A practical method for determining Green's functions using Hadamard's variational formula

Abstract

This article studies the behavior of the Green's function as a function of the length of the interval. The solution of the boundary-value problem $$\begin{gathered} \ddot u - q(t)u = - g(t), 0< t< a \hfill \\ u(0) = 0, \dot u(a) = 0 \hfill \\ \end{gathered} $$ can be expressed in the form $$u(t) = \int_0^a { G(t,y)} g(y) dy$$ The Green's functionG is viewed as a function of the interval lengtha, which leads to an effective initial-value method forG. Results of some numerical experiments are included.