Fractional sewage treatment models with impulses at variable times

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.

Keywords:

• fractional sewage treatment model
• impulses
• variable times
• existence and stability
• periodic oscillatory

AMS Subject Classifications::

• 26A33
• 34A37
• 35R12
• 35K90
• 47J35

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).