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**Numerical solution of the stationary FPK equation using Shannon wavelets.**
*(English)*
Zbl 1237.65150

Summary: The Fokker-Planck-Kolmogorov (FPK) equation governs the probability density function (p.d.f.) of the dynamic response of a particular class of linear or non-linear system to random excitation. This paper proposes a numerical method for calculating the stationary solution of the FPK equation, which is based upon the weighted residual approach using Shannon wavelets as shape functions. The method is developed here for an n -dimensional system and its relationship with the distributed approximating functional (DAF) approach is investigated. For the purposes of validation, numerical results obtained using the proposed method are compared with available exact solutions and numerical solutions for some non-linear oscillators. For the systems considered excellent results over the main body and tails of the marginal distributions are obtained. The accuracy and efficiency of the method are investigated in comparison to the finite element method (FEM).

### MSC:

65T60 | Numerical methods for wavelets |

65C20 | Probabilistic models, generic numerical methods in probability and statistics |

34F05 | Ordinary differential equations and systems with randomness |

70L05 | Random vibrations in mechanics of particles and systems |

82C80 | Numerical methods of time-dependent statistical mechanics (MSC2010) |