Green's Function
Table of Contents
Green's function of an inhomogeneous linear differential operator defined on a domain with specified initial conditions and boundar conditions is its impulse response.
That mean if \(L\) is the linear differential opertator, then
- the Green's function \(G\) is the solution of the equation \(LG = \delta\) where \(\delta\) is Dirac's delta function
- and the solution of the initial-value problem \(Ly = f\) is the convulution \(( G * f )\)
1. Convolution
Convolution of two functions \(f(t)\) and \(g(t)\) is defined as:
\begin{align} (f * g) (t) = \int_{-\infty}^{\infty} f(\tau) g(t - \tau) d\tau \end{align}Graphically it is taking the weighted average of function \(f\) by inverting the function \(g\) about \(\tau = 0\) and sliding by \(t\).
