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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