International Journal of Stochastic Analysis

International Journal of Stochastic Analysis / 2002 / Article

Open Access

Volume 15 |Article ID 541721 | 17 pages | https://doi.org/10.1155/S1048953302000060

Connections between the convective diffusion equation and the forced Burgers equation

Received01 Sep 2000
Revised01 Oct 2001

Abstract

The convective diffusion equation with drift b(x) and indefinite weight r(x), ϕt=x[aϕxb(x)ϕ]+λr(x)ϕ,(1) is introduced as a model for population dispersal. Strong connections between Equation (1) and the forced Burgers equation with positive frequency (m0), ut=2ux2uux+mu+k(x),(2) are established through the Hopf-Cole transformation. Equation (2) is a prime prototype of the large class of quasilinear parabolic equations given by ut=2ux2+(f(v))x+g(v)+h(x).(3) A compact attractor and an inertial manifold for the forced Burgers equation are shown to exist via the Kwak transformation. Consequently, existence of an inertial manifold for the convective diffusion equation is guaranteed. Equation (2) can be interpreted as the velocity field precursed by Equation (1). Therefore, the dynamics exhibited by the population density in Equation (1) show their effects on the velocity expressed in Equation (2). Numerical results illustrating some aspects of the previous connections are obtained by using a pseudospectral method.

Copyright © 2002 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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