# Solving a ODE from a diffeomorphism numerically

This is a follow up from

Example of a diffeomorphism on $\mathbb{R}^{3}$onto itself (or cube onto itself)

And

Problem getting a diffeomorphism work on $\mathbb{R}^{3}$

In short: I am trying to get a diffeomorphism of the cube onto itself. The reason I want this is because I want a transformation of the cube that I can transform back, and I need to know the exact transformation, not an approximation. However, I'd be happy to have a numerical solution of a grid on the cube, as long as the solution on those points is exact.

I tried getting a analytic function, but the one I have chosen seems to have no ODE solution.

What I want is a solution to (or any other arbitrary diffeomorphism of the cube onto itself):

\begin{align*} V(x,y,z) = \bigl( &\sin(x \cdot \pi/L)\sin(y \cdot \pi/L)\sin(z \cdot \pi/L),\\ &\sin(x \cdot \pi/L)\sin(y \cdot \pi/L)\sin(z \cdot \pi/L),\\ &\sin(x \cdot \pi/L)\sin(y \cdot \pi/L)\sin(z \cdot \pi/L)\bigr), \end{align*} being $L$ the length of the cube.

$$\frac{d}{dt} h(x,y,z,t)= V(x,y,z).$$

I need to solve $h$ for $t$ and $-t$ (read the first question for more insight). Or in other words, I need to find the inverse transformation. If I am given a grid on the "deformed" space, I need to find those points in the original cube.

However I have no idea how to do this. I am using MATLAB, if that helps to find a numerical solution.

• The second reference gives us the message "This site can’t be reached". Oct 3, 2016 at 18:34
• @HandeBruijn fixed Oct 3, 2016 at 19:15