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]=
