How to apply the Fourier transform to this problem?

In summary, the conversation discusses the approach to solving a problem involving a non-homogeneous heat equation with extra terms. The person is initially confused but then realizes that taking the Fourier transform with respect to x is not necessary. A separation of variables approach can be used and the solution ends up being a function of t times the initial conditions.
  • #1
Safder Aree
42
1
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.`
 
Physics news on Phys.org
  • #2
Safder Aree said:
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.
 
  • #3
tnich said:
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?
 
  • #4
Safder Aree said:
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.
 
  • #5
tnich said:
##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?
 
  • #6
Safder Aree said:
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
$$ -(2 \pi w)^2 \hat{u} + 4 \pi i w \hat{u} = - \hat{u}$$
?
 
  • #7
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:
  • #8
Safder Aree said:
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).
 
  • #9
The Fourier transform of a step function is ##\frac 1 {j \omega} + \pi \delta(\omega)##
 
  • #10
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.
 

Related to How to apply the Fourier transform to this problem?

1. What is the Fourier transform and how does it work?

The Fourier transform is a mathematical operation that decomposes a function or signal into its component frequencies. It works by breaking down a function into a sum of sine and cosine waves of different frequencies, amplitudes, and phases. This allows us to study the frequency content of a signal and extract useful information.

2. How can the Fourier transform be applied to real-world problems?

The Fourier transform has a wide range of applications in science and engineering. It is commonly used in signal processing, image analysis, and data compression. It can also be used in fields such as quantum mechanics, optics, and probability theory.

3. What are the limitations of the Fourier transform?

One limitation of the Fourier transform is that it assumes the signal is periodic and has a continuous spectrum. This may not be the case for all real-world signals. Additionally, the Fourier transform may not be suitable for non-linear and non-stationary signals.

4. Are there different types of Fourier transforms?

Yes, there are different types of Fourier transforms, such as the discrete Fourier transform, fast Fourier transform, and inverse Fourier transform. Each type has its own specific use and properties. For example, the fast Fourier transform is more efficient for calculating the Fourier transform of a discrete signal.

5. Can the Fourier transform be applied to multidimensional signals?

Yes, the Fourier transform can be extended to multidimensional signals, such as images and videos. This is known as the multidimensional Fourier transform and is useful for analyzing the frequency content of these signals and performing operations such as filtering and compression.

Similar threads

  • Calculus and Beyond Homework Help
Replies
5
Views
369
  • Calculus and Beyond Homework Help
Replies
6
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
830
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
6
Views
486
  • Calculus and Beyond Homework Help
Replies
12
Views
1K
Replies
0
Views
505
  • Calculus and Beyond Homework Help
Replies
8
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
421
  • Calculus and Beyond Homework Help
Replies
11
Views
2K
Back
Top