What are the benefits of studying Fourier series in physics?

[kundukoustav]

Use of Laplace Transforms!

What is the use of Laplace transforms in differential equations? I mean, why are they used?
And can you please explain the "s" domain in the Laplace Transform! I studied that it is a variable in the complex plane! But, i want to grow upon that! Please help!

 

[Vargo]

Here are some reasons why it is handy.

1. You can use Laplace transforms to turn differential equations into "algebraic" equations. You probably know this.

2. The Laplace transform treats discontinuities and things like Dirac deltas better than many other analytic methods such as power series.

3. The Laplace transform gives you a direct method of finding an explicit analytic solution to the equation in terms of integrals. The alternative method of finding independent solutions to the homogeneous equation, then finding a single solution to the nonhomogeneous equation and then matching your constants to your initial data is relatively indirect. Also, there aren't completely general methods to solve the homogeneous equation (unless the coefficients are constant).

4. If you just needed to numerically solve a specific equation, then Euler or Runge-Kutta or something like that would be great. But an analytic solution can be handy to see how the solution might vary when different parts of the problem (like coefficients, initial conditions, or w/e) change.

5. The Laplace transform is basically the generating function for the moments of the function. They use this in elementary probability texts, though I'm not sure if that isn't superseded by the use of the Fourier transform in more advanced texts.

 

[kundukoustav]

Thanks! But are they only used for non linear differential equations? Can they be used to solve any sort of non linear equations to always give an analytic solution? And what are their limitations?

Thanks in advance!

 

[kundukoustav]

Another question, why do we have to linearize before applying laplace transforms? can't they be applied directly to non linear differential equations before linearization?

 

[Vargo]

Follow the link to see a description of the type of physical problems best suited for the Laplace transform.

http://en.wikipedia.org/wiki/Linear_time_invariant

Ideally you want a linear system that does not depend on time (like a spring or a circuit or something like that), and then you apply some kind of forcing function. The solution represents the response. The Laplace Transform in particular shows how the general response to a forcing function breaks down as a superposition of responses to impulses.

You might be able to solve a nonlinear equation using a Laplace Transform if the equation had some specific form, and if you had some serious tricks up your sleeve. But in general, the LT is not well-suited to nonlinear problems. Try googling laplace transform nonlinear equations to see the stuff that comes up. This is why you have to linearize your problem first.

 

[20Tauri]

I'm not the OP, but I also have a Laplace Transform question. Where does the Laplace Transform come from and why can we use it? I understand that all the answers you can look up in a transform table come from the explicit formula for the transform, but where did we get the explicit formula in the first place?

 

[homeomorphic]

One way to think of the Laplace transform is to compare it to the Fourier transform.

The Fourier transform is sort of like Fourier series, except Fourier series can only represent a periodic function, or, alternatively, a function defined only on an interval. So, you let that interval go to infinity in length and you get the Fourier transform, rather than Fourier series. In the case of Fourier series, you only have a discrete set of frequencies, but in taking this limit, the frequency becomes a continuous variable.

The Fourier transform is a function that takes in a frequency and the function value on that frequency tells you how much of that frequency is present in the function. So, the Fourier transform is kind of like a prism that takes the white light (many different frequencies) and splits it into its components, (red, blue, green--single frequencies).

So, the Laplace transform just gives you an extra dial to turn. With the Fourier transform, you turn the dial to specify what frequency component you want. But with the Laplace transform, you have another dial that allows you to express things in terms of damped sine and cosine functions, rather than just sine and cosine. So, you put in a real part that gives you the damping, and then the imaginary part that tells you what frequency you want. So, when you put a complex number into the Laplace transform of a function, you are asking it how much of some frequency is there when you try to express the function in terms of damped sine and cosine functions. When I say damped, I meaned multiplied by e^(-a), where a is the real part of s. So, actually, it will not really be damped if a is non-positive.

 

[20Tauri]

Oh, that helps, thanks. I never really understood why we cared about Fourier series either, so those make a little more sense now too.

 

[homeomorphic]

Oh. Correction:

When I say damped, I meaned multiplied by e^(-a), where a is the real part of s. So, actually, it will not really be damped if a is non-positive.

I meant to say e^(-at).

 

[homeomorphic]

I never really understood why we cared about Fourier series either, so those make a little more sense now too.

I think the reason for caring about Fourier series is that they are sort of canonical solutions to the heat equation and the wave equation (they are eigenvalues of the relevant differential operator, and they have a certain physical interpretation in each case). So, if you can a function as a combination of them, you can solve the heat or wave equation with that function as the initial condition.

 

[20Tauri]

I think the reason for caring about Fourier series is that they are sort of canonical solutions to the heat equation and the wave equation (they are eigenvalues of the relevant differential operator, and they have a certain physical interpretation in each case). So, if you can a function as a combination of them, you can solve the heat or wave equation with that function as the initial condition.

Yeah, I never understood what we were doing with the heat equation or the wave equation in the first place. We kind of rushed through Fourier series at the very end of my differential equations class, and we didn't have time to get to the Fourier transform, so I never really got what we were doing. I knew we were using the Fourier series to approximate things (getting more accurate with more terms, etc.), but the how and why sort of escaped me.

 

[Jonnyb302]

The Fourier series is a very important thing to study for a few reasons in physics. First, you can teach Fourier series in a almost mechanical way at first that is easy to understand. Usually your first go around with learning them is just proving that some integrals are equal to zero. Then as students become more comfortable with them you can begin to talk about Fourier series in terms of more complicated mathematics, such as orthogonal series, inner products and Hilbert space.

Fourier series are like the squares of many branches of math, much easier to understand at first than a rhombus.