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Applicable Analysis
An International Journal
Volume 92, 2013 - Issue 9
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Articles

Fractional sewage treatment models with impulses at variable times

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Pages 1959-1979 | Received 25 Jul 2011, Accepted 19 Jul 2012, Published online: 11 Sep 2012
 

Abstract

In this article, two classes of sewage treatment models involving Caputo fractional derivative with impulses at variable times are established. Firstly, some new developed concepts such as critical points and noncritical points, () nonlinear functions sets, and ()-mild solutions are introduced. Secondly, local and global existence of solutions are obtained and other interesting properties such as local stability and periodic oscillatory phenomena of solutions are presented. Finally, a useful wastewater treatment model is given to demonstrate our theory results.

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Acknowledgements

The authors thank the referees for their careful reading of the manuscript and insightful comments, which helped to improve the quality of this article. They also like to acknowledge the valuable comments and suggestions from the editors, which vastly contributed to improve the presentation of this article. This work is partly supported by National Natural Science Foundation of China (10961009,10971173).

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