       Re: Partial differential equation with evolving boundary conditions

• To: mathgroup at smc.vnet.net
• Subject: [mg91498] Re: Partial differential equation with evolving boundary conditions
• From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
• Date: Sun, 24 Aug 2008 07:05:33 -0400 (EDT)
• References: <g8o868\$l6k\$1@smc.vnet.net>

```Hi,

a) the code DSolve[{\[Lambda];\[Lambda]s;emax;dutycycle;...}__]
is total useless because emax; make nothing ..

b) you can't have a first order equation on the boundary for
theta[1|0,t] you must integrate that to get
\[Theta][0, t] == Exp[-t/\[Lambda]s],
\[Theta][1, t] == Exp[-t/\[Lambda]s]

c) this is inconsistent with \[Theta][y,0]==0 and no soulution
would exist.

d) the classical way is to think about the existence and uniqueness
of the solution *before* a analytical or numeric solution
is attempt.

e)

Manipulate[s = NDSolve[{
\!\(
\*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] == Exp[-t/\[Lambda]s],
\[Theta][1, t] == Exp[-t/\[Lambda]s],
\[Theta][y, 0] == 1}, \[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,
Appearance -> "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_] :=
Which[0 <= Mod[t, 1] <= dutycycle, emax,
dutycycle < Mod[t, 1] <= 1, 0.]),
ContinuousAction -> False, ControlPlacement -> Top]

work fine.

Regards
Jens

Ingolf Dahl wrote:
> 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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