article · European Journal of Pure and Applied Mathematics
A novel non-perturbative approach offers an effective way to address scientific and technological challenges involving strongly nonlinear oscillators. The technique simplifies processing time in comparison to traditional methods by converting nonlinear ordinary differential equations into linear forms equivalent to simple harmonic motion, generating a new frequency. Numerical comparisons conducted within mathematical software validate that this formulation delivers high accuracy, outperforming several well-known approximate methodologies. Theoretical predictions align closely with numerical solution tests, reinforcing the reliability of the results. Unlike classical perturbation methods that depend on Taylor expansions to approximate restoring forces, this approach also permits direct stability analysis. Consequently, the formulation provides an accurate, dependable computational tool for analysing approximations of highly nonlinear oscillating systems in mathematical software environments.
Nonlinear oscillations appear widely across science and engineering, but modelling them accurately often requires demanding computational resources or simplifying assumptions. By converting complex nonlinear equations into simpler linear forms without relying on Taylor expansions, this mathematical technique decreases computation time while maintaining high accuracy and enabling stability checks. This provides researchers and engineers with a more reliable method for simulating complex dynamic systems.
The method could potentially be integrated into mathematical and simulation software tools used by engineers analysing nonlinear dynamic systems. Because the abstract demonstrates the approach purely through theoretical equations and numerical validation in mathematical software, this research represents early-stage foundational work. Any practical deployment in industrial design or simulation pipelines would require further applied testing within specific engineering domains.
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In this paper, we address several scientific and technological challenges with a novel non-perturbative approach (NPA), simplifying processing time compared to traditional methods. The proposed approach transforms nonlinear ordinary differential equations into linear ones, akin to simple harmonic motion, producing a new frequency. This method yields highly accurate outcomes, surpassing well-known approximate methodologies, as validated through numerical comparisons in mathematical software. The congruence between numerical solution tests and theoretical predictions further supports our findings. While classical perturbation methods rely on Taylor expansions to simplify restoring forces, NPA also enables stability analysis. Thus, for analyzing approximations of highly nonlinear oscillators in mathematical software, NPA serves as a more reliable tool.
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DOI: 10.29020/nybg.ejpam.v17i3.5339
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