Final Exam

All non-human resources are allowed.

Explain how to compute eigenvalues and eigen functions for 
Δϕ=-λ ϕ  on the partial annulus 1/3<r<2 and 0<θ<3 π/4 shown with homogeneous Dirichlet Boundary conditions on all sides.

Compute the 6 lowest eigenvalues and plot the associated eigen functions.

Graphics[{Gray,EdgeForm[Red], Disk[{0,0},2,{0, 3 π/4}],White, Disk[{0,0},1/3,{0, 3π/4}]},
Frame->True
]
Texture

Explain the orthogonality of the eigenfunctions and why it is useful.

Verify the orthogonality for your six functions.

You are given a complete set of eigenfunctions (Subscript[ϕ, n,k]) and eigenvalues (Subscript[λ, n,k]) for the Laplacian on a domain R with homogeneous Dirichlet boundary conditions.  The eigenfunctions are orthogonal with respect to a weight σ.

Explain how to solve the Heat Equation
	(Subscript[u, t])=Δu   u(0)=f1
on the domain R with the matching homogeneous boundary conditions.

Explain how to solve the Heat Equation
	(Subscript[u, t])+ γ u=Δu   u(0)=f1
on the domain R with the matching homogeneous boundary conditions. γ is a positive real constant.

Explain how to solve
	(Subscript[u, t])+ (Subscript[b, 2,12])(Subscript[ϕ, 2,12])[r,θ]=Δu   u(0,r,θ)=0
on the domain R with the matching homogeneous boundary conditions.

Explain how to solve
	(Subscript[u, t])+ (Subscript[b, 1,1])(Subscript[ϕ, 1,1])+(Subscript[b, 1,2])(Subscript[ϕ, 1,2])+(Subscript[b, 3,12])(Subscript[ϕ, 3,12])=Δu   u(0,r,θ)=f[r,θ]
on the domain R with the matching homogeneous boundary conditions.

We want to solve 
	(Subscript[u, t])=Δu   u(0)=f1
with non homogeneous Dirichlet boundary conditions u(Subscript[|, ∂R])=g.  Explain how to use a time independent function w that satisfies the boundary conditions to reduce the problem to a problem with homogeneous Dirichlet boundary conditions.