Connections between the convective diffusion equation and the forced Burgers equation

Nejib Smaoui and Fethi Belgacem

International Journal of Stochastic Analysis, 2002, vol. 15, 1-17

Abstract:

The convective diffusion equation with drift b ( x ) and indefinite weight r ( x ) , ∂ ϕ ∂ t = ∂ ∂ x [ a ∂ ϕ ∂ x − b ( x ) ϕ ] + λ r ( x ) ϕ ,       ( 1 ) is introduced as a model for population dispersal. Strong connections between Equation (1) and the forced Burgers equation with positive frequency ( m ≥ 0 ) , ∂ u ∂ t = ∂ 2 u ∂ x 2 − u ∂ u ∂ x + m u + 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 ∂ u ∂ t = ∂ 2 u ∂ x 2 + ∂ ( 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.

Date: 2002
References: Add references at CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://downloads.hindawi.com/journals/IJSA/15/541721.pdf (application/pdf)
http://downloads.hindawi.com/journals/IJSA/15/541721.xml (text/xml)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:hin:jnijsa:541721

DOI: 10.1155/S1048953302000060

Access Statistics for this article

More articles in International Journal of Stochastic Analysis from Hindawi
Bibliographic data for series maintained by Mohamed Abdelhakeem ().