Response Statistics of Single-Degree-of-Freedom Systems with Lévy Noise by Improved Path Integral Method

In this paper, we put forward an improved version of the path integral (PI) method for the response statistics of single-degree-of-freedom (SDOF) system excited by Lévy noise. To overcome the problem of large amount of calculation and storage, the PI method is simplified and parallelized, which makes the PI method for SDOF system with Lévy noise feasible and efficient. As the key to the PI method, the short-time transition probability density function (PDF) of the SDOF system is derived and verified by proving that the PI solution satisfies the corresponding fractional Fokker-Planck-Kolmogorov (FPK) equation. The fractional FPK equation, which is the governing equation of the SDOF system, is derived through the characteristic function and the Chapman-Kolmogorov equation. To solve the problem of large storage and calculations in the PI method, we simplify the one-step iteration formula and perform parallel calculations on the simplified formula. The simplification of the one-step iteration formula reduces one-fold integration, thereby reducing the storage capacity of the one-step transition matrix. Parallel calculation by domain decomposition can effectively reduce the calculation time, which can be seen from the running time of two prototypical examples. Besides, to show the effectiveness of the improved PI method, Monte Carlo solutions and analytical solutions are used as reference solutions.