Symbolic PDEs over Regions
Version 11 adds extensive support for symbolic and numerical solutions of boundary value problems over regions.
Dirichlet problem for the Laplace equation in a disk.
In[1]:=

leqn = Laplacian[u[x, y],{x, y}] == 0;
In[2]:=

dcond = DirichletCondition[u[x, y] == Sin[6 ArcTan[y/x]], True];
In[3]:=

\[CapitalOmega] = Disk[{0, 0}, 3];
Symbolically solve on the unit disk with Dirichlet boundary condition.
In[4]:=

sol = DSolveValue[{leqn, dcond},
u[x, y], {x, y} \[Element] \[CapitalOmega]]
Out[4]=

In[5]:=

Plot3D[sol, {x, y} \[Element] \[CapitalOmega], PlotRange -> All,
PlotStyle -> Hue[0.5], Exclusions -> None]
Out[5]=
