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Fractional sewage treatment models with impulses at variable times. (English) Zbl 1279.26021

Summary: 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, \(PC_{y_1}^{y_2}(PC_M^N)\) nonlinear functions sets, and \(PC_{y_1}^{y_2}(PC_M^N)\)-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.

MSC:

26A33 Fractional derivatives and integrals
34A37 Ordinary differential equations with impulses
35R12 Impulsive partial differential equations
35K90 Abstract parabolic equations
47J35 Nonlinear evolution equations
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