1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

PDE Help - Eigenfunction of a LaPlacian

  1. Oct 8, 2011 #1
    1. A function "v" where v(x,y,z)≠0 is called an eigenfunction of the Laplacian Δ (on some region Ω - with specified homogenous BC) if v satisifies the BC ad also Δv=-λv on Ω for some number λ.

    Part A: Give an example of an eigenfunction of Δ when Ω is the cube [0,∏]3 with Dirichlet BCs.
    Part B: Show that, for any eigenfunction v, the function u(t,x,y,z)=cos(√λt)v(x,y,z) is the solution for the problem:
    utt=ΔU on [0,T]XΩ with Ω=[0,∏]3
    u=0 on ∂Ω for 0≤t≤T
    {u(0,x,y,y)=v(x,y,z) for 0≤x,y,z≤∏}
    {u(0,x,y,y)=0 for 0≤x,y,z≤∏}

    2. We are studying heat equation, wave equation, eigenfunctions, d'Alamberts sol., ray-tracing, etc.

    3. Okay... I am not sure how to approach a problem of this nature. I sketched the cube from [0,∏]. I am assuming that the answer is given in part B. I can see how the cosine function is always 0 at the BC's. Any help with this process would be appreciated.
     
  2. jcsd
  3. Oct 8, 2011 #2

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Hint: Can you think of a function whose second derivative is minus a constant times itself and whose values at 0 and pi are zero?
     
  4. Oct 9, 2011 #3
    Sin(K*u)?? Second derivative would be -K2sin(K*u)

    I am not sure which variable should be inserted for "u" since u is a function of x,y,z.
     
  5. Oct 9, 2011 #4

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Your independent variables are x,y, and z. In part A you are trying to match the boundary conditions. So u = sin(x) would take care of the x = 0 and x = π boundary conditions. What about the others?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: PDE Help - Eigenfunction of a LaPlacian
  1. Pde help (Replies: 1)

  2. Laplacian help (Replies: 0)

  3. Laplacian Help (Replies: 2)

  4. PDE help (Replies: 0)

  5. Help with a pde (Replies: 1)

Loading...