I’m trying to get an analytical solution of Laplace PDE with Dirichlet boundary conditions (in polar coordinates). I managed to solve it numerically with NDSolveValue and I know there is an analytical solution and I know what it is, but I would like DSolve to return it. But DSolve returns the input.
sol = DSolve[{Laplacian[ u[\[Rho], \[CurlyPhi]], {\[Rho], \[CurlyPhi]}, "Polar"] == 0, DirichletCondition[u[\[Rho], \[CurlyPhi]] == 0, 1 <= \[Rho] <= 2 && \[CurlyPhi] == 0], DirichletCondition[u[\[Rho], \[CurlyPhi]] == 0, 1 <= \[Rho] <= 2 && \[CurlyPhi] == \[Pi]], DirichletCondition[ u[\[Rho], \[CurlyPhi]] == Sin[\[CurlyPhi]], \[Rho] == 1 && 0 <= \[CurlyPhi] <= \[Pi]], DirichletCondition[ u[\[Rho], \[CurlyPhi]] == 0., \[Rho] == 2 && 0 <= \[CurlyPhi] <= \[Pi]]}, u, {\[Rho], 1, 2}, {\[CurlyPhi], 0, \[Pi]}]; 