Help using Green’s functions in solving Differential Equations please

[Lambda96]

Homework Statement

see post

Relevant Equations

none

Hi,

unfortunately I have several problems with the following task:

I have problems with the tasks a, d and e

Unfortunately, the Green function and solving differential equations with the Green function is completely new to me

In task b, I got the following for $f_h(t)=e^{-at}$.Task a

$$\hat{L}G(t)=\Bigl( \frac{d}{dt} +a \Bigr) \Theta(t) f_h(t)$$
$$\hat{L}G(t)=\frac{d}{dt}\Theta(t) f_h(t) +a \Theta(t) f_h(t)$$
$$\hat{L}G(t)=\delta(t) f_h(t) + \Theta(t) f'_h(t) +a \Theta(t) f_h(t)$$
$$\hat{L}G(t)=\delta(t) f_h(t) -a \Theta(t) f_h(t) +a \Theta(t) f_h(t)$$
$$\hat{L}G(t)=\delta(t) f_h(t)$$

Can I now argue as follows that $\hat{L}G(t)=\delta(t)$ so when I multiply the operator by the Green function, I always get only one value. Then the following $\delta(t)=\delta(0)$ applies, so it follows that $\delta(t) f_h(t)=\delta(0) f_h(0)$ and since $f_h(0)=1$ only $\delta(t)$ remains on the left side of the equationTask d

I assumed that I should calculate the following integral.

$$\tilde{G}(\omega)= \int_{-\infty}^{\infty} dt \ e^{i \omega t} \hat{L} G(t) $$
$$ \tilde{G}(\omega)=\int_{-\infty}^{\infty} dt \ e^{i \omega t} \frac{d}{dt} G(t) +e^{i \omega t} a G(t) $$
$$ \tilde{G}(\omega)=\int_{-\infty}^{\infty} dt \ e^{i \omega t} \frac{d}{dt} G(t) +\int_{-\infty}^{\infty} dt \ e^{i \omega t} a G(t) $$

I then applied partial integration for the first integral

$$ \tilde{G}(\omega)=\biggl[ e^{i \omega t} G(t) \biggr]_{-\infty}^{\infty}-\int_{-\infty}^{\infty} dt \ i \omega e^{i \omega t} G(t) +\int_{-\infty}^{\infty} dt \ e^{i \omega t} a G(t) $$

Now, unfortunately, I don't get any further and I can't do anything with the hint from the task at the moment.
Task e

I thought that a solution may look like the following.

$$ f(t)= \int_{0}^{t} G(t)g(t) dt $$

I then calculated the following integral

$$ f(t)= \int_{0}^{t} G(t)g(t) dt $$
$$ f(t)= \int_{0}^{t} e^{-at} e^{2at} dt $$
$$ f(t)= \frac{e^{at} -1}{a}$$

If I substitute this $f(t)$ into $\hat{L}f(t)$, I get $2e^{at}-2$ but I should get $e^{2at}$.

 

[vela]

Task a

$$\hat{L}G(t)=\Bigl( \frac{d}{dt} +a \Bigr) \Theta(t) f_h(t)$$
$$\hat{L}G(t)=\frac{d}{dt}\Theta(t) f_h(t) +a \Theta(t) f_h(t)$$
$$\hat{L}G(t)=\delta(t) f_h(t) + \Theta(t) f'_h(t) +a \Theta(t) f_h(t)$$
$$\hat{L}G(t)=\delta(t) f_h(t) -a \Theta(t) f_h(t) +a \Theta(t) f_h(t)$$
$$\hat{L}G(t)=\delta(t) f_h(t)$$

Just use the property $f(t)\delta(t) = f(0)\delta(t)$.

Task d

I assumed that I should calculate the following integral.

The problem said to take the Fourier transform of the differential equation.

Task e

I thought that a solution may look like the following.

$$ f(t)= \int_{0}^{t} G(t)g(t) dt $$

Look up convolution.

 

[Lambda96]

Thanks vela for your help , with your tips I could solve the tasks now