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.

As before the solution of the θ equation is
	Θ[θ]=(Subscript[B, 1])Sin[μ θ] + (Subscript[B, 2])Cos[μ θ]
We need Θ[0]=Θ[3π/4]=0 so (Subscript[B, 2])=0  and 
	Sin[μ 3 π/4]=0 
which determines μ=4/3 n  for n=1,2,…

As before (for each n) the solution of the r equation is 
	R[r]=(Subscript[C, 1])BesselJ[4/3 n, (Sqrt[λ])r]+(Subscript[C, 2])BesselJ[4/3 n, (Sqrt[λ])r]
To satisfy R[1/3]=R[2]=0 we need to find non-trivial (Subscript[C, 1]) and (Subscript[C, 2]) satisfying
	(({{BesselJ[4/3 n,(Sqrt[λ])/3], BesselY[4/3 n,(Sqrt[λ])/3]}, {BesselJ[4/3 n,2(Sqrt[λ])], BesselY[4/3 n,2(Sqrt[λ])]}})).(({{(Subscript[C, 1])}, {(Subscript[C, 2])}}))=(({{0}, {0}}))
Thus as before we need the null space of this matrix at the values of λ where it is non-invertibe

A[n_,λ_]:= (({{BesselJ[4/3 n,(Sqrt[λ])/3], BesselY[4/3 n,(Sqrt[λ])/3]}, {BesselJ[4/3 n,2(Sqrt[λ])], BesselY[4/3 n,2(Sqrt[λ])]}}))
Plot[ 
Evaluate[
Table[ Det[A[n,λ]],{n,1,6}]
],
{λ,0, 45},
PlotRange->{-2,2},
PlotLegends->Automatic]

Computing accurate values of first sxi zeros.

Clear[λVal]
{λVal[1,1],λVal[1,2]}=λ/.FindRoot[
Det[A[1,λ]]==0, {λ,{4,16}}];
{λVal[2,1],λVal[2,2]}=λ/.FindRoot[
Det[A[2,λ]]==0, {λ,{8,22}}];
λVal[3,1]=λ/.FindRoot[Det[A[3,λ]]==0, {λ,14}];
λVal[4,1]=λ/.FindRoot[Det[A[4,λ]]==0, {λ,21}];
Plot[ 
Evaluate[
Table[ Det[A[n,λ]],{n,1,6}]
],
{λ,0, 45},
PlotRange->{-2,2},
PlotLegends->Automatic,
Epilog->{PointSize[0.02],
{Blue, Point[ {{λVal[1,1],0},{λVal[1,2],0}}]},
{Brown, Point[ {{λVal[2,1],0},{λVal[2,1],0}}]},
{Green, Point[ {λVal[3,1],0}]},
{Red, Point[ {λVal[4,1],0}]}
},
PlotLabel->NumberForm[
Sort[{λVal[1,1],λVal[1,2],λVal[2,1],λVal[2,2],λVal[3,1],λVal[4,1]}],
4]
]

I need to compute the coefficients

c[1,1]= NullSpace[A[1,λVal[1,1]]][[1]];
c[1,2]= NullSpace[A[1,λVal[1,2]]][[1]];
c[2,1]= NullSpace[A[2,λVal[2,1]]][[1]];
c[2,2]= NullSpace[A[2,λVal[2,2]]][[1]];
c[3,1]= NullSpace[A[3,λVal[3,1]]][[1]];
c[4,1]= NullSpace[A[4,λVal[4,1]]][[1]];
ϕ[n_,k_][r_,θ_]:= Sin[4/3 n θ] c[n,k].{BesselJ[4/3 n, (Sqrt[λVal[n,k]])r], BesselY[4/3 n, (Sqrt[λVal[n,k]])r]}

nks= {{1,1},{1,2},{2,1},{2,2},{3,1},{4,1}};
TableForm[
Table[
{n1,k1}=nks[[i]];
{n2,k2}=nks[[j]];
NumberForm[
Integrate[  ϕ[n1,k1][r,θ]*ϕ[n2,k2][r,θ]*r,{θ,0, 3π/4},{r,1/3,2}],
3],
{i,1,6},
{j,1,6}
],
TableHeadings->{nks,nks}
]

Table[
{n,k}=nks[[i]];
ParametricPlot3D[
{r Cos[θ], r Sin[θ], ϕ[n,k][r,θ]},
{θ,0, 3π/4},{r,1/3,2},
Mesh->None,
PlotLabel->{{n,k},λVal[n,k]},
Ticks->None],
{i,1,6}]

PlotStyle

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.