Functional Relations as a Tool for Analysing Differential Equations

This paper examines the use of functional relations as a comprehensive analytical instrument for resolving and streamlining differential equations across many categories, including linear, nonlinear, stochastic, delayed, and hybrid systems. The objective is to augment model interpretability, diminish dimensionality, and optimise computational efficiency in intricate systems. The methodology incorporates symmetric Lie analysis, stochastic calculus, operator theory, and symbolic computation. Functional relationships were established using infinitesimal symmetries, Lyapunov functionals, and moment-based analysis. Numerical and symbolic experiments were conducted utilising Maple, Mathematica, and MATLAB. Functional relations lowered model dimensionality by as much as 40% and enhanced prediction accuracy. For the Korteweg de Vries (KdV) equation, scale-invariant relationships accurately represented soliton dynamics with an error margin of less than 1.8%. In stochastic systems, functional connections among moments reduced prediction errors by 12%. In hybrid systems, piecewise invariants reduced oscillation amplitudes by 25%. Inverse problems demonstrated a 13% improvement in parameter reconstruction accuracy and an 18% reduction in calculation time. Functional relations provide a strong foundation for analysing differential equations, especially in systems marked by nonlinearity, uncertainty, or structural complexity. The results endorse the incorporation of functional relationships into control systems, digital twins, and hybrid models. Their formalisation and adaptive implementation create new opportunities for interpretable, resource-efficient modelling in applied sciences and engineering.