Computation of Green’s functions through algebraic decomposition of operators

F Adrián F Tojo

doi:10.1186/s13661-016-0671-y

Abstract

In this article, we use linear algebra to improve the computational time for obtaining Green’s functions of linear differential equations with reflection (DER). This is achieved by decomposing both the ‘reduced’ equation (the ODE associated with a given DER) and the corresponding two-point boundary conditions.

Highlights

1 Introduction Differential operators with reflection have recently been of great interest, partly due to their applications to supersymmetric quantum mechanics [ – ] or topological methods applied to nonlinear analysis [ ]
In [ ], the authors described a method to derive the Green’s function of a differential equation with constant coefficients, reflection and two-point boundary conditions. This algorithm was implemented in Mathematica in order to put it to a practical use
We have to point out that an nth-order linear differential equations with reflection (DER) is reduced to a ( n)th-order ordinary differential equation; see Theorem . and compare equations ( . ) and ( . )

Summary

Introduction

Differential operators with reflection have recently been of great interest, partly due to their applications to supersymmetric quantum mechanics [ – ] or topological methods applied to nonlinear analysis [ ].In the last years, the works in this field have been related to either obtaining eigenvalues and explicit solutions of different problems [ – ], their qualitative properties [ , ], or obtaining the associated Green’s function [ – ]. In [ ], the authors described a method to derive the Green’s function of a differential equation with constant coefficients, reflection and two-point boundary conditions. This is quite difficult when computing the Green’s functions since, in this case, we could have one, many, or no solutions of our original problem [ ]. The existence of Green’s functions for problems such as

Results

Conclusion