Stability and chemical modeling of quantifying disparities in atmospheric analysis with sustainable fractal fractional approach

Fractional-order derivative-based modeling is crucial for describing real-world forecasting problems and analyzing proposed models. It provides an advanced framework for examining intricate variations in various systems, enhancing understanding and analysis. We present a new fractional order nonlinear model for dynamics and forecasting of nitrogen oxides (NO_x) and ozone (O₃) in the atmosphere, crucial for air quality regulation and smog formation. The study compares invariant regions and solution pathways within complex reaction mechanisms impacting the ozone layer. The study uses Banach's contraction theorem, Schauder's fixed-point theorem, and fixed-point theory to study a proposed model. Fundamental equations are studied, and local sensitivity analysis is performed using MATLAB's Sim-Biology toolbox. The model is stabilized using the linear feedback regulate method, considering a fractional-order system with a managed design. Moment quintile regression is used to validate coefficient parameters for forecasting and modeling. Theoretical predictions are validated using the two-step Newton Polynomial Method, and numerical results show high agreement between theoretical analysis and numerical results. The research introduces a new method for quantifying invariant curve disparities using advanced Model Reduction Techniques (MRTs), focusing on the proximity between model predictions and actual data points. The method identifies achievable invariant regions and influential parameters for NOx and O₃ under environmental stressors. Results validate theoretical and experimental findings by employing local as well as non-singular kernels at various fractional order values and fractal dimensions to show the strong memory effect.