(*Variable&Parameters*)​​​​​​​​h=1;​​​​ri=0.5;​​​​eq=D[u[t,r,θ,z],t]-​​​​D[u[t,r,θ,z],r,r]+(1/r)D[u[t,r,θ,z],r]+​​​​1/(r^2)D[u[t,r,θ,z],θ,θ]+​​​​D[u[t,r,θ,z],z,z];​​​​​​​​(*InitialandBoundaryconditions*)​​​​​​​​ic={u[0,r,θ,z]==Exp[-r]};​​​​bc={u[t,ri,θ,z]==1,u[t,1,θ,z]==0,​​​​PeriodicBoundaryCondition[u[t,r,θ,z],θ==0,​​​​TranslationTransform[{0,0,2*Pi,0}]],​​​​PeriodicBoundaryCondition[u[t,r,θ,z],z==0,​​​​TranslationTransform[{0,0,0,h}]]};​​​​​​​​​​​​(*Solution*)​​​​​​​​sol=NDSolve[{eq==​​​​NeumannValue[0,r==ri]+NeumannValue[1,r==1],ic,bc},​​​​u[t,r,θ,z],{t,0,10},{r,ri,1},{θ,0,​​​​2*Pi},{z,0,1}];​​​​​​​​(*Plottingofsolution*)​​​​​​​​Manipulate[​​​​Plot3D[sol,{r,0,1},{θ,0,2Pi}],{t,0,10},{z,0,1}];
NDSolve
:Applying the mapping TransformationFunction[{{1,0,0,0,0},{0,1,0,0,0},{0,0,1,0,2π},{0,0,0,1,0},{0,0,0,0,1}}] of the periodic boundary condition PeriodicBoundaryCondition[u,θ0,TransformationFunction[{{1,0,0,0,0},{0,1,0,0,0},{0,0,1,0,2π},{0,0,0,1,0},{0,0,0,0,1}}]] to the coordinates {{0.5,0.,0.},{0.5,0.,0.0714286},{0.571429,0.,0.0714286},{0.571429,0.,0.},43,{0.571429,0.,0.607143},{0.5,0.,0.714286},{0.571429,0.,0.714286},287} did not result in a list of mapped coordinates.
NDSolve
:Applying the mapping TransformationFunction[{{1,0,0,0,0},{0,1,0,0,0},{0,0,1,0,2π},{0,0,0,1,0},{0,0,0,0,1}}] of the periodic boundary condition PeriodicBoundaryCondition[u,θ0,TransformationFunction[{{1,0,0,0,0},{0,1,0,0,0},{0,0,1,0,2π},{0,0,0,1,0},{0,0,0,0,1}}]] to the coordinates {{0.5,0.,0.},{0.5,0.,0.0714286},{0.571429,0.,0.0714286},{0.571429,0.,0.},43,{0.571429,0.,0.607143},{0.5,0.,0.714286},{0.571429,0.,0.714286},287} did not result in a list of mapped coordinates.
NDSolve
:Applying the mapping TransformationFunction[{{1,0,0,0,0},{0,1,0,0,0},{0,0,1,0,2π},{0,0,0,1,0},{0,0,0,0,1}}] of the periodic boundary condition PeriodicBoundaryCondition[u,θ0,TransformationFunction[{{1,0,0,0,0},{0,1,0,0,0},{0,0,1,0,2π},{0,0,0,1,0},{0,0,0,0,1}}]] to the coordinates {{0.5,0.,0.},{0.5,0.,0.0714286},{0.571429,0.,0.0714286},{0.571429,0.,0.},43,{0.571429,0.,0.607143},{0.5,0.,0.714286},{0.571429,0.,0.714286},287} did not result in a list of mapped coordinates.
General
:Further output of NDSolve::fempbcmf will be suppressed during this calculation.