{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,12]],"date-time":"2025-12-12T13:40:20Z","timestamp":1765546820753,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2021,12,4]],"date-time":"2021-12-04T00:00:00Z","timestamp":1638576000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Physical processes occurring in devices with distributed variables and a turbulent tide with a dispersion of mass and heat are often modeled using systems of nonlinear equations. Solving such a system is sometimes impossible in an analytical manner. The iterative methods, such as Newton\u2019s method, are not always sufficiently effective in such cases. In this article, a combination of the homotopy method and the parametric continuation method was proposed to solve the system of nonlinear differential equations. These methods are symmetrical, i.e., the calculations can be made by increasing or decreasing the value of the parameters. Thanks to this approach, the determination of all roots of the system does not require any iterative method. Moreover, when the solutions of the system are close to each other, the proposed method easily determines all of them. As an example of the method use a mathematical model of a non-adiabatic catalytic pseudohomogeneous tubular chemical reactor with longitudinal dispersion was chosen.<\/jats:p>","DOI":"10.3390\/sym13122324","type":"journal-article","created":{"date-parts":[[2021,12,7]],"date-time":"2021-12-07T02:48:13Z","timestamp":1638845293000},"page":"2324","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Homotopic Parametric Continuation Method for Determining Stationary States of Chemical Reactors with Dispersion"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9440-7914","authenticated-orcid":false,"given":"Marek","family":"Berezowski","sequence":"first","affiliation":[{"name":"Faculty of Chemical Engineering and Technology, Cracow University of Technology, ul. Warszawska 24, 30-155 Krak\u00f3w, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0235-0878","authenticated-orcid":false,"given":"Marcin","family":"Lawnik","sequence":"additional","affiliation":[{"name":"Department of Mathematics Applications and Methods for Artificial Intelligence, Faculty of Applied Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"5411","DOI":"10.1016\/j.ces.2010.07.003","article-title":"The application of the parametric continuation method for determining steady state diagrams in chemical engineering","volume":"65","author":"Berezowski","year":"2010","journal-title":"Chem. Eng. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Guran, L., Mitrovi\u0107, Z.D., Reddy, G.S.M., Belhenniche, A., and Radenovi\u0107, S. (2021). Applications of a Fixed Point Result for Solving Nonlinear Fractional and Integral Differential Equations. Fractal Fract., 5.","DOI":"10.3390\/fractalfract5040211"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"8","DOI":"10.5937\/vojtehg69-29517","article-title":"Some new observations on fixed point results in rectangular metric spaces with applications to chemical sciences","volume":"69","author":"Younis","year":"2021","journal-title":"Vojnoteh. Glas.\/Mil. Tech. 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Comput."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/12\/2324\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:39:52Z","timestamp":1760168392000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/12\/2324"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,4]]},"references-count":18,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2021,12]]}},"alternative-id":["sym13122324"],"URL":"https:\/\/doi.org\/10.3390\/sym13122324","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,12,4]]}}}