Nonlinear random vibration (Q1918301)

This note is concerned with the approximate solution of initial value problems for nonlinear second-order equations of type: \(u''+ \alpha g(u, u')+ \beta h(u)= f(t)\). In physical terms, these equations arise in nonlinear oscillations where \(g\) and \(h\) represent the damping and the restoring forces and \(f\) the external force. The author proposes to obtain the solution \(u\) as a series \(u= \sum_{n\geq 0} u_n\), where the terms may be computed recursively in terms of the so-called Adomian polynomials. The author claims that the above series converges (rapidly) to the solution of the initial value problem but neither the conditions nor the proof of this fact are given. An explicit application to Duffin's equation is also presented. Finally, a generalization of this method to random processes is also given.