fourier transforms

Question 1

Sufficient conditions for existence fro FT, sine and cosine transform

Answer 1

f is absolutely integrable on the appropriate interval

• FT $(-\infty,\infty)$
• cosine/ sine transform $[0,\infty)$

f and f’ piecewise cont on every finite interval

NOTE: derivation f FT of deriv you require f(x) or f’(x) $\rightarrow 0 $ as $x \rightarrow \pm \infty$

Question 2

General steps to using FT to solve PDE in u(x,t)

Answer 2

FT ode and initiaol condition to U(a,t)

solve simpler ODE in U(a,t)

IFT back to get U(x,t)

Question 3

FT of f’(x) and f’‘(x)

Answer 3

FT of f’(x)
$-i\alpha \mathcal{F}f(x)$

FT of f’‘(x)
$-\alpha^2 \mathcal{F}f(x)$

Question 4

How to know which transform to use

Answer 4

• FT $(-\infty,\infty)$
• cosine/ sine transform $[0,\infty)$

if boundary condition is :

a) a function value -> SINE T
b) derivative -> COSINE T

Question 5

Sien first and second derivative FT

Answer 5

$$\mathcal{F}_s {f’(x)} = -\alpha \mathcal{F}_c {f(x)}$$

$$\mathcal{F}_s {f’‘(x)} = \alpha f(0) -\alpha^2 \mathcal{F}_s {f(x)}$$

Question 6

cos first and second deriv FT

Answer 6

$$\mathcal{F}_c {f’(x)} = -f(0) +\alpha \mathcal{F}_s {f(x)}$$

$$\mathcal{F}_c {f’‘(x)} = - f’(0) -\alpha^2 \mathcal{F}_c {f(x)}$$