Numerical meshfree path integration method for non-linear dynamic systems

• Published in: Applied mathematics and computation Vol. 197; no. 1; pp. 426 - 434
• Main Authors: Xie, Wen-Xian, Xu, Wei, Cai, Li
• Format: Journal Article
• Language: English
• Published: New York, NY Elsevier Inc 15.03.2008; Elsevier
• Subjects: Exact sciences and technology; Mathematical analysis; Mathematics; Measure and integration; Meshfree method; Non-linear dynamic systems; Numerical analysis; Numerical analysis. Scientific computation; Numerical approximation; Ordinary differential equations; Path integration method; Sciences and techniques of general use; Meshfree method; Non-linear dynamic systems; Path integration method; Linear interpolation; Numerical integration; Differential equation; Gauss method; Probability distribution; Numerical method; Non linear system; Stochastic system; Numerical approximation; Numerical analysis; First order equation; Applied mathematics; Least squares method
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2007.07.088

We present a new numerical meshfree path integration (MPI) method for non-linear dynamic systems. The obtained MPI method can be performed in the irregular computational domain and the probability density values of the random nodes in the domain can be calculated via the MPI method and the ordinary differential equations for the first and second-order moments on the basis of Gaussian closure method. The piecewise linear interpolation based on adaptive least squares is utilized as a post processor to approximate the probability density values on arbitrary positions. The good performance of the resulting method is finally shown in the numerical examples by using three specific non-linear dynamic systems: Duffing oscillator subjected to both harmonic and stochastic excitations, Duffing–Rayleigh oscillator subjected to both harmonic and stochastic excitations, and CHEN system driven by three different Gaussian white noises.