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How to apply the Fourier transform to this problem?

Problem 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

Homework Helper
838
244
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.
 
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

Homework Helper
838
244
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.
 
##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

Homework Helper
838
244
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

Homework Helper
838
244
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.
 
Last edited:

pasmith

Homework Helper
1,667
369
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 [itex]x[/itex], 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 [itex]t[/itex] dependence one could in principle solve it by Laplace transforms).
 

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