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Re: problem with NDSolve::dvnoarg:

  • To: mathgroup at smc.vnet.net
  • Subject: [mg116546] Re: problem with NDSolve::dvnoarg:
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Sat, 19 Feb 2011 05:13:10 -0500 (EST)

tarun dutta wrote:
> n = 5;
> p = 5/1000;
> q = 1/10;
> c[-1][t] = 0; d[-1][t] = 0;
> c[n + 1][t] = 0; d[n + 1][t] = 0;
> eqn1 = Table[
>    c[i]'[t] == ((1/2)*i (i - 1) - q*i)*c[i][t] -
>      p*(Sqrt[i]*
>          c[i - 1][t]*(Sum[Sqrt[i]*d[i - 1][t]*d[i][t], {i, 0, n}]) +
>         Sqrt[i + 1]*
>          c[i + 1][
>           t]*(Sum[Sqrt[i + 1]*d[i - 1][t]*d[i][t], {i, 0, n}])), {i,
>     0, n}];
> eqn2 = Table[
>    d[i]'[t] == ((1/2)*i (i - 1) - q*i)*d[i][t] -
>      p*(Sqrt[i]*
>          d[i - 1][t]*(Sum[Sqrt[i]*c[i - 1][t]*c[i][t], {i, 0, n}]) +
>         Sqrt[i + 1]*
>          d[i + 1][
>           t]*(Sum[Sqrt[i + 1]*c[i - 1][t]*c[i][t], {i, 0, n}])), {i,
>     0, n}];
> eqn3 = Table[Sum[(c[i]^2)[t], {i, 0, n}] == 1, {i, 1}];
> eqn4 = Table[Sum[(d[i]^2)[t], {i, 0, n}] == 1, {i, 1}];
> eqns = Flatten[Join[eqn1, eqn2, eqn3, eqn4]];
> bcs = {{c[0][0] == d[0][0] == 0.004567},
>    Table[c[i][0] == 0.0000000034, {i, 1, 5}],
>    Table[d[i][0] == 0.0000000034, {i, 1, 5}]};
> var = Join[Table[c[i], {i, 0, 5}], Table[d[i], {i, 0, n}]];
> sol = NDSolve[{eqns, bcs}, var, {t, 1/10}]
> 
> NDSolve::dvnoarg:
> 
> I can not understand the meaning of this kind of error,,
> can anyone explain and give some valuable insight.
> regards,
> tarun

An addendum to my last two responses.

Once you get a viable system you learn it is overdetermined. This is 
because the algebraic invariants are not needed to make it fully 
determined. Since you want them enforced so the solution does not 
"wander" due to numerical error, you can use the Projection method to 
enforce them. I show the code below. Please observe the usage 
(d[i][t]^2) rather than (d[i]^2)[t], as the latter is what persistently 
gives rise to an ill-formed system.

n = 5;
p = 5/1000;
q = 1/10;
c[-1][t] = 0; d[-1][t] = 0;
c[n + 1][t] = 0; d[n + 1][t] = 0;
eqn1 = Table[
    c[i]'[t] == ((1/2)*i (i - 1) - q*i)*c[i][t] -
      p*(Sqrt[i]*
          c[i - 1][t]*(Sum[Sqrt[i]*d[i - 1][t]*d[i][t], {i, 0, n}]) +
         Sqrt[i + 1]*
          c[i + 1][
           t]*(Sum[Sqrt[i + 1]*d[i - 1][t]*d[i][t], {i, 0, n}])), {i,
     0, n}];
eqn2 = Table[
    d[i]'[t] == ((1/2)*i (i - 1) - q*i)*d[i][t] -
      p*(Sqrt[i]*
          d[i - 1][t]*(Sum[Sqrt[i]*c[i - 1][t]*c[i][t], {i, 0, n}]) +
         Sqrt[i + 1]*
          d[i + 1][
           t]*(Sum[Sqrt[i + 1]*c[i - 1][t]*c[i][t], {i, 0, n}])), {i,
     0, n}];
invariants = {Sum[(c[i][t]^2), {i, 0, n}] - 1,
    Sum[(d[i][t]^2), {i, 0, n}] - 1};
eqns = Flatten[Join[eqn1, eqn2]];
bcs = {{c[0][0] == d[0][0] == 0.004567},
    Table[c[i][0] == 0.0000000034, {i, 1, 5}],
    Table[d[i][0] == 0.0000000034, {i, 1, 5}]};
var = Join[Table[c[i], {i, 0, 5}], Table[d[i], {i, 0, n}]];

sol = NDSolve[Flatten[{eqns, bcs}], var, {t, 1/10},
   Method -> {Projection, "Invariants" -> invariants}]

This will give a solution.

It is, alas, almost certainly a bad solution. This is because the 
invariants are not satisfied for t==0, hence cannot be satisfied 
elsewhere. So that might indicate a problematic formulation.

Daniel Lichtblau
Wolfram Research



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