A Homotopic Approach to the Solution of the Fokker-Planck-Kolmogorov Equation
نویسندگان
چکیده
A homotopic Galerkin approach to the solution of the Fokker-Planck-Kolmogorov equation is presented. It is argued that the ideal Hilbert Space to approximate the exact solution, ψ∗, is the space L2(dψ) where dψ∗ is the probability measure induced on <n by the solution ψ∗ itself. Since this space is not accessible, it is shown that given an initial approximation sufficiently close to the exact solution, it is possible to capture the exact solution iteratively by propagating the solution over sufficiently small time intervals using the Galerkin projection method. Further, it is shown that the Fokker-Planck equation can be solved for a target nonlinear system by performing a homotopic iteration over a class of dynamical systems, Dp, indexed by a parameter p ∈ [0, 1]. The Galerkin projection method is employed to obtain recursively the solution for the desired dynamical system, D1, (corresponding to p = 1) starting from a solution that is exactly known. (for the dynamical system D0)
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