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Re: trouble with NDSolve

  • To: mathgroup at smc.vnet.net
  • Subject: [mg63642] Re: [mg63590] trouble with NDSolve
  • From: Pratik Desai <pdesai1 at umbc.edu>
  • Date: Sun, 8 Jan 2006 03:33:08 -0500 (EST)
  • References: <200601061024.FAA14623@smc.vnet.net> <43BF47B5.2040003@umbc.edu>
  • Sender: owner-wri-mathgroup at wolfram.com

Pratik Desai wrote:

> lahm wrote:
>
>> i have trouble with solving a diffusion problem with NDSolve. I would 
>> like to get a 2D concentration profile using second Fick's law 
>> dc/dt=d2c/dx2 + d2c/dy2. c is concentration and is function of t,x 
>> and y. The initial condition is concentration=1 everywhere. The 
>> boundary conditions are c=1 for x=1, y=1 and y=-1. On x=0, however, 
>> the concentration is 0, but only for -0.01<y<0.01, the other part of 
>> x=0 acts neutral, is neither a sink nor an inlet, concentration 
>> therefore is determined by the sink and the boundaries. Now when I 
>> try to imput that c=0 only at a part of x=0,  I get the message that 
>> boundary condition is not specified on a single edge of the boundary 
>> of the computational domain. NDSolve::bcedge: I tried several 
>> different approaches, but all of them have failed. There's always 
>> something wrong so mathematica always encounters some problem.  I 
>> can't make only one part of the boundary work as a sink an the other 
>> to be neutral. Please help!
>>
>>  
>>
> One of the reasons, if I am not mistaken,  the 2- D laplacian term in 
> your PDE is notoriously problemetic for a finite difference schema 
> (NDsolve uses quite a bit of FD schemes). There was a thread regarding 
> a similar issue a heat equation for a square plate...last year. Here 
> is the link anyway
> http://groups.google.com/group/comp.soft-sys.math.mathematica/browse_frm/thread/5fc408a259b51a90/4138f70b28bf6a78?lnk=st&q=Heat+equation%2C+NDSolve%2C+Nasser&rnum=1#4138f70b28bf6a78 
>
>
> Hope this helps
>
> Pratik
>
>
Actually one can solve this problem (I think) , I apologize for not 
reading your post thoroughly. Anyway here is my attempt
Just copy the stuff that appears after this line in your note book, 
should work fine....Pratik



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                    Mathematica-Compatible Notebook

This notebook can be used with any Mathematica-compatible
application, such as Mathematica, MathReader or Publicon. The data
for the notebook starts with the line containing stars above.

To get the notebook into a Mathematica-compatible application, do
one of the following:

* Save the data starting with the line of stars above into a file
  with a name ending in .nb, then open the file inside the
  application;

* Copy the data starting with the line of stars above to the
  clipboard, then use the Paste menu command inside the application.

Data for notebooks contains only printable 7-bit ASCII and can be
sent directly in email or through ftp in text mode.  Newlines can be
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NOTE: If you modify the data for this notebook not in a Mathematica-
compatible application, you must delete the line below containing
the word CacheID, otherwise Mathematica-compatible applications may
try to use invalid cache data.

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applications, contact Wolfram Research:
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(*CacheID: 232*)


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(*  CellTagsIndexPosition[      3775,        100]*)
(*WindowFrame->Normal*)



Notebook[{
Cell[BoxData[{
    \(Clear[u]\), "\[IndentingNewLine]",
    \(c = 1\), "\[IndentingNewLine]",
    \(<< Graphics`Legend`\), "\[IndentingNewLine]",
    \(u[t, 1, 1] := 1\), "\[IndentingNewLine]",
    \(u[t, \(-1\), \(-1\)] := 1\), "\[IndentingNewLine]",
    \(diffusion\  =
      Apply[Sequence, {{\[PartialD]\_t u[t, x, y] \[Equal]
              c*\[PartialD]\_\(x, x\)u[t, x, y] +
                c*\[PartialD]\_\(y, y\)u[t, x, y],
            u[t, 1, y] == u[t, \(-1\), y],
            u[t, x, 1] \[Equal] u[t, x, \(-1\)],
            u[0, x, y] \[Equal] 1 - DiscreteDelta[x, y]},
          u, {t, 0, 100}, {x, \(-1\), 1}, {y, \(-1\),
            1}}]\), "\[IndentingNewLine]",
    \(solution = NDSolve[diffusion]\), "\[IndentingNewLine]",
    \(ShowLegend[
      DensityPlot[
        Evaluate[u[0, x, y] /. solution[\([1]\)]], {x, \(-1\),
          1}, {y, \(-1\), 1}, ColorFunction -> \((GrayLevel[#] &)\),
        Mesh \[Rule] False], {GrayLevel[#] &, 10, "\< 1\>", "\<-1\>",
        LegendPosition \[Rule] {1.1, \(- .4\)}}]\), "\[IndentingNewLine]",
    \(\(ShowLegend[
        DensityPlot[
          Evaluate[u[1, x, y] /. solution[\([1]\)]], {x, \(-1\),
            1}, {y, \(-1\), 1}, ColorFunction -> \((Hue[#] &)\),
          Mesh \[Rule] False], {Hue[#] &, 10, "\< 1\>", "\<-1\>",
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WindowMargins->{{0, Automatic}, {Automatic, 0}}
]

(*******************************************************************
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Notebook[{
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]
*)



(*******************************************************************
End of Mathematica Notebook file.
*******************************************************************)




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