New Green's functions for some nonlinear oscillating systems and related PDEs

During the past three decades, the advantageous concept of the Green's function has been extended from linear systems to nonlinear ones. At that, there exists a rigorous and an approximate extension. The rigorous extension introduces the so-called backward and forward propagators, which play the same role for nonlinear systems as the Green's function plays for linear systems. The approximate extension involves the Green's formula for linear systems with a Green's function satisfying the corresponding nonlinear equation. For the numerical evaluation of nonlinear ordinary differential equations, the second approach seems to be more convenient. In this article we study a hierarchy of nonlinear partial differential equations that can be approximated by the second approach. Green's functions for particular nonlinearities are derived explicitly. Numerical error analysis in the case of exponential nonlinearity for different source functions supports the advantage of the approach.