Infinitesimal generator

Similarity variables

Reduction equation

$X_1$

$\varsigma =t,$

$u\left(x,t\right)=F\left(\varsigma \right),v\left(x,t\right)=H\left(\varsigma \right).$

$F^{″}=0,$

$H^{″}=0.$

$X_2$

$\varsigma =x,$

$u\left(x,t\right)=F\left(\varsigma \right),v\left(x,t\right)=H\left(\varsigma \right).$

$F{F}^{\prime }+H^{\prime }=0,$

$F^{\prime }+H{F}^{\prime }+F{H}^{\prime }+F^{\left(3\right)}=0.$ (D)

$X_3$

$\varsigma =t,$

$u\left(x,t\right)=x/t+F\left(\varsigma \right),v\left(x,t\right)=H\left(\varsigma \right).$

$F+\varsigma F^{\prime }=0,$

$H+\varsigma H^{\prime }+1=0.$

$X_4$

$\varsigma =x{t}^{-1/2},$

$u\left(x,t\right)=t^{-1/2}F\left(\varsigma \right),v\left(x,t\right)=-1+t^{-1}H\left(\varsigma \right).$

$F+\varsigma F^{\prime }-2F{F}^{\prime }-2H^{\prime }=0,$

$2H-2H{F}^{\prime }+\varsigma H^{\prime }-2F{H}^{\prime }-2F^{\left(3\right)}=0.$ (A)

$X_1+X_2$

$\varsigma =x-t,$

$u\left(x,t\right)=F\left(\varsigma \right),v\left(x,t\right)=H\left(\varsigma \right).$

$-F^{\prime }+F{F}^{\prime }+H^{\prime }=0,$

$F^{\prime }+H{F}^{\prime }-H^{\prime }+F{H}^{\prime }+F^{\left(3\right)}=0.$ (B)

$X_2+X_3$

$\varsigma =-t^2+2x/2,$

$u\left(x,t\right)=t+F\left(\varsigma \right),v\left(x,t\right)=H\left(\varsigma \right).$

$1+F{F}^{\prime }+H^{\prime }=0,$

$F^{\prime }+H{F}^{\prime }+F{H}^{\prime }+F^{\left(3\right)}=0.$ (C)