Sustainable Water Treatment through Fractional-Order Chemostat Modeling with Sliding Memory and Periodic Boundary Conditions: A Mathematical Framework for Clean Water and Sanitation

TL;DR

This paper introduces a novel fractional-order chemostat model with sliding memory and periodic boundary conditions to better simulate microbial pollutant degradation in sustainable water treatment, providing rigorous mathematical analysis of its solutions.

Contribution

It develops a new fractional chemostat framework incorporating sliding memory and PBCs, with proofs of existence, uniqueness, positivity, and stability of solutions, advancing mathematical modeling in water treatment.

Findings

Model captures realistic memory effects in bioprocesses.

Existence and uniqueness of solutions are established.

Solutions remain positive and bounded, ensuring physical relevance.

Abstract

This work develops and analyzes a novel fractional-order chemostat system (FOCS) with a Caputo fractional derivative (CFD) featuring a sliding memory window and periodic boundary conditions (PBCs), designed to model microbial pollutant degradation in sustainable water treatment. By incorporating the Caputo fractional derivative with sliding memory (CFDS), the model captures time-dependent behaviors and memory effects in biological systems more realistically than classical integer-order formulations. We reduce the two-dimensional fractional differential equations (FDEs) governing substrate and biomass concentrations to a one-dimensional FDE by utilizing the PBCs. The existence and uniqueness of non-trivial, periodic solutions are established using the Caratheodory framework and fixed-point theorems, ensuring the system's well-posedness. We prove the positivity and boundedness of solutions, demonstrating that substrate concentrations remain within physically meaningful bounds and biomass concentrations stay strictly positive, with solution trajectories confined to a biologically feasible invariant set. Additionally, we analyze non-trivial equilibria under constant dilution rates and derive their stability properties. The rigorous mathematical results confirm the viability of FOCS models for representing memory-driven, periodic bioprocesses, offering a foundation for advanced water treatment strategies that align with Sustainable Development Goal 6 (Clean Water and Sanitation).