Is my change of variables of a PDE correct?

I've calculated a lot and checked the first derivatives with wolframalpha. Still I'm not sure if I have done everything correctly, could someone have a look please?

Original PDE: \begin{align*} u_t- \frac{\lambda}{2}u_{zz}=0 \end{align*} New coordinate: \begin{align*} y_i=\frac{1}{\pi} \arctan(\gamma z+c)+\frac{1}{2} \Leftrightarrow z=\frac{\tan(\frac{1}{2}(2 \pi y- \pi))-c}{\gamma} \end{align*} First derivatives: \begin{align*} y_z= \frac{\gamma}{\pi((c+\gamma z)^2+1)}\\ z_y=-\frac{2 \pi}{\gamma (\cos(2 \pi y)-1)} \end{align*} u differentiated using chaining rule, derivation of differential operators: \begin{align*} u_z=u_y y_z=u_y \frac{\gamma}{\pi((c+\gamma z)^2+1)}\\ \Rightarrow \frac{\partial}{\partial z}= \frac{\partial}{\partial y}\frac{\gamma}{\pi((c+\gamma z)^2+1)} \end{align*} \begin{align*} u_y= u_z z_y = u_z (-\frac{2 \pi}{\gamma (\cos(2 \pi y)-1)})\\ \Rightarrow \frac{\partial}{\partial y} = \frac{\partial}{\partial z} (-\frac{2 \pi}{\gamma (\cos(2 \pi y)-1)}) \end{align*} To ease notiation \begin{align*} A:= y_z= \frac{\gamma}{\pi((c+\gamma z)^2+1)}. \end{align*} 2nd derivative using product rule: \begin{align*} u_{zz} &= \frac{\partial}{\partial z} u_z = \frac{\partial}{\partial z} (u_y \frac{\gamma_i}{\pi((c+\gamma z)^2+1)})\\ &=A(\frac{\partial}{\partial y}(u_y A)) =A(u_{yy} A+u_yA_y)=A(u_{yy}A+u_y\cdot 0)=A^2 u_{yy} \end{align*} Inserting into original PDE: \begin{align*} u_t- \frac{\lambda}{2}A^2 u_{yy}=0 \Leftrightarrow u_t - \frac{\lambda}{2} (\frac{\gamma}{\pi((c+\gamma z)^2+1)})^2 u_{yy}=0 \end{align*} Now $z=\frac{\tan(\frac{1}{2}(2 \pi y- \pi))-c}{\gamma}$ should be inserted into the equation.

• I've at least displayed the image on the same page, but if you wrote it in $\LaTeX$ you'll find you can easily copy the code here, between $...$ or $$...$$ for displayed formulas. This isn't foolproof since MathJax isn't $\LaTeX$, but what you've written looks compatible. On a different matter, I'm curious as to why you would use this change of variables? – GPerez Jun 5 '15 at 15:01
• Okay now another question, what are the sub-indices $i$ for? Do you have more than just $(t,z)$ as variables? – GPerez Jun 5 '15 at 15:40