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MathGroup Archive 2000

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Re: Problem with partial differential equations

  • To: mathgroup at smc.vnet.net
  • Subject: [mg23502] Re: Problem with partial differential equations
  • From: Roland Franzius <Roland.Franzius at uos.de>
  • Date: Tue, 16 May 2000 02:44:51 -0400 (EDT)
  • References: <8etp9i$n1n@smc.vnet.net> <BH0R4.93$5j.5351@ralph.vnet.net> <84CS4.206$5j.13403@ralph.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Hi Max, I don't see a problem.  

In:

P1[x_, t_] := 2*x^2; 
P2[x_, t_] := 2*(x + t)^2; 
Diff = 0.2; 
G[x_] := Exp[-x^2]; 

solution = n /. First[NDSolve[
      {Derivative[0, 1][n][x, t] == Diff*Derivative[2, 0][n][x, t] + 
         n[x, t]*Derivative[2, 0][P1][x, t] + 
         Derivative[1, 0][n][x, t]*Derivative[1, 0][P1][x, t], 
	n[x, 0] == G[x], 
	n[-3., t] == G[-3.], 
       n[3., t] == G[3.]}, 
	n, {x, -3, 3}, {t, 0, 4}, 
      StartingStepSize -> 0.05]]; 
Plot3D[solution[x, t], {x, -2, 2}, {t, 0, 4}]; 

Out:  -SurfaceGrafics-

This has been copied and pasted from below with replacement  = -> := in
the first 4 lines

My diffusion problem solution is looking quite similar:

Needs["Graphics`Arrow`"]; 

Diffuse[name_,density0_]:= name= 
y/.
NDSolve[ {D[y[t, x], t] == D[y[t, x], x, x],  y[0, x] == density0[x], 
Derivative[0, 1][y][t, 0] == 0, Derivative[0, 1][y][t, Pi] == 0}, 
y,  {t, 0, 4}, {x, 0, Pi}, MaxSteps -> 400][[1]]

Diffuse[YD,#*Sin[4*#]^2&]

pl = Plot3D[
	YD[t, x],{x, 0.01, Pi - 0.1}, {t, 0, 0.05}, PlotPoints -> 50, Mesh ->
False,
       PlotRange -> {-0.5, 3}, Boxed -> False,
	Axes -> None, DisplayFunction -> Identity];

AxesAndText = 
Graphics[{RGBColor[0., .6, 0.],
	Arrow[{0.1, 0.24}, {0.4, 0.14}],
	Text["Ort", {0.33, 0.12}],
	RGBColor[1., 0, 0],
	Arrow[{0.75, 0.1}, {0.85, 0.3}],
	Text["Zeit", {0.8, 0.3}],
	RGBColor[0, 0, 1.],
	Arrow[{0.04, 0.36}, {0.04, 0.6}],
	Text["Temp", {0.10, 0.58}],
	RGBColor[0., 0., 0.],
	Text["Tempt", {0.47, 0.35}],
	RGBColor[0., 0., 0.],
	Text["Temp0", {0.24, 0.55}],
	Text["Abl0", {0.2, 0.5}],
	Text["Abl0", {0.86, 0.45}]}];


Show[pl, AxesAndText]

good luck

roland 

ps: only by chance I just saw your reply at
comp.soft-sys.math.mathematica

Max Ulbrich wrote:
> 
> Hi Roland, I tried, but I still get an error message like
> 
> NDSolve::ndnum:
>     The right-hand side of the differential equation does not evaluate
> to a number at t == 0.
> 
> Has anyone an idea what is wrong?
> 
> Max
> 
> Roland Franzius wrote:
> 
> > Hi Max,
> > seems to work if you use delayed definitions (at evalutation time)
> >
> > > P1[x_,t_]:=2x^2;
> > > P2[x_,t_]:=2(x+t)^2;
> > > Diff=0.2;
> > > G[x_]:=Exp[-x^2];
> >
> > probably a problem of fixing variables to numbers
> >
> > Max Ulbrich wrote:
> > >
> > > Hi,
> > >
> > > I have a problem with a partial differential equation.
> > > I want to solve a diffusion problem in a time-dependent potential.
> > > It works with the potential P1, but not with P2.
> > > Can someone help me?
> > >
> > > P1[x_,t_]=2x^2;
> > > P2[x_,t_]=2(x+t)^2;
> > > Diff=0.2;
> > > G[x_]=Exp[-x^2];
> > > solution=n/.First[
> > >       NDSolve[{Derivative[0,1][n][x,t]==
> > >
> > > Diff*Derivative[2,0][n][x,t]+n[x,t]*Derivative[2,0][P1][x,t]+
> > >
> > > Derivative[1,0][n][x,t]*Derivative[1,0][P1][x,t],n[x,0]==G[x],
> > >           n[-3,t]==G[-3],n[3,t]==G[3]},n,{x,-3,3},{t,0,4},
> > >         StartingStepSize->0.05]];
> > > Plot3D[solution[x,t],{x,-2,2},{t,0,4}];
> > >
> > > Max Ulbrich
> > > ulbrich at biochem.mpg.de
> >
> > --
> > Roland Franzius
> >
> >   +++ exactly <<n>> lines of this message have value <<FALSE>> +++

-- 
Roland Franzius
Theor. Physik FB Physik, Univ. Osnabrueck

  +++ exactly <<n>> lines of this message have value <<FALSE>> +++


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