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Re: Changing variables within a differential equation

  • To: mathgroup at
  • Subject: [mg105223] Re: Changing variables within a differential equation
  • From: schochet123 <schochet123 at>
  • Date: Wed, 25 Nov 2009 02:30:10 -0500 (EST)
  • References: <hdttaa$h4h$>

On Nov 17, 12:18 pm, riccardo benini <riccardo.ben... at>
> given a (partial) differential equation F(y''[x], y'[x], y[x], x)==0 (where
> y and x may be though as multidimensional),
> how can I correctly set up achangeofvariablesy -> y[g],  x -> x[t],
> and get an equation in g[t]:   F2(g''[t], g'[t], g[t], t)==0?

The first step is to express first derivatives of new variables with
respect to old ones
 in terms of first derivatives of old variables with respect to new
ones. To do this define

  newvars_List] := (dsub[oldvars, newvars] =
    With[{newofold = Through[newvars[Sequence @@ oldvars]],
      oldofnew = Through[oldvars[Sequence @@ newvars]]},
          0 == D[(newofold /. Thread[oldvars -> oldofnew]) - newvars,
            var], {var, newvars}] /. Thread[oldofnew -> oldvars]],
       Flatten[Table[D[newofold, var], {var, oldvars}]]]]]) /;
  Length[newvars] == Length[oldvars]

The second step is to express the first derivative of an arbitrary
function with respect to an old variable
 in terms of derivatives with respect to new variables, which can be
done by

transD[f_, oldvars_List, newvars_List, oldvar_] :=
 With[{newofold = Through[newvars[Sequence @@ oldvars]],
    tmpvars = Table[Unique[], {Length[oldvars]}]},
   D[f /. Thread[oldvars -> tmpvars] /. Thread[newvars -> newofold],
       oldvar] /. Thread[tmpvars -> oldvars] /.
     Thread[newofold -> newvars] /. dsub[oldvars, newvars]] /;
  Length[oldvars] == Length[newvars]

Now define your partial differential equation using transD. For
example, the Laplacian is

laplacian[f_, oldvars_, newvars_] :=
 Plus @@ Table[
   transD[transD[f, oldvars, newvars, var], oldvars, newvars,
    var], {var, oldvars}]

To check, calculate

    f[r, theta, phi], {x, y, z}, {r, theta,
     phi}] /. {x -> (#1 Cos[#2] Sin[#3] &),
    y -> (#1 Sin[#2] Sin[#3] &), z -> (#1 Cos[#3] &)}]]

and compare to the known formula for the Laplacian in spherical
coordinates, which may be found at


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