A general form of Krylov–Bogoliubov–Mitropolskii method for solving nonlinear partial differential equations

Abstract

A general asymptotic solution can be obtained for a class of partial differential equations with small nonlinearities whose dominant linear part involves an nth order, n = 2 , 3 , … , time derivative. The method used is an extension of the Krylov–Bogoliubov–Mitropolskii (KBM) method. The formulation as well as the determination of the solution is quite easy. Many authors have extended the KBM method to investigate some physical and mechanical oscillating systems, modelled by either second-order hyperbolic type partial differential equations or certain partial differential equations with third-order time derivative. They mainly extended the method to investigate individual problems. On the contrary, the proposed solution covers various types of nonlinear problems modelled by partial differential equations whose linear part involves second-, third-, etc. order time derivative. Substituting n = 2 , 3 into the general formula, it can be shown that the formula readily becomes to those extended by several authors. The method is illustrated with a physical problem whose linear part involves a third-order time derivative.