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

TL;DR

This paper explores new Green's functions for certain nonlinear oscillating systems and related PDEs, deriving explicit solutions and analyzing numerical errors to improve understanding and computation of nonlinear differential equations.

Contribution

It introduces a hierarchy of nonlinear PDEs with explicit Green's functions and demonstrates the effectiveness of an approximate approach for numerical evaluation.

Findings

Explicit Green's functions derived for specific nonlinearities

Numerical error analysis supports the approximate method's advantages

Improved computational techniques for nonlinear differential equations

Abstract

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.