(adsbygoogle = window.adsbygoogle || []).push({}); 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:

u_{tt}=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.

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# Homework Help: PDE Help - Eigenfunction of a LaPlacian

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