Green's functions for higher order nonlinear equations

TL;DR

This paper extends the Green's function method to higher order nonlinear differential equations, enabling exact solutions under certain conditions and demonstrating its effectiveness through specific examples and numerical analysis.

Contribution

It introduces a generalized Green's function approach for higher order nonlinear equations, expanding the applicability of the method beyond second order cases.

Findings

Green's function solutions are possible for higher order nonlinear equations with homogeneous nonlinear terms.

The method is effective for Boussinesq and Korteweg-de Vries equations, requiring higher order Green functions.

Numerical error analysis confirms the accuracy and practicality of the proposed solutions.

Abstract

The well-known Green's function method has been recently generalized to nonlinear second order differential equations. In this paper we study possibilities of exact Green's function solutions of nonlinear differential equations of higher order. We show that, if the nonlinear term satisfies a generalized homogeneity property, then the nonlinear Green's function can be represented in terms of the homogeneous solution. Specific examples and a numerical error analysis support the advantage of the method. We show how, for the Bousinesq and Kortweg-de Vries equations, we are forced to introduce higher order Green functions to obtain the solution to the inhomogeneous equation. The method proves to work also in this case supporting our generalization that yields a closed form solution to a large class of nonlinear differential equations, providing also a formula easily amenable to numerical evaluation.