{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T12:54:19Z","timestamp":1740142459371,"version":"3.37.3"},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2022,4,9]],"date-time":"2022-04-09T00:00:00Z","timestamp":1649462400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2022,4,9]],"date-time":"2022-04-09T00:00:00Z","timestamp":1649462400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100007129","name":"Natural Science Foundation of Shandong Province","doi-asserted-by":"publisher","award":["ZR2019MA019"],"award-info":[{"award-number":["ZR2019MA019"]}],"id":[{"id":"10.13039\/501100007129","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":[[2022,6]]},"DOI":"10.1007\/s40314-022-01838-x","type":"journal-article","created":{"date-parts":[[2022,4,9]],"date-time":"2022-04-09T04:02:52Z","timestamp":1649476972000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Exact delta shock wave solution to the Riemann problem for the three-component chromatography model"],"prefix":"10.1007","volume":"41","author":[{"given":"Zhijian","family":"Wei","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6483-6497","authenticated-orcid":false,"given":"Meina","family":"Sun","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,4,9]]},"reference":[{"key":"1838_CR1","doi-asserted-by":"publisher","first-page":"125267","DOI":"10.1016\/j.jmaa.2021.125267","volume":"502","author":"E Abreu","year":"2021","unstructured":"Abreu E, De la Cruz R, Lambert W (2021) Riemann problem and delta-shock solutions for a Keyfitz-Kranzer system with a forcing term. J Math Anal Appl 502:125267","journal-title":"J Math Anal Appl"},{"key":"1838_CR2","doi-asserted-by":"publisher","first-page":"1890","DOI":"10.1137\/090754686","volume":"41","author":"L Ambrosio","year":"2009","unstructured":"Ambrosio L, Crippa G, Figalli A, Spinolo LV (2009) Some new well-posedness results for continuity and transport equations, and applications to the chromatography system. SIAM J Math Anal 41:1890\u20131920","journal-title":"SIAM J Math Anal"},{"key":"1838_CR3","doi-asserted-by":"publisher","first-page":"925","DOI":"10.1137\/S0036141001399350","volume":"34","author":"GQ Chen","year":"2003","unstructured":"Chen GQ, Liu H (2003) Formation of $$\\delta $$-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids. SIAM J Math Anal 34:925\u2013938","journal-title":"SIAM J Math Anal"},{"key":"1838_CR4","doi-asserted-by":"publisher","first-page":"475","DOI":"10.1016\/j.jmaa.2011.04.002","volume":"380","author":"H Cheng","year":"2011","unstructured":"Cheng H, Yang H (2011) Delta shock waves in chromatography equations. J Math Anal Appl 380:475\u2013485","journal-title":"J Math Anal Appl"},{"key":"1838_CR5","doi-asserted-by":"publisher","first-page":"333","DOI":"10.1016\/j.jde.2004.12.011","volume":"211","author":"VG Danilov","year":"2005","unstructured":"Danilov VG, Shelkovich VM (2005) Dynamics of propagation and interaction of $$\\delta $$-shock waves in conservation law systems. J Differ Equ 211:333\u2013381","journal-title":"J Differ Equ"},{"key":"1838_CR6","doi-asserted-by":"crossref","unstructured":"De la Cruz R, Santos M, Abreu E (2021) Interaction of delta shock waves for a nonsymmetric Keyfitz-Kranzer system of conservation laws. Monatsh Math 194:737\u2013766","DOI":"10.1007\/s00605-021-01524-w"},{"key":"1838_CR7","doi-asserted-by":"publisher","first-page":"110","DOI":"10.1016\/j.na.2014.04.016","volume":"106","author":"L Guo","year":"2014","unstructured":"Guo L, Pan L, Yin G (2014) The perturbed Riemann problem and delta contact discontinuity in chromatography equations. Nonlinear Anal TMA 106:110\u2013123","journal-title":"Nonlinear Anal TMA"},{"key":"1838_CR8","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1007\/s40314-020-01282-9","volume":"39","author":"I Jegdic","year":"2020","unstructured":"Jegdic I (2020) Numerical solutions in chromatography using large time step and overlapping grids methods. Comp Appl Math 39:235","journal-title":"Comp Appl Math"},{"key":"1838_CR9","doi-asserted-by":"publisher","first-page":"5463","DOI":"10.1088\/1361-6544\/aae04b","volume":"31","author":"H Kalisch","year":"2018","unstructured":"Kalisch H, Mitrovic D, Teyekpiti V (2018) Existence and uniqueness of singular solutions for a conservation law arising in magnetohydrodynamics. Nonlinearity 31:5463\u20135483","journal-title":"Nonlinearity"},{"key":"1838_CR10","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1108\/EUM0000000004104","volume":"6","author":"DM Lu","year":"1996","unstructured":"Lu DM, Simpson HC, Gilchrist A (1996) The application of split-coefficient matrix method to transient two phase flows. Int J Numer Meth Heat Fluid flow 6:63\u201376","journal-title":"Int J Numer Meth Heat Fluid flow"},{"key":"1838_CR11","doi-asserted-by":"publisher","first-page":"721","DOI":"10.1007\/s11856-018-1672-6","volume":"225","author":"YG Lu","year":"2018","unstructured":"Lu YG (2018) Global weak solutions for the chromatography system. Israel J Math 225:721\u2013741","journal-title":"Israel J Math"},{"key":"1838_CR12","doi-asserted-by":"publisher","first-page":"2002","DOI":"10.1016\/j.chroma.2010.01.059","volume":"1217","author":"M Mazzotti","year":"2010","unstructured":"Mazzotti M, Tarafder A, Cornel J, Gritti F, Guiochon G (2010) Experimental evidence of a delta-shock in nonlinear chromatography. J Chromatogr A 1217:2002\u20132012","journal-title":"J Chromatogr A"},{"key":"1838_CR13","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1016\/j.aml.2019.01.008","volume":"92","author":"SMA Mohamed","year":"2019","unstructured":"Mohamed SMA, Nedeljkov M (2019) Simplified chormatography model and inverse of split delta shocks. Appl Math Lett 92:49\u201353","journal-title":"Appl Math Lett"},{"key":"1838_CR14","doi-asserted-by":"publisher","first-page":"489","DOI":"10.1007\/s00205-009-0281-2","volume":"197","author":"M Nedeljkov","year":"2010","unstructured":"Nedeljkov M (2010) Shadow waves: entropies and interactions for delta and singular shocks. Arch Ration Mech Anal 197:489\u2013537","journal-title":"Arch Ration Mech Anal"},{"key":"1838_CR15","doi-asserted-by":"publisher","first-page":"1677","DOI":"10.1080\/00036810903569515","volume":"90","author":"B Nilsson","year":"2011","unstructured":"Nilsson B, Shelkovich VM (2011) Mass, momentum and energy conservation laws in zero-pressure gas dynamics and delta-shocks. Appl Anal 90:1677\u20131689","journal-title":"Appl Anal"},{"key":"1838_CR16","doi-asserted-by":"publisher","first-page":"705","DOI":"10.4310\/CMS.2018.v16.n3.a5","volume":"16","author":"L Pan","year":"2018","unstructured":"Pan L, Han X, Li T, Guo L (2018) The generalized Riemann problem and instability of delta shock to the chromatography equations. Commun Math Sci 16:705\u2013734","journal-title":"Commun Math Sci"},{"key":"1838_CR17","volume-title":"First-order partial differential equations, volume 2: theory and application of hyperbolic systems of quasilinear equations","author":"HK Rhee","year":"2001","unstructured":"Rhee HK, Aris R, Amundson NR (2001) First-order partial differential equations, volume 2: theory and application of hyperbolic systems of quasilinear equations. Dover Publications, New York"},{"key":"1838_CR18","doi-asserted-by":"publisher","first-page":"1187","DOI":"10.1007\/s10884-017-9594-2","volume":"30","author":"COR Sarrico","year":"2018","unstructured":"Sarrico COR, Paiva A (2018) Delta shock waves in the shallow water system. J Dyn Differ Equ 30:1187\u20131198","journal-title":"J Dyn Differ Equ"},{"key":"1838_CR19","first-page":"667","volume":"2","author":"VM Shelkovich","year":"2012","unstructured":"Shelkovich VM (2012) One class of systems of conservation laws admitting delta-shocks. Hyperplic Probl Theory Numer Appl 2:667\u2013674","journal-title":"Hyperplic Probl Theory Numer Appl"},{"key":"1838_CR20","doi-asserted-by":"publisher","first-page":"76","DOI":"10.1080\/14029251.2015.996442","volume":"22","author":"C Shen","year":"2015","unstructured":"Shen C (2015) The asymptotic behaviors of solutions to the perturbed Riemann problem near the singular curve for the chromatography system. J Nonlinear Math Phys 22:76\u2013101","journal-title":"J Nonlinear Math Phys"},{"issue":"N654","key":"1838_CR21","first-page":"1","volume":"137","author":"W Sheng","year":"1999","unstructured":"Sheng W, Zhang T (1999) The Riemann problem for the transportation equations in gas dynamics. Mem Am Math Soc 137(N654):1\u201377","journal-title":"Mem Am Math Soc"},{"key":"1838_CR22","doi-asserted-by":"publisher","first-page":"2641","DOI":"10.3934\/cpaa.2020115","volume":"19","author":"A Sen","year":"2020","unstructured":"Sen A, Raja Sekhar T (2020) Delta shock wave and wave interactions in a thin film of a perfectly soluble anti-surfactant solution. Commun Pure Appl Anal 19:2641\u20132653","journal-title":"Commun Pure Appl Anal"},{"key":"1838_CR23","doi-asserted-by":"publisher","first-page":"425","DOI":"10.1090\/S0033-569X-2011-01207-3","volume":"69","author":"M Sun","year":"2011","unstructured":"Sun M (2011) Delta shock waves for the chromatography equations as self-similar viscosity limits. Q Appl Math 69:425\u2013443","journal-title":"Q Appl Math"},{"key":"1838_CR24","doi-asserted-by":"publisher","first-page":"631","DOI":"10.1016\/j.aml.2013.01.002","volume":"26","author":"M Sun","year":"2013","unstructured":"Sun M (2013) Interactions of delta shock waves for the chromatography equations. Appl Math Lett 26:631\u2013637","journal-title":"Appl Math Lett"},{"key":"1838_CR25","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1016\/j.aml.2019.04.015","volume":"96","author":"M Sun","year":"2019","unstructured":"Sun M (2019) The multiplication of distributions in the study of delta shock wave for the nonlinear chromatography system. Appl Math Lett 96:61\u201368","journal-title":"Appl Math Lett"},{"key":"1838_CR26","doi-asserted-by":"publisher","first-page":"103068","DOI":"10.1016\/j.nonrwa.2019.103068","volume":"53","author":"M Sun","year":"2020","unstructured":"Sun M (2020) Concentration and cavitation phenomena of Riemann solutions for the isentropic Euler system with the logarithmic equation of state. Nonlinear Anal RWA 53:103068","journal-title":"Nonlinear Anal RWA"},{"key":"1838_CR27","doi-asserted-by":"publisher","first-page":"781","DOI":"10.1090\/S0002-9947-1983-0716850-2","volume":"280","author":"B Temple","year":"1983","unstructured":"Temple B (1983) Systems of conservation laws with invariant submanifolds. Trans Am Math Soc 280:781\u2013795","journal-title":"Trans Am Math Soc"},{"key":"1838_CR28","doi-asserted-by":"publisher","first-page":"143","DOI":"10.1090\/conm\/017\/16","volume":"17","author":"B Temple","year":"1983","unstructured":"Temple B (1983) Systems of conservation laws with coinciding shock and rarefaction curves. Contemp Math 17:143\u2013151","journal-title":"Contemp Math"},{"key":"1838_CR29","doi-asserted-by":"publisher","first-page":"766","DOI":"10.1016\/j.jmaa.2016.03.001","volume":"439","author":"C Tsikkou","year":"2016","unstructured":"Tsikkou C (2016) Singular shocks in a chromatography model. J Math Anal Appl 439:766\u2013797","journal-title":"J Math Anal Appl"},{"key":"1838_CR30","doi-asserted-by":"publisher","first-page":"1451","DOI":"10.1007\/s00033-013-0300-x","volume":"64","author":"G Wang","year":"2013","unstructured":"Wang G (2013) One-dimensional nonlinear chromatography system and $$\\delta -$$shock waves. Z Angew Math Phys 64:1451\u20131469","journal-title":"Z Angew Math Phys"},{"key":"1838_CR31","doi-asserted-by":"publisher","first-page":"391","DOI":"10.1007\/s40840-015-0178-1","volume":"39","author":"X Wang","year":"2016","unstructured":"Wang X, Guo L, Yin G (2016) Interactions of elementary waves for the nonlinear chromatography equations. Bull Malays Math Sci Soc 39:391\u2013407","journal-title":"Bull Malays Math Sci Soc"},{"key":"1838_CR32","doi-asserted-by":"publisher","first-page":"4195","DOI":"10.1007\/s40840-021-01161-4","volume":"44","author":"Z Wei","year":"2021","unstructured":"Wei Z, Sun M (2021) Riemann problem and wave interactions for a Temple-class hyperbolic system of conservation laws. Bull Malays Math Sci Soc 44:4195\u20134221","journal-title":"Bull Malays Math Sci Soc"},{"key":"1838_CR33","doi-asserted-by":"publisher","first-page":"5951","DOI":"10.1016\/j.jde.2012.02.015","volume":"252","author":"H Yang","year":"2012","unstructured":"Yang H, Zhang Y (2012) New developments of delta shock waves and its applications in systems of conservation laws. J Differ Equ 252:5951\u20135993","journal-title":"J Differ Equ"},{"key":"1838_CR34","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1007\/s00033-015-0599-6","volume":"67","author":"Q Zhang","year":"2016","unstructured":"Zhang Q (2016) Interactions of delta shock waves and stability of Riemann solutions for nonlinear chromatography equations. Z Angew Math Phys 67:15","journal-title":"Z Angew Math Phys"},{"key":"1838_CR35","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1007\/s10473-019-0106-9","volume":"39B","author":"T Zhang","year":"2019","unstructured":"Zhang T, Sheng W (2019) Global solutions of the perturbed Riemann problem for the chromatography equations. Acta Math Sci 39B:57\u201382","journal-title":"Acta Math Sci"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-022-01838-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-022-01838-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-022-01838-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,6,9]],"date-time":"2022-06-09T21:15:19Z","timestamp":1654809319000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-022-01838-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,9]]},"references-count":35,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2022,6]]}},"alternative-id":["1838"],"URL":"https:\/\/doi.org\/10.1007\/s40314-022-01838-x","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"type":"print","value":"2238-3603"},{"type":"electronic","value":"1807-0302"}],"subject":[],"published":{"date-parts":[[2022,4,9]]},"assertion":[{"value":"19 October 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 December 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 March 2022","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 April 2022","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"133"}}