title: Self-similar solutions to a density-dependent reaction-diffusion model.
abstract: In this paper, we investigated a density-dependent reaction-diffusion equation, u(t)=(u(m))(xx)+u-u(m). This equation is known as the extension of the Fisher or Kolmogoroff-Petrovsky-Piscounoff equation, which is widely used in population dynamics, combustion theory, and plasma physics. By employing a suitable transformation, this equation was mapped to the anomalous diffusion equation where the nonlinear reaction term was eliminated. Due to its simpler form, some exact self-similar solutions with compact support have been obtained. The solutions, evolving from an initial state, converge to the usual traveling wave at a certain transition time. Hence, the connection between the self-similar solution and the traveling wave solution is quite clear from these results. Moreover, the solutions were found in a manner that propagates either to the right or to the left. Furthermore, the two solutions form a symmetric solution, expanding in both directions. Applications to spatiotemporal pattern formation in biological populations is discussed.
authors.affiliation: Division of Physics, School of Science, University of Phayao, Mueang Phayao, Phayao 56000, Thailand. waipot.ng@up.ac.th
authors.forename: Waipot
authors.initials: W
authors.lastname: Ngamsaad
authors.forename: Kannika
authors.initials: K
authors.lastname: Khompurngson
mesh_terms: D000465:Algorithms; D003198:Computer Simulation; D004058:Diffusion; D008956:Models, Chemical