Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 321 / 2 / c / ii / Solution

For $m=4,n=2$, the equation is the one-dimensional porous medium equation $S_{\tau }=(S^3{)}_{xx}$. Use a similarity solution $S={\tau }^{-1/4}f(\xi )$ and $\xi =x{\tau }^{-1/4}$. ThenThe no-mass-flux condition at $x=0$ sets the integration constant to zero:With $f(0)=1$,ConsequentlyIts edge is $R=12r_0{\tau }^{1/2}\propto t^{1/2}$. Moreover,is time-independent. This agrees with part (b), since $2-m+2n=2$.