Re: Transform differential equation by tranformation rule

*To*: mathgroup at smc.vnet.net*Subject*: [mg111101] Re: Transform differential equation by tranformation rule*From*: Maxim <m.r at inbox.ru>*Date*: Tue, 20 Jul 2010 03:44:14 -0400 (EDT)*References*: <i1s6t2$a0a$1@smc.vnet.net>

On Jul 17, 7:15 am, TrinhDao <hackerdarkr... at yahoo.com> wrote: > 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... Here's a kind of a brute force way to do the change of variables: cov = Solve[ D[U[x, y], ##] == D[u[r, z], ##, NonConstants -> {r, z}]& @@@ {{}, {x}, {y}, {x, x}, {x, y}, {y, y}}, D[u[r, z], ##] & @@@ {{}, {r}, {z}, {r, r}, {r, z}, {z, z}}]; Derivative[0, 2][u][r, z] + Derivative[1, 0][u][r, z]/r + Derivative[2, 0][u][r, z] /. cov[[1]] /. {r -> Cos[x] Sinh[y], z -> Sin[x] Cosh[y]} /. (NonConstants -> _) -> Sequence[] // Simplify Out[2]= 2 (Coth[y] Derivative[0, 1][U][x, y] + Derivative[0, 2][U][x, y] - Tan[x] Derivative[1, 0][U][x, y] + Derivative[2, 0][U][x, y])/ (Cos[2 x] + Cosh[2 y]) It differs from your transformed equation in the coefficient before Uy (Coth instead of Tanh). You might have mixed up Sinh and Cosh in your replacement rules for r and z. Maxim Rytin m.r at inbox.ru