TITLE:
Numeric Solution of the Fokker-Planck-Kolmogorov Equation
AUTHORS:
Claudio Floris
KEYWORDS:
Stochastic Differential Equations; Markov Vectors; Fokker-Planck-Kolmogorov Equation; Finite Element Numeric Solution; Modified Hermite Weighting Functions; Spline Interpolation
JOURNAL NAME:
Engineering,
Vol.5 No.12,
November
26,
2013
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.