ON SOME PRACTICAL APPLICATIONS OF CUBIC EQUATIONS IN SCIENCE AND ENGINEERING ANALYSIS

Abstract

Cubic polynomial equations arise frequently in engineering and applied physics as reduced forms of fundamental governing relations. This paper develops a unified analytical and computational framework for nine representative contexts, including Rayleigh surface waves, cubic equations of state, large-deflection membrane bending, combined bending and tension in elastic members, principal stress analysis, quantum-well eigenvalue problems, signal autocorrelation, cnoidal-wave solutions of the Korteweg–de Vries equation, and travelling waves in the FitzHugh–Nagumo model. For each case, the governing cubic is derived or systematically reformulated, and the physical interpretation of admissible roots is established. Particular attention is given to a numerically stable Lagrange-based formulation, which provides improved robustness and accuracy compared with classical closed-form and iterative approaches across different root regimes. The Rayleigh-wave secular cubic is analysed in detail, including the critical Poisson ratio  that separates distinct solution structures. Representative examples from wave propagation, thermodynamics, and stress analysis are used to compare performance with Newton–Raphson and companion-matrix eigenvalue methods. The results demonstrate consistent numerical stability and physical interpretability. The study further highlights an underlying geometric structure of cubic equations in the complex plane, revealing a unifying perspective across diverse engineering applications.