       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

```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

```

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