How to apply the Fourier transform to this problem?

[Safder Aree]

Homework Statement

$$ u_{tt}(x,t) + 2u_t(x,t) = -u(x,t), -\infty < x < \infty, t> 0$$
$$u(x,t), as |x| \rightarrow \infty, t>0$$
$$u(x,0) = f(x), u_t(x,0)= g(x) , -\infty < x < \infty $$

Relevant Equations

Fourier Transform equation:
$$ \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x) e^{iwx} dx $$

I am struggling to figure out how to approach this problem. I've only solved a homogenous heat equation $$u_t = u_{xx}$$ using a Fourier transform, where I can take the Fourier transform of both sides then solve the general solution in Fourier terms then inverse transform. However, since this question has extra terms I'm a little confused.$$ u_{tt}(x,t) + 2u_t(x,t) = -u(x,t), -\infty < x < \infty, t> 0$$
$$u(x,t), as |x| \rightarrow \infty, t>0$$
$$u(x,0) = f(x), u_t(x,0)= g(x) , -\infty < x < \infty $$
Any advice and or guidance would be greatly appreciated. All I would do is take the Fourier transform of all the terms but from there I don't think I know what to do.`

 

[tnich]

I am struggling to figure out how to approach this problem. I've only solved a homogenous heat equation $$u_t = u_{xx}$$ using a Fourier transform, where I can take the Fourier transform of both sides then solve the general solution in Fourier terms then inverse transform. However, since this question has extra terms I'm a little confused.$$ u_{tt}(x,t) + 2u_t(x,t) = -u(x,t), -\infty < x < \infty, t> 0$$
$$u(x,t), as |x| \rightarrow \infty, t>0$$
$$u(x,0) = f(x), u_t(x,0)= g(x) , -\infty < x < \infty $$
Any advice and or guidance would be greatly appreciated. All I would do is take the Fourier transform of all the terms but from there I don't think I know what to do.`

Try it and see what occurs to you when you look at the result.
Since there are no partials with respect to x, you can consider x constant and just look at the Fourier transform with respect to t.

 

[Safder Aree]

Try it and see what occurs to you when you look at the result.
Since there are no partials with respect to x, you can consider x constant and just look at the Fourier transform with respect to t.

So my understanding is that then applying the transform leads to:

$$ -(2 \pi w)^2 \hat{u_t} + 4 \pi i w \hat{u_t} = - fourier(constant)$$

Is the Fourier of a constant a dirac delta function?

 

[tnich]

So my understanding is that then applying the transform leads to:

$$ -(2 \pi w)^2 \hat{u_t} + 4 \pi i w \hat{u_t} = - fourier(constant)$$

Is the Fourier of a constant a dirac delta function?

$u(x,t)$ is a function of t, so I don't think it should be constant.

 

[Safder Aree]

$u(x,t)$ is a function of t, so I don't think it should be constant.

Oh no you're right,

$$ -(2 \pi w)^2 \hat{u_t} + 4 \pi i w \hat{u_t} = - \hat{u}(x,t)$$
Would that be right? Where would I go from here?

 

[tnich]

Oh no you're right,

$$ -(2 \pi w)^2 \hat{u_t} + 4 \pi i w \hat{u_t} = - \hat{u}(x,t)$$
Would that be right? Where would I go from here?

Wouldn't it be
$$ -(2 \pi w)^2 \hat{u} + 4 \pi i w \hat{u} = - \hat{u}$$
?

 

[tnich]

I have worked it through. I think it actually makes a lot more sense to do the Fourier transform with respect to x. That gives you a differential equation in t that is easy to solve. Once you have that solution, you can apply the boundary conditions and transform back. It does seem to me that you could do this problem more easily without the Fourier transforms.

 

[pasmith]

I am struggling to figure out how to approach this problem. I've only solved a homogenous heat equation $$u_t = u_{xx}$$ using a Fourier transform, where I can take the Fourier transform of both sides then solve the general solution in Fourier terms then inverse transform. However, since this question has extra terms I'm a little confused.$$ u_{tt}(x,t) + 2u_t(x,t) = -u(x,t), -\infty < x < \infty, t> 0$$
$$u(x,t), as |x| \rightarrow \infty, t>0$$
$$u(x,0) = f(x), u_t(x,0)= g(x) , -\infty < x < \infty $$
Any advice and or guidance would be greatly appreciated. All I would do is take the Fourier transform of all the terms but from there I don't think I know what to do.`

As written the PDE contains no derivative with respect to $x$, so it is effectively an uncountable number of uncoupled linear constant coefficient first order ODEs and there is no need to do any transforms (although as it's an initial value problem for the $t$ dependence one could in principle solve it by Laplace transforms).

 

[osilmag]

The Fourier transform of a step function is $\frac 1 {j \omega} + \pi \delta(\omega)$

 

[Henryk]

There is no differentiation with respect to x, so a simple separation of variables works.
Furthermore, the equation is homogenous with respect to t and the function can be written as $(u(x,t) = exp(-st) f(x)$, Substitute that into the equation and you get the following
$(s^2 - 2s +1)f(x) = 0$ and that gives you s = 1.
So the solution to the equation is
$u(x,t) = exp(-t)f(x) = exp(-t)u(x,0)$
No need for Fourier transform.