PX275 - H9 - FT and differential equations

considering a function, $f (x)$ , and its (partial) derivatives
the fourier transform:

\begin{aligned} F (\frac{d f}{d x}) & = \int_{- \infty}^{\infty} e^{- i k x} \frac{d f}{d x} d x \\ = [e^{- i k x} f (x)]_{- \infty}^{+ \infty} + i k \int_{- \infty}^{\infty} e^{- i k x} f (x) d x \end{aligned}

note: $f (x)$ needs to be well-behaved and integrable so that $F (f (x))$ can sensibly be considered
if $lim_{x \to \pm \infty} f (x) = 0$ , so the first term is zero

F (\frac{d f}{d x}) = i k \int_{- \infty}^{\infty} e^{- i k x} f (x) d x = i k \tilde{f} (k)

similarly, for the second derivative:

\begin{aligned} F (\frac{d^{2} f}{d x^{2}}) & = \int_{- \infty}^{\infty} e^{- i k x} \frac{d^{2} f}{d x^{2}} d x \\ = \underset{0}{\underset{⏟}{{[e^{- i k x} \frac{d f}{d x}]}_{- \infty}^{+ \infty}}} + \underset{i k F (f^{'} (x)) = i k (i k \tilde{f} (k))}{\underset{⏟}{i k \int_{- \infty}^{\infty} e^{- i k x} \frac{d f}{d x} d x}} \\ = - k^{2} \tilde{f} (k) \end{aligned}

general formula for the FT of the

n^{t h}

derivative

F (\frac{d^{n} f}{d x^{n}}) = (i k)^{n} \tilde{f} (k)

this proves to be a power tool for handling differential equations
the derivatives of a function can be converted into the FT of the function
the original PDE can be turned into an algebraic expression involving $\tilde{f} (k)$ , which tends to be easier to solve
the solution for $\tilde{f} (k)$ can be transformed back into $f (x)$

example

considering a differential equation:

\frac{d^{2} f}{d x^{2}} + α \frac{d f}{d x} + β f (x) = g (x)

taking the FT of both sides:

\begin{matrix} - k^{2} \tilde{f} (k) + α i k \tilde{f} (k) + β \tilde{f} (k) = \tilde{g} (k) \\ (- k^{2} + α i k + β) \tilde{f} (k) = \tilde{g} (k) \end{matrix}

the second order ODE has been turned into an algebraic expression for $\tilde{f} (k)$ , and:

f (x) = F^{- 1} (\tilde{f} (k))