Mathematical Modeling and Optimization Framework for Complex Engineering Systems Using Nonlinear Analysis

This paper presents a comprehensive mathematical modeling and optimization framework for analyzing complex engineering systems governed by nonlinear relationships and multi-parameter interactions. The study develops an integrated approach based on differential equations, functional approximations, and constrained optimization techniques to capture system behaviour with high accuracy. The methodology incorporates sensitivity analysis to identify dominant variables and applies stability assessment to ensure model robustness across varying boundary conditions. A systematic simulation strategy is employed to validate theoretical predictions and quantify deviations arising from uncertainties in measurement and material properties. The results demonstrate that the proposed mathematical framework enhances prediction accuracy, reduces computational effort, and offers a generalized formulation adaptable to multiple engineering applications. The study contributes to improved decision-making in system design and performance optimization, while also establishing a foundation for future work on hybrid analytical–numerical models.