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

**References**:**Re: Partial differential equation with evolving boundary conditions***From:*Jens-Peer Kuska <kuska@informatik.uni-leipzig.de>