4515 Exam 1

Compute the eigenfunctions and eigenvalues for -Δϕ=λ ϕ  on the semi circular domain 0<θ<π and 0<r<1.

Homogeneous Dirichlet conditions all round the boundary.

Homogenous Dirichlet boundary conditions on the circular boundary and homogeneous Neumann conditions on the straight edge(s).

Homogeneous Neumann conditions on the circular boundary and homogeneous Dirichlet conditions on the straight edge(s).

Compute the eigenfunctions and eigenvalues of -Δϕ=λ ϕ with Dirichlet boundry conditions on the circular cylinder 0<r<(Subscript[r, max]), 0≤θ<2π, and 0<z<(Subscript[z, max]).   Do the pieces in the order θ, z, then r.

For each of the problems in Q1 plot the four eigenfunctions with the lowest eigenvalues.