Solve a Dirichlet Problem for the Laplace Equation

Specify the Laplace equation in 2D.

In[1]:=

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

Prescribe a Dirichlet condition for the equation in a rectangle.

In[2]:=

\[CapitalOmega] = Rectangle[{0, 0}, {1, 2}];

In[3]:=

dcond = DirichletCondition[ u[x, y] == Piecewise[{{UnitTriangle[2 x - 1], y == 0 || y == 2}}, 0], True];

Solve the Dirichlet problem.

In[4]:=

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

Out[4]=

Extract the first 300 terms from the Inactive sum.

In[5]:=

asol = sol /. {\[Infinity] -> 300} // Activate;

Visualize the solution on the rectangle.

In[6]:=

Plot3D[asol // Evaluate, {x, y} \[Element] \[CapitalOmega], PlotRange -> All, PlotTheme -> "Business"]

Out[6]=