Applying Convolution to a PDE with a Fourier Transform

[member 428835]

Homework Statement

$$u_{xx}=u_t+u_x$$ subject to $u(x,0)=f(x)$ and $u$ and $u_x$ tend to 0 as $x\to\pm\infty$.

Homework Equations

Fourier Transform

The Attempt at a Solution

Taking the Fourier transform of the PDE yields
$$
(\omega^2-i\omega) F\{u\}= \partial_t F\{u\}\implies\\
F\{u\} = \exp(t/4)F\{f(\omega)\}\exp((\omega-i/2)^2t)\implies\\
u = \exp(t/4)\int_{-\infty}^\infty f(\omega)\exp((\omega-i/2)^2t)\exp(i\omega x) \, d\omega
$$ From here I don't know how to apply convolution. Can anyone help?

 

[Orodruin]

What is the function that has $\exp((\omega-i/2)^2t)$ as its Fourier transform?

Edit: Also, $i^2 = -1$ ... You should have $\exp(-(\omega+i/2)^2t)$.

 

[member 428835]

What is the function that has $\exp((\omega-i/2)^2t)$ as its Fourier transform?

Edit: Also, $i^2 = -1$ ... You should have $\exp(-(\omega+i/2)^2t)$.

Thanks for the correction. And a Gaussian transforms another Gaussian I believe, which is why I casted it in that form. I know $$F\left( \sqrt{\frac{\pi}{\alpha}}\exp(-x^2/4\alpha)\right) = \exp(-\alpha\omega^2).$$ Then the inverse transform of $\exp(-(\omega-i/2)^2t)$ where $\alpha = 1/t$ should be $\sqrt{\pi t}\exp(-t(x-i/2)^2/4)$. What do you think?

 

[Orodruin]

I think you are guessing rather than performing the proper contour integral in the complex plane.

 

[member 428835]

Yea I was looking at a formula that looked applicable. To work it out we'd have $$\int_{-\infty}^\infty \exp(-(\omega-i/2)^2t)\exp(i\omega x)\,d\omega = \int_{-\infty}^\infty \exp(-(\omega-i/2)^2t+i\omega x)\,d\omega\\
=\sqrt{\frac{\pi}{t}}\exp\left( -\frac{x(2t+x)}{4t} \right)$$
How does that look for the inverse Fourier transform?

 

[Orodruin]

... and multiplying this by the exponent you already extracted gives you? (In fact, it is preferable not to complete the square until you have the Fourier integral...)

 

[Orodruin]

You might also want to double check the normalisation of your Fourier transform and that of the table you are getting your transforms from.

 

[member 428835]

... and multiplying this by the exponent you already extracted gives you? (In fact, it is preferable not to complete the square until you have the Fourier integral...)

Gotcha, so this gives us $$\sqrt{\frac{\pi}{t}}\exp\left( \frac{(t-x)^2}{4t} \right).$$ But how do I apply this to use convolution?

 

[Orodruin]

You should still double check the normalisation. And there is now a missing sign in the exponent.

Then use that the Fourier transform of a convolution is the product of the Fourier transforms.

 

[member 428835]

Shoot, we should have $$\sqrt{\frac{\pi}{t}}\exp\left( -\frac{(t+x)^2}{4t} \right)\exp(-t/2).$$ Can you elaborate on what you mean by "use that the FOurier transform of a convolution is the product of the Fourier transforms?

 

[Orodruin]

Your conclusion in post 8 was closer. Note that the sign of the exponent with the t also changes due to your original sign error. Again, the best thing is to wait with completing the square until you have the inverse FT integral.

If you are not familiar with the convolution theorem, I suggest reading up on it. It should be explained in any introductory textbook on Fourier transforms. Or you can start on Wikipedia https://en.m.wikipedia.org/wiki/Convolution_theorem

Note the issues with normalisation.

 

[member 428835]

Shoot, yea I made an algebraic error. For completeness the equation after transforming the PDE should read $$
-(\omega^2+i\omega) F\{u\}= \partial_t F\{u\}\implies\\
F\{u\} = F\{f(\omega)\}\exp(-(\omega^2+i\omega)t)\implies\\
u = \int_{-\infty}^\infty F\{f(\omega)\}\exp(-(\omega^2+i\omega)t)\exp(i\omega x) \, d\omega\\
= \exp\left(- \frac{(x-t)^2}{4t} \right)\int_{-\infty}^\infty F\{f(\omega)\}\exp\left( -t\left(i\frac{x-t}{2t}-\omega\right)^2\right) \, d\omega$$ Proceeding (if this is correct), convolution seems to imply I should take the inverse Fourier transform of $F\{f(\omega)\}$ and $\exp\left( -t\left(i\frac{x-t}{2t}-\omega\right)^2\right)$ and multiply these together?

 

[Orodruin]

Up to a factor of $1/2\pi$ depending on normalisation of the Fourier transform.

Note that the Green's function of the heat equation is
$$
\frac{1}{\sqrt{4\pi t}} e^{-\frac{x^2}{4t}}.
$$
The difference in your case is the appearance of the convection term. You can get rid of it using an appropriate change of variables $\tau = t$, $\xi = x-t$. This is also the procedure you would obtain using the method of characteristics. If you do this, you will get the standard heat equation $u_\tau = u_{\xi\xi}$. Reinserting your original coordinates into the resulting Green's function immediately gives
$$
G(x,t) = \frac{1}{\sqrt{4\pi t}} e^{-\frac{(x-t)^2}{4t}}.
$$