Wolfram Language

Core Geometry

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.

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leqn = Laplacian[u[x, y],{x, y}] == 0;
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dcond = DirichletCondition[u[x, y] == Sin[6 ArcTan[y/x]], True];
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\[CapitalOmega] = Disk[{0, 0}, 3];

Symbolically solve on the unit disk with Dirichlet boundary condition.

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sol = DSolveValue[{leqn, dcond}, u[x, y], {x, y} \[Element] \[CapitalOmega]]
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Plot3D[sol, {x, y} \[Element] \[CapitalOmega], PlotRange -> All, PlotStyle -> Hue[0.5], Exclusions -> None]
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