Monte Carlo method for solution of systems ODE

Abstract

An application of the Monte Carlo method to solution of problems Cauchy for system of linear and nonlinear ordinary differential equations is considered. the Monte Carlo method is topical for solution of large systems of equations and in the cases with given functions are not smooth enough. A system is reduced to an equivalent system of integral equations of the Volterra type. For linear systems this transformation allows to avoid restrictions, connected with a convergence of a majorizing process. Examples of estimates of functionals of solutions are given, and a behavior of their dispersions are discussed. In the general case an interval of solution is divided into finite subintervals in that nonlinear functions are approximated by polynomials. An obtained integral equations are solved by using branching Markov chains with an absorption. Appearing problems of parallel algorithms are discussed. As an example an one dimensional cubic equation is considered. A method of levels of branching process is discussed in details. a comparison of numerical results with a solution, obtained by the Runge Kutta method, is presented.