Is sin[(n+1/2)x] also an eigenfunction for this problem?

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Discussion Overview

The discussion revolves around the eigenfunctions and eigenvalues related to the partial differential equation ut - 4uxx = 0, with specified boundary conditions. Participants explore whether sin[(n+1/2)x] qualifies as an eigenfunction under the given conditions.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant states that the eigenvalues are (n+1/2)² with n being a non-negative integer, and that cos[(n+1/2)x] is an eigenfunction.
  • The same participant questions whether sin[(n+1/2)x] is also an eigenfunction and seeks clarification on this point.
  • Another participant challenges the correctness of the boundary conditions, suggesting they imply odd functions but may not ensure smoothness at the boundaries.
  • A subsequent reply confirms the boundary conditions are correct as per a PDE course but does not provide further clarification on the implications.
  • Another participant expresses uncertainty about the physical context of the boundary conditions, indicating a lack of clarity on their origin.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the boundary conditions or the status of sin[(n+1/2)x] as an eigenfunction. Multiple competing views remain regarding the implications of the boundary conditions.

Contextual Notes

The discussion highlights potential limitations in understanding the boundary conditions and their implications for the eigenfunctions. There is an unresolved question regarding the smoothness of the solution at the boundaries.

kingwinner
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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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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:
Yes, I double checked that these are the correct boundary conditions. It is from a PDE course.
 
In that case I don't know. What physical situation do such conditions come from?
 

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