I'm looking for some documentation about solving DEs using Fourier Transformation (not Fourier Series).
In particular, I have this one DE that I solved using another technique. Somebody mentioned that it can be solved using FT as well:
$\varepsilon - \frac{1}{2}l^2 \frac{d^2 \varepsilon}{dx^2} = \delta(x)$
Where $\varepsilon$ is a function of $x$ only, $l$ is a scalar constant and $\delta(x)$ is the Dirac "function".
The solution for the homogeneous case is
$\varepsilon = A \; exp \left[ -\frac{|x|\sqrt2}{l} \right]$
If I remember correctly, taking some derivatives resulted in the final solution of $A = \dfrac{1}{l \sqrt{2}}$.
Ok, now with FT. I found this site, the example looks a little like my DE -- with in my case $g(t) = \delta(x)$, which is convenient considering the convolution in the end.
Without the constant $\frac{1}{2}l^2$ I get
$Y(f) - (2\pi i f)^2 Y(f) = \left( 1 - (2 \pi i f)^2 \right)\;Y(f) = G(f)$
So
$Y(f) = \frac{G(f)}{1+4 \pi^2 f^2}$.
According to the site (would like to know how to compute it myself),
$F^{-1}\left( \frac{1}{1+4 \pi^2 f^2} \right) = \frac{e^{-|t|}}{2}$
Taking the convolution with the Dirac "function" is again the same function, $\dfrac{e^{-|t|}}{2} = exp \left[ \dfrac{-|t|}{2} \right]$.
So to conclude:
- Any documentation about solving DEs using FT is more than welcome
- How to do the same, now with the constant $\frac{1}{2}l^2$
- How to compute the inverse Fourier Transform of expressions like above (I know it involves an integral to infinity, but I never learned how to solve integrals like that)