# Green’s Function¶

## Green’s Function for Second Order Differential Equations¶

A general form of second order differential equations is written as

$L[y] \equiv y'' + p(x) y' + q(x) y = f(x),$

for $$a<x<b$$ with boundary conditions

$B_1[y] = B_2[y] = 0.$

The solution to it is

$y(x) = \int _a ^b G(x\vert \xi) f(\xi) d\xi$

where $$G(x\vert \xi)$$ is the Green’s function. The Green’s function is an impulse response of the dynamical system, i.e.,

$L[G(x\vert \xi)] = \delta(x-\xi)$

with the two boundary conditions,

$B_1[G] = B_2[G] = 0.$

First order differential of Green’s function $$G'(x\vert \xi)$$ has a jump discontinuity at $$x=\xi$$. This is expected since we have to have a dirac delta type of response at $$x=\xi$$.

### Examples¶

For a 2nd order differential equation,

$y''(x) = f(x), \qquad y(0)= y(1)=0,$

the Green function is

$G''(x\vert \xi) = \delta(x-\xi), \qquad G(0\vert \xi) = G(1\vert \xi) = 0.$

We already know the two solutions to the homogeneous equation, $$y=1$$ or $$y=x$$. However, only the second solution can satisfy the boundary conditions. Then the Green’s function should have these properties,

$\begin{split}G(x\vert \xi) = \begin{cases} c_1+c_2 x, &\quad x<\xi \\ d_1+d_2 x. & \quad x>\xi \end{cases}\end{split}$

The boundary conditions give us

$\begin{split}G(x\vert \xi) = \begin{cases} c x, &\quad x<\xi \\ d(x-1). & \quad x>\xi \end{cases}\end{split}$

We also know that the Green’s functon must be continuous, we then require

$c\xi = d (\xi -1).$

Using the discontinuity of the first order derivative of the Green’s function, we get

$d_x d (x-1)- d_x cx = 1.$

With all these equation, we determine the Green’s function,

$\begin{split}G(x\vert\xi) = \begin{cases} (\xi -1 ) x , & \qquad x<\xi \\ \xi(x-1). & \qquad x>\xi \end{cases}\end{split}$

To get the solution to $$y$$, we integrate over $$\xi$$,

$\begin{split}y(x) &= \int_0^1 G(x\vert \xi) f(\xi) d\xi \\ & = (x-1)\int_0^x \xi f(\xi) d\xi + x \int_x^1 (\xi -1) f(\xi) d\xi .\end{split}$

It is that easy. This is the super power of the Green’s function.

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