Functional form Transformed form $w\left(X,t\right)=u\left(X,t\right)v\left(X,t\right)$ ${W}_{k}\left(X\right)={U}_{k}\left(X\right){V}_{k}\left(X\right)$ $w\left(X,t\right)=\alpha \stackrel{˙}{u}\left(X,t\right)$ ${W}_{k}\left(X\right)=\alpha {\stackrel{˙}{U}}_{k}\left(X\right)$ , $\alpha$ is constant $w\left(X,t\right)=u\left(X,t\right)\stackrel{˙}{v}\left(X,t\right)$ ${W}_{k}\left(X\right)={\sum }_{i=0}^{k}\text{ }{U}_{i}\left(X,t\right){\stackrel{˙}{V}}_{k-i}\left(X,t\right)$ $w\left(x,t\right)=\frac{{\partial }^{r}}{\partial {t}^{r}}u\left(X,t\right)$ ${W}_{k}\left(X\right)=\left(k+1\right)\cdots \left(k+r\right){\stackrel{˙}{U}}_{k}\left(X\right)=\frac{\left(k+r\right)!}{k!}{\stackrel{˙}{U}}_{k}\left(X\right)$ $w\left(X,t\right)=\frac{{\partial }^{{r}_{1}+{r}_{2}+\cdots +{r}_{n}}}{\partial {x}_{1}^{{r}_{1}}\partial {x}_{2}^{{r}_{2}}\cdots \partial {x}_{n}^{{r}_{n}}}u\left(X,t\right)$ ${W}_{k}\left(x\right)=\frac{{\partial }^{{r}_{1}+{r}_{2}+\cdots +{r}_{n}}}{\partial {x}_{1}^{{r}_{1}}\partial {x}_{2}^{{r}_{2}}\cdots \partial {x}_{n}^{{r}_{n}}}{U}_{k}\left(X\right)$