  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 exist a rigorous and an approximate extensions. 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 non-linearities are derived
explicitly. Numerical error analysis in the case of exponential non-linearity
for different source functions supports the advantage of the approach.
