BLUES function method in computational physics

TL;DR

The paper introduces the BLUES function method, a novel computational approach for solving nonlinear differential equations in physics, enabling exact or approximate solutions using Green's functions and related linear DEs.

Contribution

It presents the BLUES function concept, extending Green's function techniques to nonlinear DEs, with practical methods for exact and approximate solutions.

Findings

Demonstrated the method with analytical examples

Validated the approach through numerical computations

Suggested potential applications in various physics problems

Abstract

We introduce a computational method in physics that goes "beyond linear use of equation superposition" (BLUES). A BLUES function is defined as a solution of a nonlinear differential equation (DE) with a delta source that is at the same time a Green's function for a related linear DE. For an arbitrary source, the BLUES function can be used to construct an exact solution to the nonlinear DE with a different, but related source. Alternatively, the BLUES function can be used to construct an approximate piecewise analytical solution to the nonlinear DE with an arbitrary source. For this alternative use the related linear DE need not be known. The method is illustrated in a few examples using analytical calculations and numerical computations. Areas for further applications are suggested.