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Transform differential equation by tranformation rule

  • To: mathgroup at smc.vnet.net
  • Subject: [mg111016] Transform differential equation by tranformation rule
  • From: TrinhDao <hackerdarkrose at yahoo.com>
  • Date: Sat, 17 Jul 2010 08:15:08 -0400 (EDT)

Dear Mathematica users,

I want to transform one different equation (variable r,z) to another
different equation (variable x,y).

The origin equation is :
equation = Derivative[0, 2][u][r, z] +  Derivative[1, 0][u][r, z]/r +
Derivative[2, 0][u][r, z] == 0

The transform rule is :

r = Cos[x]Sinh[y]
z = Sin[x] Cosh[y]

============

Now, my task is simple, apply this transform to differential equation.
I do like this :

transformRule = {r[x][y]-> Cos[x]Sinh[y], z[x][y]-> Sin[x] Cosh[y]}
dtrules=Join @@ ({#,D[#,t],D[#,{t,2}]} & /@ transformRule)

But it seems that something is wrong...

=========

And the analytical result is :

anaSolution = 1 / (Sin^2[x] + Sinh^2[y] ) * ( Derivative[2, 0][u][x,y]
+  -Tan[x]*Derivative[1, 0][u][x, y]  +  Derivative[0, 2][u][x, y] +
Tanh[y]*Derivative[0, 1][u][x, y] )


==========

Can someone do this ? It seems simple but infact, when i touch it, it
made the complex result. So what i want is how to obtain the
anaSolution.

Waiting for ur help...



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