Comparison of the two slab solutions for short times. What error is made by using Eq$.11\mathrm{.}5-10($based on semi$-$infinite slab$)$ instead of Eq$.11\mathrm{.}5-38($based on the slab of finite thickness$),$ when at$/$b $0\mathrm{.}1$ and for a position $0\mathrm{.}9$ of the way from the midplane to the slab surface? Using the graphically presented

$\frac{T_1-T}{T_1-T_0}=2{\displaystyle \underset{n=0}{\overset{}{}}}\frac{(-1{)}^n}{(n+\frac{1}{2})}$exp$[-(n+\frac{1}{2}{)}^2{}^2\frac{t}{b^2}]cos(n+\frac{1}{2})\frac{y}{b}$

Fig. $12\mathrm{.}1-1.$ Temperature profiles for unsteady$-$state heat conduction in a slab of finite thickness $2b.$ The initial temperature of the slab is $T_0,$ and $T_1$ is the temperature imposed at the slab surfaces for time $t>0.[$H$.$ S$.$ Carslaw and J$.$ C$.$ Jaeger, Conduction of Heat in Solids, $2$nd edition, Oxford University Press $(1959),$ p$.101.]$

Fig. $12\mathrm{.}1-2.$ Temperature profiles for unsteady$-$state heat conduction in a cylinder of radius $R.$ The initial temperature of the cylinder is $T_0,$ and $T_1$ is the temperature imposed at the cylinder surface for time $t>0.[$H$.$ S$.$ Carslaw and J$.$ C$.$ Jaeger, Conduction of Heat in Solids, $2$nd edition, Oxford University Press $(1959),$ p$.200.]$

Fig. $12\mathrm{.}1-3.$ Temperature profiles for unsteady$-$state heat conduction in a sphere of radius $R.$ The initial temperature of the sphere is $T_0,$ and $T_1$ is the temperature imposed at the sphere surface for time $t>0.[$H$.$ S$.$ Carslaw and J$.$ C$.$ Jaeger, Conduction of Heat in Solids, $2$nd edition, Oxford University Press $(1959),$ p$.234.]$solution $($Figure $11\mathrm{.}5-1)$ for making the comparison.

or

$\frac{T-T_0}{T_1-T_0}=1-$erf$(\frac{y}{\sqrt[\phantom{2}]{4t}})$