Mathematical Analysis of Biodegradation Model under Nonlocal Operator in Caputo Sense

• Published in: Mathematics (Basel) Vol. 9; no. 21; p. 2787
• Main Authors: Alqahtani, Rubayyi T., Ahmad, Shabir, Akgül, Ali
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01-11-2021
• Subjects: Biodegradation; Biomass; Boundary conditions; Caputo operator; Euler method; Fixed points (mathematics); Functional analysis; Mathematical analysis; Mathematical models; Mathematics; Ordinary differential equations; Pollutants; Sludge; Steady state models; Wastewater treatment; Water treatment
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math9212787

To lower the concentration of organic pollutants in the effluent stream, wastewater must be treated before being discharged into the environment. The question of whether wastewater treatment facilities can successfully reduce the concentration of micropollutants found in their influent streams is becoming increasingly pressing. The removal of micropollutants in treatment plants is investigated using a model that incorporates biodegradation and sorption as the key processes of micropollutant removal. This article provides the mathematical analysis of the wastewater model that describes the removal of micropollutant in treatment plants under a non-local operator in Caputo sense. The positivity of the solution is presented for the Caputo fractional model. The steady state’s solution of model and their stability is presented. The fixed point theorems of Leray–Schauder and Banach are used to deduce results regarding the existence of the solution of the model. Ulam–Hyers (UH) types of stabilities are presented via functional analysis. The fractional Euler method is used to find the numerical results of the proposed model. The numerical results are illustrated via graphs to show the effects of recycle ratio and the impact of fractional order on the evolution of the model.