Functional form

Transformed form

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

$W_k\left(X\right)=U_k\left(X\right)V_k\left(X\right)$

$w\left(X,t\right)=\alpha \dot{u}\left(X,t\right)$

$W_k\left(X\right)=\alpha {\dot{U}}_k\left(X\right)$ , $\alpha$ is constant

$w\left(X,t\right)=u\left(X,t\right)\dot{v}\left(X,t\right)$

$W_k\left(X\right)={\displaystyle {\sum }_{i=0}^k}{U}_i\left(X,t\right){\dot{V}}_{k-i}\left(X,t\right)$

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

$W_k\left(X\right)=\left(k+1\right)\cdots \left(k+r\right){\dot{U}}_k\left(X\right)=\frac{\left(k+r\right)!}{k!}{\dot{U}}_k\left(X\right)$

$w\left(X,t\right)=\frac{{\partial }^{r_1+r_2+\cdots +r_n}}{\partial x_1^{r_1}\partial x_2^{r_2}\cdots \partial x_n^{r_n}}u\left(X,t\right)$

$W_k\left(x\right)=\frac{{\partial }^{r_1+r_2+\cdots +r_n}}{\partial x_1^{r_1}\partial x_2^{r_2}\cdots \partial x_n^{r_n}}{U}_k\left(X\right)$