- #1
romeo6
- 54
- 0
Hi,
I have a basic ODE:
[tex]y''(x)+\frac{1}{4}y'(x)=f(x)[/tex]
on 0<x<L
With Boundary conditions:
[tex]y(0)=y(L)=0[/tex]
For which I would like to construct a Green Function.
Rather than just plain ask for help, I'll show you what I've been thinking and maybe someone wiser can help/correct me:
We must solve:
[tex]G''(x,x')+\frac{1}{4}G'(x,x'')=\delta(x-x')[/tex]
Since G vanishes at the boundaries we can expand as a Fourier sine series:
[tex]G(x,x')=\sum_{n=1}^\infty \gamma_n sin\frac{n\pi x}{L}[/tex]
and:
[tex]\delta(x-x')=\sum_{n=1}^\infty A_n(x')sin\frac{n\pi x}{L}[/tex]
Integrating the delta function I get:
[tex]A_n(x')=\frac{2}{L}sin\frac{n\pi x}{L}[/tex]
I take the first and second derivative of the equation for G
and plug everything into the differential equation for G:
[tex]\sum_{n=1}^\infty \gamma_n (\frac{-n^2 \pi^2}{L^2})sin\frac{n\pi x}{L}+\frac{1}{4} \sum_{n=1}^\infty \gamma_n (\frac{n \pi}{L})cos\frac{n\pi x}{L}=\sum_{n=1}^\infty \frac{2}{L}sin\frac{n\pi x'}{L}sin\frac{n\pi x}{L}[/tex]
Now I am not sure what to do from here to get the required Green function...can someone guide me please?
I have a basic ODE:
[tex]y''(x)+\frac{1}{4}y'(x)=f(x)[/tex]
on 0<x<L
With Boundary conditions:
[tex]y(0)=y(L)=0[/tex]
For which I would like to construct a Green Function.
Rather than just plain ask for help, I'll show you what I've been thinking and maybe someone wiser can help/correct me:
We must solve:
[tex]G''(x,x')+\frac{1}{4}G'(x,x'')=\delta(x-x')[/tex]
Since G vanishes at the boundaries we can expand as a Fourier sine series:
[tex]G(x,x')=\sum_{n=1}^\infty \gamma_n sin\frac{n\pi x}{L}[/tex]
and:
[tex]\delta(x-x')=\sum_{n=1}^\infty A_n(x')sin\frac{n\pi x}{L}[/tex]
Integrating the delta function I get:
[tex]A_n(x')=\frac{2}{L}sin\frac{n\pi x}{L}[/tex]
I take the first and second derivative of the equation for G
and plug everything into the differential equation for G:
[tex]\sum_{n=1}^\infty \gamma_n (\frac{-n^2 \pi^2}{L^2})sin\frac{n\pi x}{L}+\frac{1}{4} \sum_{n=1}^\infty \gamma_n (\frac{n \pi}{L})cos\frac{n\pi x}{L}=\sum_{n=1}^\infty \frac{2}{L}sin\frac{n\pi x'}{L}sin\frac{n\pi x}{L}[/tex]
Now I am not sure what to do from here to get the required Green function...can someone guide me please?