{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T00:34:20Z","timestamp":1759970060755,"version":"build-2065373602"},"reference-count":42,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,3]],"date-time":"2025-01-03T00:00:00Z","timestamp":1735862400000},"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>Approximate analytical solutions to doubly degenerate reaction-diffusion models pertinent to population dynamics and chemical kinetics have been developed. The double integral-balance method applied to preliminary transformed models and by a direct integration approach has provided physically reasonable results. The model equation scaling has revealed the time and length scales, as well as the characteristic velocity of the process and the Fourier number as the controlling dimensionless group.<\/jats:p>","DOI":"10.3390\/sym17010069","type":"journal-article","created":{"date-parts":[[2025,1,3]],"date-time":"2025-01-03T10:17:23Z","timestamp":1735899443000},"page":"69","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Sharp Front Approach Solutions to Some Doubly Degenerate Reaction-Diffusion Models"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7957-8192","authenticated-orcid":false,"given":"Jordan","family":"Hristov","sequence":"first","affiliation":[{"name":"Department of Chemical Engineering, University of Chemical Technology and Metallurgy, 8 Kliment Ohridsky, blvd., 1756 Sofia, Bulgaria"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1111\/j.1469-1809.1937.tb02153.x","article-title":"The wave of advance of advantageous genes","volume":"7","author":"Fisher","year":"1937","journal-title":"Ann. Eugen."},{"key":"ref_2","first-page":"1","article-title":"Etude de l\u2019equation de la diffusion avec croissance de la quantite de la matiere e son application a un problem biologique","volume":"1","author":"Kolmogorov","year":"1937","journal-title":"Mosc. Univ. Bull. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"466","DOI":"10.1016\/j.physa.2007.04.040","article-title":"Traveling wave solutions for a reaction-diffusion model for bacterial growth","volume":"383","author":"Mansour","year":"2007","journal-title":"Phys. A"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"240","DOI":"10.1016\/j.apm.2006.11.013","article-title":"Traveling wave solutions of a nonlinear reaction-diffusion-chemotaxis model for bacterial pattern formation","volume":"32","author":"Mansour","year":"2008","journal-title":"Appl. Math. Model."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1017\/S144618111100054X","article-title":"Traveling wave solutions for doubly degenerate reaction-diffusion equations","volume":"52","author":"Mansour","year":"2010","journal-title":"ANZIAM J."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1016\/0893-9659(94)90051-5","article-title":"An approximation to a sharp type solution of density-dependent reaction-diffusion equation","volume":"7","author":"Maini","year":"1994","journal-title":"Appl. Math. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1006\/jdeq.1995.1055","article-title":"Traveling wave phenomena in some degenerate reaction-diffusion equations","volume":"117","author":"Maini","year":"1995","journal-title":"J. Differ. Equ."},{"key":"ref_8","unstructured":"Stewart, W.G., Ray, W.H., and Couley, C.C. (1980). Density-dependent interaction-diffusion systems. Dynamics and Modeling of Reactive Systems, Academic Press."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1016\/0022-5193(80)90024-7","article-title":"Some exact solutions to non-linear diffusion problem in population genetics and combustion","volume":"85","author":"Newman","year":"1980","journal-title":"J. Theor. Biol."},{"key":"ref_10","unstructured":"Zeldovich, Y.B., and Raizer, Y.P. (1967). Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Academic Press."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"697","DOI":"10.1016\/S0008-8846(96)85006-X","article-title":"Application of Danckwerts\u2019s solution to simultaneous diffusion and chemical reaction to concrete","volume":"26","author":"Tumidajski","year":"1996","journal-title":"Cem. Concr. Res."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"665","DOI":"10.1016\/0009-2509(65)80003-3","article-title":"Unsteady mass transfer with chemical reaction. Part II: More general initial and boundary conditions","volume":"20","author":"Pao","year":"1965","journal-title":"Chem. Eng. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1016\/0025-5564(77)90062-1","article-title":"On the diffusion in biological populations","volume":"33","author":"Gurtin","year":"1977","journal-title":"Math. Biosci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1007\/BF03167096","article-title":"Travelling wave solutions to some dependent diffusion equations","volume":"3","author":"Hosono","year":"1986","journal-title":"Jap. J. Appl. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/j.wavemoti.2012.06.003","article-title":"Analysis of propagating fronts in a nonlinear diffusion model with chemotaxis","volume":"50","author":"Mansour","year":"2013","journal-title":"Wave Motion"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"835","DOI":"10.1016\/S0092-8240(79)80020-8","article-title":"Explicit solutions of Fisher\u2019s equation for a special wave speed","volume":"41","author":"Ablowitz","year":"1979","journal-title":"Bull. Math. Biol."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"660","DOI":"10.1007\/s10958-021-05260-2","article-title":"Method of monotone solutions for reaction-diffusion equations","volume":"253","author":"Volpert","year":"2021","journal-title":"J. Math. Sci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"284","DOI":"10.1016\/j.amc.2008.12.089","article-title":"Haar wavelet method for solving Fisher\u2019s equation","volume":"211","author":"Hariharan","year":"2009","journal-title":"Appl. Math. Comp."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1016\/S0377-0427(01)00356-9","article-title":"Numerical study of Fisher\u2019s reaction-diffusion equations by the Sinc collocation method","volume":"137","year":"2001","journal-title":"J. Comp. Appl. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1007\/s11075-006-9052-4","article-title":"Power series solutions for KPP equations","volume":"43","author":"Boumenir","year":"2006","journal-title":"Numer. Algorithms"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2017","DOI":"10.1137\/15M1039006","article-title":"Self-similar solutions for reversing interfaces in the slow diffusion equation with strong absorption","volume":"15","author":"Foster","year":"2016","journal-title":"SIAM J. Appl. Dyn. Sys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"635","DOI":"10.1007\/s00231-015-1579-2","article-title":"Integral solutions to transient nonlinear heat (mass) diffusion with a power-law diffusivity: A semi-infinite medium with fixed boundary conditions","volume":"52","author":"Hristov","year":"2016","journal-title":"Heat Mass Transf."},{"key":"ref_23","first-page":"67","article-title":"On certain nonstationary motions of liquids and gases in porous media","volume":"16","author":"Barenblatt","year":"1952","journal-title":"Appl. Math. Mech."},{"key":"ref_24","unstructured":"Zeldovich, Y.B., and Kompaneets, A.S. (1950). On the theory of heat propagation for temperature dependent thermal conductivity. Collection Commemorating the 70th Anniversary of A. F. Joffe, Izv. Akad. Nauk SSSR. (In Russian)."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1093\/qjmam\/12.4.407","article-title":"Diffusion from an instantaneous point source with concentration-dependent coefficient","volume":"12","author":"Pattle","year":"1959","journal-title":"Q. J. Mech. Appl. Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1093\/imamat\/40.2.73","article-title":"High-Order Nonlinear Diffusion","volume":"40","author":"Smyth","year":"1988","journal-title":"IMA J. Appl. Math."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1007\/BF00128865","article-title":"Similarity solutions for nonlinear diffusion-a new integration procedure","volume":"23","author":"Hill","year":"1989","journal-title":"J. Eng. Math."},{"key":"ref_28","first-page":"22","article-title":"The heat-balance integral method by a parabolic profile with unspecified exponent: Analysis and benchmark exercises","volume":"13","author":"Hristov","year":"2009","journal-title":"Therm. Sci."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/0022-0396(91)90021-Z","article-title":"Traveling waves and finite propagation in a reaction-diffusion equation","volume":"93","author":"Vazquez","year":"1991","journal-title":"J. Differ. Equ."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"7647","DOI":"10.1016\/j.jde.2017.08.025","article-title":"Fisher-KPP problem with doubly nonlinear diffusion","volume":"263","author":"Audrito","year":"2017","journal-title":"J. Differ. Equ."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Ghazaryan, A.R., Lafortune, S., and Manukian, V. (2023). Introduction to Traveling Waves, Chapman and Hall\/CRC. [1st ed.].","DOI":"10.1201\/9781003147619"},{"key":"ref_32","first-page":"335","article-title":"The heat balance integral and its application to problems involving a change of phase","volume":"80","author":"Goodman","year":"1958","journal-title":"Trans. ASME"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/S0065-2717(08)70097-2","article-title":"Application of Integral Methods to Transient Nonlinear Heat Transfer","volume":"Volume 1","author":"Irvine","year":"1964","journal-title":"Advances in Heat Transfer"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/0893-9659(95)00047-T","article-title":"Merging traveling waves for the porous-Fisher\u2019s equation","volume":"12","author":"Witelski","year":"1995","journal-title":"Appl. Math. Lett."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"3367","DOI":"10.1088\/0305-4470\/38\/15\/009","article-title":"A Fisher\/KPP-type equation with density-dependent diffusion and convection: Travelling wave-solutions","volume":"38","author":"Gilding","year":"2005","journal-title":"J. Phys. A"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1017\/S0022112069000176","article-title":"Finite bandwidth, finite amplitude convection","volume":"38","author":"Newell","year":"1969","journal-title":"J. Fluid. Mech."},{"key":"ref_37","unstructured":"Pelce, P. (1988). A theory of thermal propagation of flames. Dynamics of Curved Fronts, Academic Press."},{"key":"ref_38","first-page":"77","article-title":"Painleve analysis of a class of nonlinear diffusion equations","volume":"9","author":"Chandrasekaran","year":"1996","journal-title":"Int. J. Stoch. Anal."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"2061","DOI":"10.1109\/JRPROC.1962.288235","article-title":"An active pulse transmission line simulating nerve axon","volume":"50","author":"Nagumo","year":"1962","journal-title":"Proc. IRE"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1007\/s12043-012-0504-1","article-title":"On the sarp front-type solution of the Nagumo equation with nonlinear diffusion and convection","volume":"80","author":"Mansour","year":"2013","journal-title":"Pramana"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1016\/0001-8708(70)90023-X","article-title":"Nagumo\u2019s equation","volume":"4","author":"McKean","year":"1970","journal-title":"Adv. Math."},{"key":"ref_42","unstructured":"Crank, J. (1975). The Mathematics of Diffusion, Clarendon Press. [2nd ed.]."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/1\/69\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:22:35Z","timestamp":1759918955000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/1\/69"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,3]]},"references-count":42,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["sym17010069"],"URL":"https:\/\/doi.org\/10.3390\/sym17010069","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,1,3]]}}}