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MathGroup Archive 2000

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NDSolve: Distinguishing equations

  • To: mathgroup at smc.vnet.net
  • Subject: [mg21574] NDSolve: Distinguishing equations
  • From: Anthony Foglia <afoglia at hal.physics.ucsb.edu>
  • Date: Sat, 15 Jan 2000 02:04:17 -0500 (EST)
  • Organization: University of California, Santa Barbara
  • Sender: owner-wri-mathgroup at wolfram.com

I have a set of three PDEs that I want to solve using NDSolve (in
Mathematica 4).  Unfortunately I seem to keep getting two errors.  One is
that there are no intial values specified (even though there are enough for
me to solve it exactly, which I have already done).  So if I add more
initial values, I get a message stating ""a clear distinction cannot be
found between the PDE's, initial values, and boundary conditions."

So is there some general order or rules for the equations I give NDSolve to
solve to avoid the chances of this second error?

For more specifics, below are the three equations, (some constant setting,)
and the NDSolve commands and errors.  My goal is to add a non-linear term
to these equations and solve them, but first I want to understand NDSolve
enough to get this to work.  Thanks for any help.

In[1]:= eq1 = \[Rho] D[v[y, t], t] == D[\[Sigma][y, t], y]

In[2]:= \!\(eq2 = 
    D[\[Sigma][y, t], 
        t]\  == \ \[Mu]\ D[v[y, t], 
            y] - \(\(2\ \[Mu]\)\/\[Tau]\) \((\[Lambda]\ \[Sigma][y, 
                  t] - \[CapitalDelta][y, t])\)\)

In[3]:= \!\(eq3 = 
    D[\[CapitalDelta][y, t], 
        t] == \(1\/\[Tau]\) \((\[Lambda]\ \[Sigma][y, t] -
\[CapitalDelta][y, 
              t])\)\)

In[4]:= \[Rho] = 1; \[Mu] = 1; \[Lambda] = 1; \[Tau] = 1; h = 1;

In[5]:= \[Omega] = \[Pi]/15;

In[6]:= \!\(\(\[Epsilon]\_m =  .01;\)\)

In[7]:= Clear[\[Sigma], \[CapitalDelta], v]

In[8]:= \!\(NDSolve[{eq1, eq2, eq3, v[0, t] == \(-v[h, t]\), 
      v[0, t] == \(1\/2\) h\ \[Epsilon]\_m\ \[Omega]\ E\^\(I\ \[Omega]\
t\), 
      v[0, 0] == \(1\/2\) 
          h\ \[Epsilon]\_m\ \[Omega]}, {\[Sigma], \[CapitalDelta], v}, {y,
0, 
      h}, {t, 0, 10*2  \[Pi]/\[Omega]}]\)

NDSolve::"ivnone": "No initial values specified."

In[9]:= \!\(NDSolve[{eq1, eq2, eq3, 
      v[0, t] == \(1\/2\) h\ \[Epsilon]\_m\ \[Omega]\ E\^\(I\ \[Omega]\
t\), 
      v[h, t] == \(-\(1\/2\)\) 
          h\ \[Epsilon]\_m\ \[Omega]\ E\^\(I\ \[Omega]\ t\), 
      v[0, 0] == \(1\/2\) h\ \[Epsilon]\_m\ \[Omega], \[Sigma][y, 0] == 
        0, \[CapitalDelta][y, 0] == 0}, {\[Sigma], \[CapitalDelta], v}, {y, 
      0, h}, {t, 0, 10*2  \[Pi]/\[Omega]}]\)

\!\(\*
  RowBox[{\(NDSolve::"pdivbc"\), \(\(:\)\(\ \)\), "\<\"In
\\!\\({\\(\\(\\(\\(\
\\(\\(\\*SuperscriptBox[\\\"v\\\", TagBox[\\((0, 1)\\), Derivative], \
Rule[MultilineFunction, None]]\\)\\)[\\(\\(y, t\\)\\)]\\)\\) == \
\\(\\(\\(\\(\\*SuperscriptBox[\\\"\[Sigma]\\\", TagBox[\\((1, 0)\\), \
Derivative], Rule[MultilineFunction, None]]\\)\\)[\\(\\(y, \
t\\)\\)]\\)\\)\\)\\), \\(\\(\[LeftSkeleton] 6 \[RightSkeleton]\\)\\), \
\\(\\(\\(\\(\[CapitalDelta][\\(\\(y, 0\\)\\)]\\)\\) == 0\\)\\)}\\), a clear
\
distinction cannot be found between the PDE's, initial values, and boundary
\
conditions.\"\>"}]\)


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