Partial differential equation with evolving boundary conditions

*To*: mathgroup at smc.vnet.net*Subject*: [mg91477] Partial differential equation with evolving boundary conditions*From*: "Ingolf Dahl" <ingolf.dahl at telia.com>*Date*: Sat, 23 Aug 2008 01:41:41 -0400 (EDT)*Organization*: Goteborg University*Reply-to*: <ingolf.dahl at telia.com>

Best friends! I am trying to solve a partial differential equation (in principle the heat equation) with boundary conditions that also evolve with time. For instance this works for me: Manipulate[s=NDSolve[{\[Lambda];emax;dutycycle;\!\( \*SubscriptBox[\(\[PartialD]\), \(y, y\)]\ \(\[Theta][y, t]\)\)+elecy[t]*(5-\[Theta][y,t])*y*(1-y)==\[Lambda]*\!\( \*SubscriptBox[\(\[PartialD]\), \(t\)]\(\[Theta][y, t]\)\), \[Theta][y,0]==0, -(\!\( \*SubscriptBox[\(\[PartialD]\), \(y1\)]\(\[Theta][y1, t]\)\)/.{y1->0})*0.1+\[Theta][0,t]==0,+( \!\( \*SubscriptBox[\(\[PartialD]\), \(y1\)]\(\[Theta][y1, t]\)\)/.{y1->1})*0.1+\[Theta][1,t]== 0},\[Theta],{y,0,1},{t,0,2}];Plot3D[\[Theta][y,t]/.s,{y,0,1},{t,0,2},PlotSty le->Automatic,PlotRange->{0,5}],{{emax,25.,"emax"},0,100,Appearance->"Labele d"},{{dutycycle,0.25,"dutycycle"},0,1,Appearance->"Labeled"}, {{\[Lambda],1.,"\[Lambda]"},0.001,10,Appearance->"Labeled"},Initialization:> (elecy[t_]:=Which[0<=Mod[t,1]<=dutycycle,emax,dutycycle<Mod[t,1]<=1,0.]), ContinuousAction->False,ControlPlacement->Top] but this does not work Manipulate[s=NDSolve[{\[Lambda];\[Lambda]s;emax;dutycycle;\!\( \*SubscriptBox[\(\[PartialD]\), \(y, y\)]\ \(\[Theta][y, t]\)\)+elecy[t]*(5-\[Theta][y,t])*y*(1-y)==\[Lambda]*\!\( \*SubscriptBox[\(\[PartialD]\), \(t\)]\(\[Theta][y, t]\)\), \[Theta][0,t]==-\[Lambda]s*\!\( \*SubscriptBox[\(\[PartialD]\), \(t\)]\(\[Theta][0, t]\)\),\[Theta][1,t]==-\[Lambda]s*\!\( \*SubscriptBox[\(\[PartialD]\), \(t\)]\(\[Theta][1, t]\)\), \[Theta][y,0]==0},\[Theta],{y,0,1},{t,0,2}];Plot3D[\[Theta][y,t]/.s,{y,0,1}, {t,0,2},PlotStyle->Automatic,PlotRange->{0,5}],{{emax,25.,"emax"},0,100,Appe arance->"Labeled"},{{dutycycle,0.25,"dutycycle"},0,1,Appearance->"Labeled"}, {{\[Lambda],1.,"\[Lambda]"},0.001,10,Appearance->"Labeled"},{{\[Lambda]s,1., "\[Lambda]s"},0.001,10,Appearance->"Labeled"},Initialization:>(elecy[t_]:=Wh ich[0<=Mod[t,1]<=dutycycle,emax,dutycycle<Mod[t,1]<=1,0.]), ContinuousAction->False,ControlPlacement->Top] I obtain the error message NDSolve::bdord: Boundary condition \[Theta][0,t]+1. (\[Theta]^(0,1))[0,t] should have derivatives of order lower than the differential order of the partial differential equation. >> plus a lot more. As I understand it, the second case should be solvable in principle. Is this error a deficiency of NDSolve, or have I made some mistake? This problem can be seen as a partial differential equation (in y and t) coupled to two ordinary differential equations (for the boundary conditions, only in t). What is the best way to solve such problems? This is the simplification of a more involved case, where the boundary conditions also are coupled to the partial differential equation, but I have tried to boil down the problem here. Best regards Ingolf Dahl Sweden