Green's Function of Linear ODE

[member 428835]

Homework Statement

Find Green's function of $u''+u=f$.

Homework Equations

What we all know.

The Attempt at a Solution

Let Greens function be $G$. Then $G''+G=\delta(x-x_0)$. This admits solutions superimposed of sine and cosine. Let's split the function at $x=x_0$. Then we require four boundary conditions. We are typically given 1 at each end point (2 total), we know $G$ is continuous, and lastly there is a jump discontinuity at $x=x_0$. This is where I'm stuck. In determining this fourth BC, would we integrate the ODE from $x_0-$ to $x_0+$? This would imply $u'_+-u'_- + \int_-^+G = 1$.

Is there an easier way?

 

[Ray Vickson]

Homework Statement

Find Green's function of $u''+u=f$.

Homework Equations

What we all know.

The Attempt at a Solution

Let Greens function be $G$. Then $G''+G=\delta(x-x_0)$. This admits solutions superimposed of sine and cosine. Let's split the function at $x=x_0$. Then we require four boundary conditions. We are typically given 1 at each end point (2 total), we know $G$ is continuous, and lastly there is a jump discontinuity at $x=x_0$. This is where I'm stuck. In determining this fourth BC, would we integrate the ODE from $x_0-$ to $x_0+$? This would imply $u'_+-u'_- + \int_-^+G = 1$.

Is there an easier way?

That is as easy as it gets: $\int_{-x_0}^{+x_0} G \, dx = 0$, so $G'(+x_0) - G'(-x_0) = 1$.

 

[Orodruin]

we know $G$ is continuous,

This is often stated without motivation. It may or may not be true depending on the differential equation.

and lastly there is a jump discontinuity at $x=x_0$. This is where I'm stuck. In determining this fourth BC, would we integrate the ODE from $x_0-$ to $x_0+$? This would imply $u'_+-u'_- + \int_-^+G = 1$.

Is there an easier way?

I never liked the "integrate from $x_{0-}$ to $x_{0+}$ argument. I find it much more instructive and transparent to make an ansatz of the form $G(x,x_0) = \theta(x-x_0)g_+(x) + \theta(x_0 - x) g_-(x)$ and insert it into the differential equation and then start identifying terms.

However, if you do want to use the integration approach, it is the differential equation for the Green's function that you want to integrate. This does not contain any u (exchange your u+ and u- for G+ and G-). Also the integral of the function itself vanishes in the limit.

 

[member 428835]

Thanks everyone! Yea, I made a few typos and had a mind blank.