{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,9]],"date-time":"2026-02-09T22:54:26Z","timestamp":1770677666050,"version":"3.49.0"},"reference-count":57,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,6,29]],"date-time":"2023-06-29T00:00:00Z","timestamp":1687996800000},"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>We utilize the travelling-wave Ansatz to obtain novel analytical solutions to the linear diffusion\u2013reaction equation. The reaction term is a function of time and space simultaneously, firstly in a Lorentzian form and secondly in a cosine travelling-wave form. The new solutions contain the Heun functions in the first case and the Mathieu functions for the second case, and therefore are highly nontrivial. We use these solutions to test some non-conventional explicit and stable numerical methods against the standard explicit and implicit methods, where in the latter case the algebraic equation system is solved by the preconditioned conjugate gradient and the GMRES solvers. After this verification, we also calculate the transient temperature of a 2D surface subjected to the cooling effect of the wind, which is a function of space and time again. We obtain that the explicit stable methods can reach the accuracy of the implicit solvers in orders of magnitude shorter time.<\/jats:p>","DOI":"10.3390\/computation11070127","type":"journal-article","created":{"date-parts":[[2023,6,30]],"date-time":"2023-06-30T00:43:30Z","timestamp":1688085810000},"page":"127","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Analytical and Numerical Results for the Diffusion-Reaction Equation When the Reaction Coefficient Depends on Simultaneously the Space and Time Coordinates"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7140-3868","authenticated-orcid":false,"given":"Ali Habeeb","family":"Askar","sequence":"first","affiliation":[{"name":"Institute of Physics and Electrical Engineering, University of Miskolc, 3515 Miskolc, Hungary"},{"name":"Department of Fluid and Heat Engineering, University of Miskolc, 3515 Miskolc, Hungary"},{"name":"Mechanical Engineering Department, University of Technology\u2014Iraq, Baghdad 10066, Iraq"}]},{"given":"\u00c1d\u00e1m","family":"Nagy","sequence":"additional","affiliation":[{"name":"Institute of Physics and Electrical Engineering, University of Miskolc, 3515 Miskolc, Hungary"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6206-3910","authenticated-orcid":false,"given":"Imre Ferenc","family":"Barna","sequence":"additional","affiliation":[{"name":"Wigner Research Center for Physics, 1051 Budapest, Hungary"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0439-3070","authenticated-orcid":false,"given":"Endre","family":"Kov\u00e1cs","sequence":"additional","affiliation":[{"name":"Institute of Physics and Electrical Engineering, University of Miskolc, 3515 Miskolc, Hungary"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Jacobs, M.H. 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