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Fourier Transformation of ODE

1. 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:

fourier_ode.JPG



2. Homework Equations

a) The Fourier Transform

b) The Inverse Fourier Transform

3. The Attempt at a Solution

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

[tex]\mathrm{i} k \hat{f}(k) - 2\hat{f}(k) = \hat{\phi}(k)[/tex]

I then solve for the transformed function:

[tex]\hat{f}(k) = \frac{\hat{\phi}(k)}{\mathrm{i}k-2}[/tex]

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

[tex]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) [/tex]

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?
 

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Answers and Replies

stevendaryl
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[tex]\hat{f}(k) = \frac{\hat{\phi}(k)}{\mathrm{i}k-2}[/tex]

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

[tex]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) [/tex]
I think you're using an invalid property of fourier transforms. If [itex]F^{-1}[/itex] means the inverse Fourier transform, then it's not true that

[itex]F^{-1}(\hat{A}(k) \hat{B}(k)) = F^{-1}(\hat{A}(k)) F^{-1}(\hat{B}(k))[/itex]

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

[itex]\hat{\phi}(k) = \int \phi(x) e^{-ikx} dx = \int_0^{\infty} e^{-2x -ikx} dx[/itex]

What is the value of that integral?
 
Ray Vickson
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1. 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


2. Homework Equations

a) The Fourier Transform

b) The Inverse Fourier Transform

3. The Attempt at a Solution

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

[tex]\mathrm{i} k \hat{f}(k) - 2\hat{f}(k) = \hat{\phi}(k)[/tex]

I then solve for the transformed function:

[tex]\hat{f}(k) = \frac{\hat{\phi}(k)}{\mathrm{i}k-2}[/tex]

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

[tex]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) [/tex]

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)##:
$$(a \circ b) (x) = \int_{-\infty}^{\infty} a(x-y) b(y) \, dy.$$
 

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