Fourier Transformation of ODE

[The_Sleeper]

Homework Statement

I am to solve an ODE using the Fourier Transform, however I am quite inexperienced in using this method so I'd like some advice:

Homework Equations

a) The Fourier Transform

b) The Inverse Fourier Transform

The Attempt at a Solution

I started by applying the Fourier Transform to the equation, this gives me:

$$\mathrm{i} k \hat{f}(k) - 2\hat{f}(k) = \hat{\phi}(k)$$

I then solve for the transformed function:

$$\hat{f}(k) = \frac{\hat{\phi}(k)}{\mathrm{i}k-2}$$

At this point I want to invert this function in order to find the solution to the ODE:

$$f(x) = F^{-1}\left( \frac{\hat{\phi}(k)}{\mathrm{i}k-2} \right) = F^{-1}\left( \hat{\phi}(k) \right) F^{-1}\left( \frac{1}{\mathrm{i}k-2} \right) = \phi(x) F^{-1}\left( \frac{1}{\mathrm{i}k-2} \right)$$

This is where I'm a bit unsure of how to proceed. As I said in the beginning I'm quite inexperienced with this. So I am unsure if I have made any mistakes or wrong assumptions up till this point. How do I proceed with the inversion of the function?

 

[stevendaryl]

$$\hat{f}(k) = \frac{\hat{\phi}(k)}{\mathrm{i}k-2}$$

At this point I want to invert this function in order to find the solution to the ODE:

$$f(x) = F^{-1}\left( \frac{\hat{\phi}(k)}{\mathrm{i}k-2} \right) = F^{-1}\left( \hat{\phi}(k) \right) F^{-1}\left( \frac{1}{\mathrm{i}k-2} \right) = \phi(x) F^{-1}\left( \frac{1}{\mathrm{i}k-2} \right)$$

I think you're using an invalid property of Fourier transforms. If $F^{-1}$ means the inverse Fourier transform, then it's not true that

$F^{-1}(\hat{A}(k) \hat{B}(k)) = F^{-1}(\hat{A}(k)) F^{-1}(\hat{B}(k))$

But you don't need that. You just need to compute $\hat{\phi(k)}$ and then compute $F^{-1}(\frac{\hat{\phi}(k)}{ik - 2})$

$\hat{\phi}(k) = \int \phi(x) e^{-ikx} dx = \int_0^{\infty} e^{-2x -ikx} dx$

What is the value of that integral?

 

[Ray Vickson]

Homework Statement

I am to solve an ODE using the Fourier Transform, however I am quite inexperienced in using this method so I'd like some advice:

View attachment 222522

Homework Equations

a) The Fourier Transform

b) The Inverse Fourier Transform

The Attempt at a Solution

I started by applying the Fourier Transform to the equation, this gives me:

$$\mathrm{i} k \hat{f}(k) - 2\hat{f}(k) = \hat{\phi}(k)$$

I then solve for the transformed function:

$$\hat{f}(k) = \frac{\hat{\phi}(k)}{\mathrm{i}k-2}$$

At this point I want to invert this function in order to find the solution to the ODE:

$$f(x) = F^{-1}\left( \frac{\hat{\phi}(k)}{\mathrm{i}k-2} \right) = F^{-1}\left( \hat{\phi}(k) \right) F^{-1}\left( \frac{1}{\mathrm{i}k-2} \right) = \phi(x) F^{-1}\left( \frac{1}{\mathrm{i}k-2} \right)$$

This is where I'm a bit unsure of how to proceed. As I said in the beginning I'm quite inexperienced with this. So I am unsure if I have made any mistakes or wrong assumptions up till this point. How do I proceed with the inversion of the function?

You made a fatal error: if $a(x)$ and $b(x)$ have F.Ts $A(k)$ and $B(k)$ it is not the case that $\text{F.T.}(a b) = A(k) B(k)$. In fact,
$$A(k) B(k) = \text{F.T.} (a \circ b )(k) = A(k) B(k), $$
where $a \circ b$ is the convolution of $a(\cdot)$ and $b(\cdot)$:
$$(a \circ b) (x) = \int_{-\infty}^{\infty} a(x-y) b(y) \, dy.$$