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Journal of Applied Mathematics and Stochastic Analysis
Volume 15 (2002), Issue 1, Pages 53-69

Connections between the convective diffusion equation and the forced Burgers equation

Kuwait University, Department of Mathematics and Computer Science, P.O. Box 5969, Safat 13060, Kuwait

Received 1 September 2000; Revised 1 October 2001

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.


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.