Fourier method | eigenfunctions

In summary, the conversation discusses using the separation of variables/Fourier method to solve a PDE with given boundary conditions. The resulting eigenvalues and eigenfunctions are mentioned, but there is a question about whether an additional eigenfunction exists. The conversation ends without a clear answer to the question.
  • #1
kingwinner
1,270
0
Use separation of variables/Fourier method to solve
ut - 4uxx = 0, -pi<x<pi, t>0
u(-pi,t) = -u(pi,t), ux(-pi,t) = -ux(pi,t), t>0.
=============================

What I got is that (n+1/2)2 are eigenvalues (n=0,1,2,3,...) and cos[(n+1/2)x)] is an eigenfunction.
Instead of two sets of eigenvalues, there is only one set. I cannot find another set of eigenvalues.
My question is: is sin[(n+1/2)x)] also an eigenfunction for this problem? Why or why not?

Thanks for any help!
 
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  • #2
Are you sure those are the correct boundary conditions? What problem led you to those boundary conditions? They seem to imply only that u and du/dx are odd functions of x, not that u is smooth at pi.
 
Last edited:
  • #3
Yes, I double checked that these are the correct boundary conditions. It is from a PDE course.
 
  • #4
In that case I don't know. What physical situation do such conditions come from?
 

1. What is the Fourier method in mathematics?

The Fourier method is a mathematical technique that is used to represent a complex signal as a sum of simpler sinusoidal functions. It is named after French mathematician Joseph Fourier, who first introduced the concept in the early 19th century.

2. How does the Fourier method work?

The Fourier method uses a mathematical tool called the Fourier transform to break down a complex signal into its constituent frequencies. This is achieved by representing the signal as a sum of infinite sinusoidal functions, known as eigenfunctions, each with a different frequency and amplitude.

3. What are eigenfunctions in the context of the Fourier method?

Eigenfunctions, also known as eigenvectors, are the building blocks of the Fourier method. They are a set of sinusoidal functions with different frequencies and amplitudes that, when combined, can represent any complex signal. These eigenfunctions have the property that they do not change when multiplied by a constant, making them useful in the Fourier method.

4. What are the applications of the Fourier method?

The Fourier method has a wide range of applications in various fields such as signal processing, image and audio compression, data analysis, and solving differential equations. It is also used in physics and engineering to study and analyze complex systems and signals.

5. Are there any limitations to the Fourier method?

While the Fourier method is a powerful tool, it does have some limitations. It assumes that the signal being analyzed is periodic, which may not always be the case. It also requires an infinite number of eigenfunctions to accurately represent a signal, which is not feasible in practical applications. Additionally, the Fourier method may not be suitable for analyzing non-linear signals or systems.

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