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Re: Partial differential equation with evolving boundary conditions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg91540] Re: Partial differential equation with evolving boundary conditions
  • From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
  • Date: Tue, 26 Aug 2008 03:29:13 -0400 (EDT)
  • Organization: Uni Leipzig
  • References: <g8o868$l6k$1@smc.vnet.net> <200808241105.HAA15454@smc.vnet.net> <g8u33a$qkg$1@smc.vnet.net>
  • Reply-to: kuska at informatik.uni-leipzig.de

Hi,

Ingolf Dahl wrote:
> Hi Jens-Peer,
> Thanks for your interest and your comments
> 
> My problematic code was a little damaged by the copy-and-paste. I try again:
> 
> Manipulate[s = NDSolve[{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, 
>   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]
> 
> Interleaved comments to your answer follow below
> 
> Best regards
> 
> Ingolf Dahl
>  
> 
>> -----Original Message-----
>> From: Jens-Peer Kuska [mailto:kuska at informatik.uni-leipzig.de] 
>> Sent: den 24 augusti 2008 13:06
>> To: mathgroup at smc.vnet.net
>> Subject:  Re: Partial differential equation with 
>> evolving boundary conditions
>>
>> Hi,
>>
>> a) the code DSolve[{\[Lambda];\[Lambda]s;emax;dutycycle;...}__]
>>     is total useless because emax; make nothing ..
> 
> The emax; command is there to tell Manipulate to recalculate NDSolve each
> time emax is changed. Maybe there is a better way to accomplish this?


Yes define elecy[] with two extra parameters for emax, and dutycycle.

> 
>> 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]
> 
> That is a reformulation of my question, but it that of mathematical
> necessity or is it just necessary for NDSolve?

It is mathematical necessary because otherwise you have
at the boundary *two* differential equations for the t-dependence
and both can't be satisfyed. At x==1 you would get two equations

NDSolve[{-lambdas*theta[1]'[t]=theta[1][t],
          theta[1]'[t]==something},__]

you can't satisfy *both*.

> 
> I wanted to avoid this approach, because my real problem is slightly more
> involved, and then straightforward integration is not possible.
> 
> By the way, the solutions you suggest are incomplete, they should read
> 
>     \[Theta][0, t] == integrationconstant0* Exp[-t/\[Lambda]s],
>     \[Theta][1, t] == integrationconstant1* Exp[-t/\[Lambda]s]
> 
>> c) this is inconsistent with \[Theta][y,0]==0 and no soulution
>>     would exist.
> 
> and then with integrationconstant0 = integrationconstant1 = 0 there is
> trivial consistence with 
> \[Theta][y,0]==0 
> 
>>
>> d) the classical way is to think about the existence and uniqueness
>>     of the solution *before* a analytical or numeric solution
>>     is attempt.
> 
> Sometimes I prefer other ways, all roads should lead to Rome. Sometimes it
> is very illuminating to try to find a constructive solution. In this way I
> was able to formulate a question appropriate for MathGroup, where all
> answers use to be nice. 
> 
> But with \[Theta][0, t]==0 and  \[Theta][1, t] == 0 there should exist
> solutions

In this case yes

, and thus also in this case(?) I do not think that the rule
> ""Boundary condition ... should have derivatives of order lower than the
> differential order of the partial differential equation" is applicable in
> this case. 

And why it should not be applicable ? where is the
mathematical proof for this.

Regards
   Jens


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