and Hall\/CRC (2000)","DOI":"10.1201\/9781482285727"},{"key":"850_CR10","first-page":"178","volume":"10","author":"V Franklin","year":"2013","unstructured":"Franklin, V., Paramasivam, M., Miller, J.J.H., Valarmathi, S.: 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 (2013)","journal-title":"Int. J. Numer. Anal. Model."},{"key":"850_CR11","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.cam.2006.06.005","volume":"206","author":"JL Gracia","year":"2007","unstructured":"Gracia, J.L., Lisbona, F.: A uniformly convergent scheme for a system of reaction\u2013diffusion equations. J. Comp. Appl. Math. 206, 1\u201316 (2007)","journal-title":"J. Comp. Appl. Math."},{"key":"850_CR12","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1007\/s10444-008-9086-3","volume":"32","author":"JL Gracia","year":"2010","unstructured":"Gracia, J.L., Lisbona, F., O\u2019Riordan, E.: A coupled system of singularly perturbed parabolic reaction-diffusion equations. Adv. Comput. Math. 32, 43\u201361 (2010)","journal-title":"Adv. Comput. Math."},{"key":"850_CR13","doi-asserted-by":"publisher","first-page":"616","DOI":"10.1093\/imanum\/drp032","volume":"31","author":"N Kopteva","year":"2011","unstructured":"Kopteva, N., Savescu, S.B.: Pointwise error estimates for a singularly perturbed time-dependent semilinear reaction-diffusion problem. IMA J. Numer. Anal. 31, 616\u2013639 (2011)","journal-title":"IMA J. Numer. Anal."},{"key":"850_CR14","doi-asserted-by":"publisher","first-page":"787","DOI":"10.1007\/s00211-011-0395-y","volume":"119","author":"N Kopteva","year":"2011","unstructured":"Kopteva, N., Stynes, M.: Stabilised approximation of interior-layer solutions of a singularly perturbed semilinear reaction-diffusion problem. Numer. Math. 119, 787\u2013810 (2011)","journal-title":"Numer. Math."},{"key":"850_CR15","unstructured":"Ladyzhenskaya, O.A., Solonnikov, V.A., Uraltseva, N.N.: Linear and Quasilinear Equations of Parabolic Type Translations of Mathematical Monographs, vol. 23. American Mathematical Society, Providence, R.I (1967)"},{"key":"850_CR16","first-page":"121","volume":"73","author":"T Linss","year":"2004","unstructured":"Linss, T., Madden, N.: Accurate solution of a system of coupled singularly perturbed reaction-diffusion equations. Computing 73, 121\u2013133 (2004)","journal-title":"Computing"},{"key":"850_CR17","volume-title":"Nonlinear Parabolic and Elliptic Equations","author":"CV Pao","year":"1992","unstructured":"Pao, C.V.: Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York (1992)"},{"key":"#cr-split#-850_CR18.1","unstructured":"Shishkin, G.I., Shishkina, L.P.: Approximation of a system of singularly perturbed parabolic reaction-diffusion equations in a rectangle, Zh. Vychisl. Mat. Mat. Fiz. 48, 660-673 (2008) (in Russian)"},{"key":"#cr-split#-850_CR18.2","unstructured":"translation in Comput. Math. Math. Phys. 48, 627-640 (2008)"},{"key":"850_CR19","doi-asserted-by":"publisher","first-page":"251","DOI":"10.3846\/1392-6292.2008.13.251-261","volume":"13","author":"LP Shishkina","year":"2008","unstructured":"Shishkina, L.P., Shishkin, G.I.: Robust numerical method for a system of singularly perturbed parabolic reaction-diffusion equations on a rectangle. Math. Model. Anal. 13, 251\u2013261 (2008)","journal-title":"Math. Model. Anal."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-019-00850-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-019-00850-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-019-00850-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,12,3]],"date-time":"2020-12-03T02:34:35Z","timestamp":1606962875000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-019-00850-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,12,4]]},"references-count":20,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2020,11]]}},"alternative-id":["850"],"URL":"https:\/\/doi.org\/10.1007\/s11075-019-00850-3","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,12,4]]},"assertion":[{"value":"21 February 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 November 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 December 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}