Abstract
The solution of an n-dimensional stochastic differential equation driven by Gaussian white noises is a Markov vector. In this way, the transition joint probability density function (JPDF) of this vector is given by a deterministic parabolic partial differential equation, the so-called Fokker-Planck-Kolmogorov (FPK) equation. There exist few exact solutions of this equation so that the analyst must resort to approximate or numerical procedures. The finite element method (FE) is among the latter, and is reviewed in this paper. Suitable computer codes are written for the two fundamental versions of the FE method, the Bubnov-Galerkin and the Petrov-Galerkin method. In order to reduce the computational effort, which is to reduce the number of nodal points, the following refinements to the method are proposed: 1) exponential (Gaussian) weighting functions different from the shape functions are tested; 2) quadratic and cubic splines are used to interpolate the nodal values that are known in a limited number of points. In the applications, the transient state is studied for first order systems only, while for second order systems, the steady-state JPDF is determined, and it is compared with exact solutions or with simulative solutions: a very good agreement is found.
Highlights
It is widely recognized that many types of agencies acting on engineering structures and equipments have random characteristics
In the finite element (FE) procedure the state space is discretized into a number of finite regions: the finite element mesh consists of a grid of points at which the joint probability density function (JPDF) p(z,t) is to be computed
This paper aims at giving a contribution to some questions regarding the FE method for solving the FPK equatio n, which remains unanswered or partially answered
Summary
It is widely recognized that many types of agencies acting on engineering structures and equipments have random characteristics. The severe limitations of the FPK equation approach resides in that exact analytical solutions are available in a restricted number of cases, and mostly in the steady state. In [41], two types of variational approaches are presented, both of which are aimed at finding approximate values of the non-zero eigenvalues of the FPK operator (the first eigenvalue is zero, and it corresponds to the steady state PDF). The former one constructs an approximate Hermitian operator, and the other is based on a perturbation expansion. In order to contain the computing time, the transient state is studied for first order systems only, while for second order systems the steady-state JPDF is determined
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