{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T20:19:37Z","timestamp":1774469977713,"version":"3.50.1"},"reference-count":23,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2024,11,1]],"date-time":"2024-11-01T00:00:00Z","timestamp":1730419200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>Four formulations based on the Kirchhoff transformation and time linearization for the numerical study of one-dimensional reaction\u2013diffusion equations, whose heat capacity, thermal inertia and reaction rate are only functions of the temperature, are presented. The formulations result in linear, two-point boundary-value problems for the temperature, energy or heat potential, and may be solved by either discretizing the second-order spatial derivative or piecewise analytical integration. In both cases, linear systems of algebraic equations are obtained. The formulation for the temperature is extended to two-dimensional, nonlinear reaction\u2013diffusion equations where the resulting linear two-dimensional operator is factorized into a sequence of one-dimensional ones that may be solved by means of any of the four formulations developed for one-dimensional problems. The multidimensional formulation is applied to a two-dimensional, two-equation system of nonlinearly coupled advection\u2013reaction\u2013diffusion equations, and the effects of the velocity and the parameters that characterize the nonlinear heat capacities and thermal conductivity are studied. It is shown that clockwise-rotating velocity fields result in wave stretching for small vortex radii, and wave deceleration and thickening for counter-clockwise-rotating velocity fields. It is also shown that large-core, clockwise-rotating velocity fields may result in large transient periods, followed by time intervals of apparent little activity which, in turn, are followed by the propagation of long-period waves.<\/jats:p>","DOI":"10.3390\/computation12110218","type":"journal-article","created":{"date-parts":[[2024,11,1]],"date-time":"2024-11-01T04:59:54Z","timestamp":1730437194000},"page":"218","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Finite Difference Methods Based on the Kirchhoff Transformation and Time Linearization for the Numerical Solution of Nonlinear Reaction\u2013Diffusion Equations"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1578-5352","authenticated-orcid":false,"given":"Juan I.","family":"Ramos","sequence":"first","affiliation":[{"name":"Escuela de Ingenier as Industriales, Universidad de M\u00e1laga, Doctor Ortiz Ramos, s\/n, 29071 M\u00e1laga, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2024,11,1]]},"reference":[{"key":"ref_1","unstructured":"Kirchhoff, G. 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Mechanics of Flow Similarities, Springer Nature Switzerland AG.","DOI":"10.1007\/978-3-030-42930-0"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1057","DOI":"10.1016\/S0960-0779(00)00072-2","article-title":"Propagation of spiral waves in anisotropic media: From waves to stripes","volume":"12","author":"Ramos","year":"2001","journal-title":"Chaos Solitons Fractals"}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/12\/11\/218\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:26:24Z","timestamp":1760113584000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/12\/11\/218"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,1]]},"references-count":23,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2024,11]]}},"alternative-id":["computation12110218"],"URL":"https:\/\/doi.org\/10.3390\/computation12110218","relation":{},"ISSN":["2079-3197"],"issn-type":[{"value":"2079-3197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,11,1]]}}}