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rexasaurus
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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.
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.