Explicit solutions of a two-dimensional fourth-order nonlinear diffusion equation
type: Item
instance of: scholarly article
title: Explicit solutions of a two-dimensional fourth-order nonlinear diffusion equation
zbMATH Open document ID: 1028.35037
DOI: 10.1016/S0895-7177(03)00015-3
author: S. I. Betelu
author: J. R. King
published in: Mathematical and Computer Modelling
publication date: 2003-10-15T00:00:00Z
review text: A two-dimensional fourth-order nonlinear diffusion equation of the form \[ \frac{\partial u}{\partial t}+\nabla\left(u\nabla\nabla^2u\right)=0 \] is considered. Polynomial solutions of the form \(u(t,x,y)= c(t)\left(1-x^2/a(t)-y^2/b(t)\right)^2\), where the time evolution \(a(t)\), \(b(t)\) is defined parametrically, are constructed. Elliptic (\(a>b>0\)) and hyperbolic (\(a>0,b<0\)) interfaces are discussed. In the case of parabolic interface solutions have the form \(u(t,x,y)=c(t)\left(x-a(t)-y^2/b(t) \right)^2\). Extensions of the obtained solutions to three and higher dimensions are noted.
Mathematics Subject Classification ID: 35C05
Mathematics Subject Classification ID: 35K25
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 1993298
zbMATH Keywords: interfaces
zbMATH Keywords: polynomials
reviewed by: Valery V. Karachik
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1993298