Kuba's MoreCalculus package - an example

Question

First of all, congratulations to @Kuba for this excellent package.

I did not know where to report the following example, so I thought I would make a post about it. I understand that this might be one of the coordinate transformations that are causing issues to Solve, however, this coordinate tranformation appears a lot and I think it might be useful if this can be resolved somehow.

The mathematical description of the problem is given below and is taken directly from this paper - see eq.(5.14) on page 31 and also eq.(A.2) on page 36.

The equation is

\begin{equation} \frac{1}{\sigma^2} \partial_{\sigma} (\sigma^2 \partial_{\sigma}V) + \partial^2_{\eta} V = 0 \end{equation}

The change of variables $(\eta, \sigma) \leftrightarrow (\rho, w)$ is given by

\begin{equation} \begin{aligned} \sigma &= \rho \cos w \, ,\\ \eta &= \rho \sin w \, . \end{aligned} \end{equation}

The result should be given by

\begin{equation} \frac{1}{\rho^2} (2 \cot w \partial_w V + \partial^2_w V + 3 \rho \partial_{\rho} V) + \partial^2_{\rho} V = 0 \end{equation}

The command that I used is shown below

DChange[D[σ^2*D[V[σ, η], σ], σ] + D[V[σ, η], {η, 2}] == 0, 
  {ρ == η^2/Sqrt[η^2 + σ^2] + σ^2/Sqrt[η^2 + σ^2], 
   w == ArcTan[σ/Sqrt[η^2 + σ^2], η/Sqrt[η^2 + σ^2]]}, {σ, η}, {ρ, w}, 
  {V[σ, η]}]

I tried also to input the change of variables in the way that is stated above; that is

DChange[D[σ^2*D[V[σ, η], σ], σ] + D[V[σ, η], {η, 2}] == 0, 
  {σ == ρ*Cos[w], η == ρ*Sin[w]}, {σ, η}, {ρ, w}, {V[σ, η]}]

None of the above produces the desired result.

Answer 1

As noted by Kuba, there is a typo in the PDE. Then, to obtain 2nd PDE in the question the needed rule is

\begin{equation} \begin{aligned} \color{red}{\eta} &= \rho \cos w \,\\ \color{red}{\sigma} &= \rho \sin w \,\\ \end{aligned} \end{equation}

Finally, your transformation rule is improper. Try the following:

Assuming[{ρ > 0, -Pi < w < Pi}, 
 DChange[1/σ^2 D[σ^2 D[V[σ, η], σ], σ] + D[V[σ, η], {η, 2}] == 0, 
   "Cartesian" -> "Polar", {η, σ}, {ρ, w}, {V[σ, η]}]]

Or the following:

Assuming[{ρ > 0, -π < w < π}, 
 DChange[D[σ^2 D[V[σ, η], σ], σ]/σ^2 + D[V[σ, η], {η, 2}] == 0, 
         {Sqrt[η^2 + σ^2] == ρ, w == ArcTan[η, σ]}, {η, σ}, {ρ, w}, {V[σ, η]}]]

Answer 2

Thanks and sorry but I won't have time to update the package soon nor I were doing any calculus for couple of years so here are just my quick notes (which do not solve everything):

You forgot about 1/σ^2 in your equation.

The question is what to assume to make:

Solve[{σ == ρ*Cos[w], η == ρ*Sin[w]}, {ρ,  w}]

to return only the second result. I tried ρ > 0 but that makes it stuck.

Regardless,

DChange[
  1/σ^2 D[σ^2*D[V[σ, η], σ], σ] + D[V[σ, η], {η, 2}] == 0,
  {σ==ρ*Cos[w],η==ρ*Sin[w]},
  {σ,η},{ρ,w},{V[σ,η]}
] //
  ReplaceAll[C[1]->0] //
  Refine[#,{ρ>0}]&    // (*Get rid of Sqrt[ρ^2] ?justified*)
  ReplaceAll[ArcTan[-Cos[w],-Sin[w]]->w] // (*maybe can be automatic with smart Assumptions*)
  Map[-#/ρ &] // Expand // (* tidy up*)
  TraditionalForm

Closer but not yet there