# Solve Axisymmetric PDEs

Solve a PDE over a axisymmetric region.

 In:= XSubscript[r, 1] = 1; Subscript[r, 2] = 2; \[CapitalOmega] = ImplicitRegion[ True, {{r, Subscript[r, 1], Subscript[r, 2]}, {z, 0, 1/2}}];

Specify a Neumann value of 100 W/m2(watt per square meter) at the inner edge and a temperature of 10 °C (degrees) at the outer edge, with a thermal conduction coefficient k of 10 W/m/C (watt per meter per degree).

 In:= Xk = 100; nv = -100; dc = 10; uif = NDSolveValue[{k/r D[ r D[u[r, z], r], r] + k D[u[r, z], {z, 2}] == NeumannValue[nv, r == 1], DirichletCondition[u[r, z] == 10, r == 2]}, u, {r, z} \[Element] \[CapitalOmega]];

Plot the solution.

 In:= Xdp = DensityPlot[uif[r, z], {r, z} \[Element] \[CapitalOmega], ColorFunction -> "Temperature"]
 Out= Verify the solution.

 In:= XanalyticalSolution[r_, z_] = (dc + nv*Subscript[r, 1]/k Log[r/Subscript[r, 2]]); Plot[uif[r, 0] - analyticalSolution[r, z], {r, Subscript[r, 1], Subscript[r, 2]}]
 Out= Plot the solution in 3D.

 Out= ## Mathematica

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