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Qualitative analysis and solutions of bounded travelling waves for the fluidized-bed modelling equation

Published online by Cambridge University Press:  30 March 2010

Weiguo Zhang ,
Lanyun Bian  and
Yan Zhao
Weiguo Zhang
Affiliation:
College of Science, University of Shanghai for Science and Technology, Shanghai 200093, People's Republic of China ([email protected])
Lanyun Bian
Affiliation:
College of Science, University of Shanghai for Science and Technology, Shanghai 200093, People's Republic of China ([email protected])
Yan Zhao
Affiliation:
College of Science, University of Shanghai for Science and Technology, Shanghai 200093, People's Republic of China ([email protected])
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Abstract

We apply the theory of planar dynamical systems to carry out a qualitative analysis for the planar dynamical system corresponding to the fluidized-bed modelling equation. We obtain the global phase portraits of this system under various parameter conditions and the existence conditions of bounded travelling-wave solutions of this equation. According to the discussion on relationships between the behaviours of bounded travelling-wave solutions and the dissipation coefficients ε and δ, we find a critical value λ0 for arbitrary travelling-wave velocity υ. This equation has a unique damped oscillatory solution as ∥ε + δυ∥ < λ0 and ∥ε + δυ∥ ≠ 0, while it has a unique monotone kink profile solitary-wave solution as ∥ε + δυ∥ > λ0. By means of the undetermined coefficients method, we obtain the exact bell profile solitary-wave solution and monotone kink profile solitary-wave solution. Meanwhile, we obtain the approximate damped oscillatory solution. We point out the positions of these solutions in the global phase portraits. Finally, based on integral equations that reflect the relationships between the approximate damped oscillatory solutions and the implicit exact damped oscillatory solutions, error estimates for the approximate damped oscillatory solutions are presented.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 140 , Issue 2 , April 2010 , pp. 241 - 257
Copyright
Copyright © Royal Society of Edinburgh 2010

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