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 |
Keywords:
fractional sewage treatment model; impulses; variable times; existence and stability; periodic oscillatoryReferences:
| [1] | DOI: 10.1142/0906 · doi:10.1142/0906 |
| [2] | Yang T, Impulsive Control Theory (2001) |
| [3] | DOI: 10.1155/9789775945501 · doi:10.1155/9789775945501 |
| [4] | Xiang X, Southeast Asian Bull. Math. 30 pp 367– (2006) |
| [5] | DOI: 10.1080/02331930500530401 · Zbl 1101.45002 · doi:10.1080/02331930500530401 |
| [6] | DOI: 10.1016/j.na.2009.01.139 · Zbl 1238.34079 · doi:10.1016/j.na.2009.01.139 |
| [7] | DOI: 10.1007/s00025-010-0057-x · Zbl 1209.34095 · doi:10.1007/s00025-010-0057-x |
| [8] | Wang J, Math. Commun. 16 pp 235– (2011) |
| [9] | K. Diethelm,The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010 · Zbl 1215.34001 · doi:10.1007/978-3-642-14574-2 |
| [10] | Kilbas AA, North-Holland Mathematics Studies 204 (2006) |
| [11] | Lakshmikantham V, Theory of Fractional Dynamic Systems (2009) |
| [12] | Miller KS, An Introduction to the Fractional Calculus and Differential Equations (1993) |
| [13] | Podlubny I, Fractional Differential Equations (1999) |
| [14] | Tarasov VE, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media (2010) |
| [15] | DOI: 10.1007/s10440-008-9356-6 · Zbl 1198.26004 · doi:10.1007/s10440-008-9356-6 |
| [16] | Ahmad B, Topol. Methods Nonlinear Anal. 35 pp 295– (2010) |
| [17] | DOI: 10.1016/j.na.2009.07.033 · Zbl 1187.34026 · doi:10.1016/j.na.2009.07.033 |
| [18] | DOI: 10.1016/j.jmaa.2007.06.021 · Zbl 1209.34096 · doi:10.1016/j.jmaa.2007.06.021 |
| [19] | DOI: 10.1016/j.amc.2009.12.062 · Zbl 1191.34098 · doi:10.1016/j.amc.2009.12.062 |
| [20] | DOI: 10.1016/j.nonrwa.2010.06.013 · Zbl 1214.34010 · doi:10.1016/j.nonrwa.2010.06.013 |
| [21] | DOI: 10.1016/j.cnsns.2011.02.003 · Zbl 1223.45007 · doi:10.1016/j.cnsns.2011.02.003 |
| [22] | DOI: 10.1016/j.amc.2011.05.071 · Zbl 1242.34140 · doi:10.1016/j.amc.2011.05.071 |
| [23] | DOI: 10.7494/OpMath.2011.31.3.341 · Zbl 1228.26012 · doi:10.7494/OpMath.2011.31.3.341 |
| [24] | DOI: 10.1016/S0022-247X(02)00583-8 · Zbl 1026.34008 · doi:10.1016/S0022-247X(02)00583-8 |
| [25] | DOI: 10.1016/j.nonrwa.2010.05.029 · Zbl 1260.34017 · doi:10.1016/j.nonrwa.2010.05.029 |
| [26] | DOI: 10.1016/j.na.2009.01.105 · Zbl 1175.34082 · doi:10.1016/j.na.2009.01.105 |
| [27] | DOI: 10.1016/j.na.2009.01.202 · Zbl 1177.34084 · doi:10.1016/j.na.2009.01.202 |
| [28] | DOI: 10.1007/s10440-008-9356-6 · Zbl 1198.26004 · doi:10.1007/s10440-008-9356-6 |
| [29] | DOI: 10.1016/j.nahs.2009.01.008 · Zbl 1193.34056 · doi:10.1016/j.nahs.2009.01.008 |
| [30] | DOI: 10.1016/j.na.2009.08.046 · Zbl 1187.34108 · doi:10.1016/j.na.2009.08.046 |
| [31] | DOI: 10.1016/j.na.2010.11.007 · Zbl 1227.34009 · doi:10.1016/j.na.2010.11.007 |
| [32] | DOI: 10.1016/j.camwa.2009.05.016 · Zbl 1189.34007 · doi:10.1016/j.camwa.2009.05.016 |
| [33] | DOI: 10.1016/j.na.2009.01.138 · Zbl 1238.34113 · doi:10.1016/j.na.2009.01.138 |
| [34] | Wang J, Topol. Methods Nonlinear Anal. 38 pp 17– (2011) |
| [35] | DOI: 10.1016/j.camwa.2009.06.026 · Zbl 1189.34154 · doi:10.1016/j.camwa.2009.06.026 |
| [36] | Michalski MW, Instytut Matematyczny (1993) |
| [37] | Henry D, Geometric Theory of Semilinear Parabolic Equations (1981) · doi:10.1007/BFb0089647 |
| [38] | DOI: 10.1016/j.jmaa.2006.05.061 · Zbl 1120.26003 · doi:10.1016/j.jmaa.2006.05.061 |
| [39] | Ahmed NU, Semigroup Theory with Application to Systems and Control (1991) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
