An innovative step-by-step stochastic linearization procedure for solving deterministic nonlinear differential equations

Abstract

Deterministic nonlinear differential equations are widely used in engineering to model the response of different mechanical systems. In most cases, exact analytical solutions are not available, and approximate techniques are therefore required. Deterministic linearization methods constitute a common class of approximate techniques. However, the development of methods that combine high accuracy with computational efficiency remains a challenging task.

In this paper, an innovative procedure based on a stochastic framework is proposed for solving deterministic nonlinear differential equations. The key idea is to replace the initial condition by a Gaussian random variable with small variance and mean equal to the deterministic initial value. This transformation turns the deterministic response into a stochastic process whose probability density function evolves according to the diffusion-free Fokker-Planck-Kolmogorov equation. Starting from this formulation, the evolution equation of the first-order statistics is derived, and an innovative numerical method, based on a step-by-step stochastic linearization, is introduced. The method is based on a subdivision of the mean value of the nonlinear term in different steps, so that within each step a closed-form linearized solution can be obtained together with the corresponding time instant. It is shown that, for sufficiently small variance of the initial condition, the mean value of the stochastic process coalesces with the solution of the original deterministic nonlinear equation.

The reliability of the proposed procedure is assessed through numerical applications to Kelvin-Voigt-type models with one and two nonlinear springs. The results demonstrate excellent agreement with both direct Runge-Kutta integration and quasi-linearization method, thereby confirming their accuracy and effectiveness. Additionally, it is worth highlighting the enhanced performance of the proposed procedure, as it strongly reduces computational time.