Inverse dynamic problems, the Focker— Plank—Kolmogorov equation and the Laplace iterative process

Abstract

Abstract - In this paper the Focker-Plank-Kolmogorov equation based on the Laplace iteration process in the two-dimensional case is reduced to the system of differential equations for the right-hand sides of the dynamic system and system of equations on probability density. In some cases corollary systems can be integrated that allows to obtain the exact formulas for the right-hand sides of the dynamic system and probability density. To deduce some solutions group analysis of differential equations and invariant solutions are used [9, 10]. The solution obtained are functionally arbitrary. In the paper determination of probability density and right-hand sides of the dynamic system by the most probable motion path is also considered. Supposing probability density known, we obtain a differential equation of the second order in the partial derivatives of the right-hand sides of the dynamic system. At last, in the multidimensional case, we use the Focker-Plank-Kolmogorov equation to obtain an equation on the potential of the dynamic system forces.