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]:=
![Click for copyable input](assets.en/symbolic-pdes-over-regions/In_75.png)
leqn = Laplacian[u[x, y],{x, y}] == 0;
In[2]:=
![Click for copyable input](assets.en/symbolic-pdes-over-regions/In_76.png)
dcond = DirichletCondition[u[x, y] == Sin[6 ArcTan[y/x]], True];
In[3]:=
![Click for copyable input](assets.en/symbolic-pdes-over-regions/In_77.png)
\[CapitalOmega] = Disk[{0, 0}, 3];
Symbolically solve on the unit disk with Dirichlet boundary condition.
In[4]:=
![Click for copyable input](assets.en/symbolic-pdes-over-regions/In_78.png)
sol = DSolveValue[{leqn, dcond},
u[x, y], {x, y} \[Element] \[CapitalOmega]]
Out[4]=
![](assets.en/symbolic-pdes-over-regions/O_52.png)
In[5]:=
![Click for copyable input](assets.en/symbolic-pdes-over-regions/In_79.png)
Plot3D[sol, {x, y} \[Element] \[CapitalOmega], PlotRange -> All,
PlotStyle -> Hue[0.5], Exclusions -> None]
Out[5]=
![](assets.en/symbolic-pdes-over-regions/O_53.png)