Consider a semi$-$infinite medium with thermal diffusivity $$ at an initial temperature of

$T_i.$ For $t>0,$ a constant temperature, $T_0,$ is applied to the surface $(x=0).$

a$)$ Determine a $3$rd order polynomial approximation for the temperature profile $T(\frac{x}{}).$

b$)$ Determine an expression for the penetration depth, $(t),$ using the integral method of

weighted residuals and compare to the exact solution given in class $($Ex $7\mathrm{.}1$ part $b)).$

c$)$ Determine an expression for the heat transfer coefficient defined according to

$q_x(x=0)=h(T_0-T_i)$ and compare to the exact solution.