Test Problems: 2D

Test Problems on Rectangular domain 0<x<a and 0<y<b.

Separation of Variables gives solutions on a rectangular domain for 
	Δu=f
with a variety of boundary conditions.

For homgeneous Dirichlet conditions on all four sides
	u=(∑_(i,j))(Subscript[α, i,j])Sin[π x/a]Sin[π y/b]
and you can work out the f by differentiating.

For homgeneous Neumann conditions on all four sides
	u=(∑_(i,j))(Subscript[β, i,j])Cos[π x/a]Cos[π y/b]
and you can work out the f by differentiating.

For homgeneous Neumann conditions on the two x sides and homogeneous Dirichlet conditions on the two y-sides
	u=(∑_(i,j))(Subscript[β, i,j])Cos[π x/a]Sin[π y/b]
and you can work out the f by differentiating.

You can work out forms for other possibilities for the derivatives.

Test Problems on Quarter-Circular domain 0<r<a and 0<θ<π/2.

Separation of Variables gives solutions on circular domains for 
	Δu=f
with a variety of boundary conditions.

For Homogeneous Dirichlet conditions on all three sides
	u=(∑_(i,j))(Subscript[α, i,j])(Subscript[J, i])[(Subscript[γ, i,j]) r/a]Sin[2θ]
and you can work out the f by differentiating.  In the above (Subscript[J, i]) is the Bessel J function of order i and (Subscript[γ, i,j]) is the jth root of (Subscript[J, i]).

a=2;
{i,j}={1,1};
γ=N[BesselJZero[2i,j]];
ParametricPlot3D[
{r Cos[θ], r Sin[θ], BesselJ[2i, γ r/a] Sin[2i θ]},
{r, 0, a},
{θ, 0, π/2}
]

Clear[i,j,r,a,θ,γ]
<<VectorAnalysis`
SetCoordinates[Cylindrical[r,θ,z]]
FullSimplify[Laplacian[  BesselJ[2i, γ r/a] Sin[2i θ]]]

Much Simpler Test Problems 2D/3D

1) Pick a solution that satisfies your boundary conditions.
2) Compute the Laplacian
3) Use your computed Laplacian for the RHS.

uSol[r_,θ_]:= (r^2)(1-r) Sin[2 θ];
ParametricPlot3D[{r Cos[θ], r Sin[θ], uSol[r,θ]},{r, 0, 1},{θ,0, π/2},
BoxRatios->{1,1,0.5}
]
Simplify[Laplacian[ uSol[r,θ]]]