Operator method for the perturbative solution of ordinary differential equations

Abstract

We adapt the Feynman operator calculus to the Lie operators connected with the solution of ordinary, coupled, first order differential equations, particularly those arising from non-linear oscillations or from interactions between modes. Combining this calculus with the method of averaging we obtain approximate solutions which are algebraic and exponential functions of the appropriate expansion parameter rather than power series of this parameter. In addition to illustrating the method for the damped harmonic oscillator and for the van der Pol oscillator we present an approximate solution of the Lotka-Volterra problem of two competing species. The solutions are compared numerically with those of a variant of the Bogoliubov-Krylov-Mitropolsky (BKM) method of averaging. The bulk of the analytic calculations both for the operator algorithm and for the BKM method have been automated.