# Solve Partial Differential Equations over Regions

Solve partial differential equations numerically over full-dimensional regions in 1D, 2D, and 3D. The method used is primarily based on finite elements and allows for Dirichlet, Neumann, and Robin boundary conditions, as well as time-varying equations.

Solve a Poisson equation over a disk and with zero boundary conditions.

 In[1]:= Xusol = NDSolveValue[{\!\( \*SubsuperscriptBox[\(\[Del]\), \({x, y}\), \(2\)]\(u[x, y]\)\) == 1, DirichletCondition[u[x, y] == 0, True]}, u, {x, y} \[Element] Disk[]]
 Out[1]=
 In[2]:= XPlot3D[usol[x, y], {x, y} \[Element] Disk[]]
 Out[2]=

Solve a Poisson equation over a more complicated region.

 In[4]:= Xusol = NDSolveValue[{\!\( \*SubsuperscriptBox[\(\[Del]\), \({x, y}\), \(2\)]\(u[x, y]\)\) == 10, DirichletCondition[u[x, y] == 0, True]}, u, {x, y} \[Element] \[ScriptCapitalR]]
 Out[4]=
 In[5]:= XPlot3D[usol[x, y], {x, y} \[Element] \[ScriptCapitalR], PlotPoints -> 150, ImageSize -> 550, PlotRange -> {All, All, {All, -2}}, Mesh -> None, BoxRatios -> {1, 0.273, 0.4}, PlotTheme -> "Marketing", PlotStyle -> Opacity[0.5]]
 Out[5]=

## Mathematica

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