Mathematical Analysis of Biodegradation Model under Nonlocal Operator in Caputo Sense
Abstract
:1. Introduction and Motivation
- and denote the soluble substrate, heterotroph (particulate biomass), soluble micropollutants, and particulate micropollutants, respectively.
- and represent the substrate concentration in the feed, the flow rate through bioreactor, and the volume of the bioreactor, respectively.
- and denote the recycle concentration factor, heterotroph concentration, and the recycle ratio, respectively.
- and denote the decay coefficient of particulate biomass, soluble micropollutants concentration, and the total suspended solids, respectively.
- represent the biological removal rate and residence time.
- and represent the Henry coefficient of micropollutant, the flow rate of aeration, and the concentration of particulate micropollutants in the feed.
2. Mathematical Analysis
2.1. Theoretical Analysis
2.1.1. Steady States and Its Stability
- The proposed model has two steady states. The first one is the washout branch (), when . The washout branch is given byThe second one is the no-washout branch (), when The no-washout branch is given by:Now, we present the local stability (LS) of the steady states. Consider the Jacobian matrix as:
- For the LS of the washout solution branch, consider the eigenvalues of at the washout branch as:Since The eigenvalue whenIt follows that whenOtherwise, for , the WSS is LS if is enough low:
- For an LS of no-washout solution branch, from the second equation of model (8), we haveFor the no-washout branch soFor the above equation, the characteristic polynomail of is as follows:The no-washout solution is of interest when all solution are positive. So, in these cases, the coefficients of are positive. From the above equations, we see that , , and To prove that considerThus so no-washout solution branch is LS.
2.1.2. Applications of Fixed Point Theory to the Existence of Solution
2.1.3. Ulam–Hyers Stability
- for
- for
2.2. Computational Analysis
3. Numerical Simulations
4. Conclusions
- Using more generalized nonolocal operators with different kernels.
- Involving fuzziness and uncertainty in the model.
- Controllability and chaotic behaviour of the model.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Compounds | |||
---|---|---|---|
Highly biodegradable | |||
Erythromycin (ERY) | |||
Ibuprofen (IBP) | |||
Roxithromycin(ROX) | |||
Naproxen(NPX) | |||
Slowly biodegradable | |||
Trimethoprim(TMP) | |||
Sulfamethoxazole(SMX) | |||
Fluoxetine(FLX) |
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Alqahtani, R.T.; Ahmad, S.; Akgül, A. Mathematical Analysis of Biodegradation Model under Nonlocal Operator in Caputo Sense. Mathematics 2021, 9, 2787. https://doi.org/10.3390/math9212787
Alqahtani RT, Ahmad S, Akgül A. Mathematical Analysis of Biodegradation Model under Nonlocal Operator in Caputo Sense. Mathematics. 2021; 9(21):2787. https://doi.org/10.3390/math9212787
Chicago/Turabian StyleAlqahtani, Rubayyi T., Shabir Ahmad, and Ali Akgül. 2021. "Mathematical Analysis of Biodegradation Model under Nonlocal Operator in Caputo Sense" Mathematics 9, no. 21: 2787. https://doi.org/10.3390/math9212787
APA StyleAlqahtani, R. T., Ahmad, S., & Akgül, A. (2021). Mathematical Analysis of Biodegradation Model under Nonlocal Operator in Caputo Sense. Mathematics, 9(21), 2787. https://doi.org/10.3390/math9212787