Functional Form

Transformed Form

$u\left(x,y,t\right)$

$U_k\left(x,y\right)=\frac{1}{k!}{\left[\frac{{\partial }^k}{\partial t^k}u\left(x,y,t\right)\right]}_{t=0}$

$w\left(x,y,t\right)=au\left(x,y,t\right)\pm bv\left(x,y,t\right)$

$W_k\left(x,y\right)=a{U}_k\left(x,y\right)\pm b{V}_k\left(x,y\right)$

$w\left(x,y,t\right)=u\left(x,y,t\right)v\left(x,y,t\right)$

$w\left(x,y,t\right)=\frac{{\partial }^s}{\partial t^s}u\left(x,y,t\right)$

$W_k\left(x,y\right)=\frac{\left(k+s\right)!}{k!}{U}_{k+s}\left(x,y\right)$

$w\left(x,y,t\right)=\frac{{\partial }^2}{\partial y^2}u\left(x,y,t\right)$

$W_k\left(x,y\right)=\frac{{\partial }^2}{\partial y^2}{U}_k\left(x,y\right)$

$w\left(x,y,t\right)=\frac{{\partial }^2}{\partial x^2}u\left(x,y,t\right)$

$W_k\left(x,y\right)=\frac{{\partial }^2}{\partial x^2}{U}_k\left(x,y\right)$