Problem $2(25$ pts$).$ During a thermal treatment, a steel sphere of radius $R$ is initially

heated to the temperature $T_0.$ The sphere is then quenched in water at temperature $T_{w}T(t)\frac{d}{dt}([$ internal $],[$ energy $],[of$ the sphere $])=([$ heat $],[$ escaping $],[$ the sphere $])q=-h(T-T_w)hcT(t)\frac{dT}{dt}=f(T,t)T_w.$

The amount $of$ water $is$ sufficient $to$ assume that its temperature $T_w$ remains constant during

the quenching process.

$(a)[5pts]$ The temperature $T(t)of$ the sphere during the quenching process $is$ governed $by$

the energy balance equation

$\frac{d}{dt}([$ internal $],[$ energy $],[of$ the sphere $])=([$ heat $],[$ escaping $],[$ the sphere $])$

Assume that the heat escaping a unit area $of$ the sphere follows the law $of$ convection,

that $isq=-h(T-T_w),$ where $his$ the constant heat transfer coefficient. Letting $cbe$

the constant heat capacity $of$ the sphere, derive $an$ ODE for $T(t)in$ the form

$\frac{dT}{dt}=f(T,t)$

$(b)[7pts]$ Find the analytical solution $of$ the ODE found $in(a).$

$(c)[7pts]$ Write pseudo$-$code that implements the ODE found $in(a)$ using the following

methods. For each method specify $ifitis$ explici$\frac{t}{i}$mplicit and $singl\frac{e}{m}$ulti$-$step.

explicit Euler method

implicit Euler method

$AM1$

$BDF2$

$(d)[3pts]$ Explain the difference between implicit and explicit methods. What are advan$-$

tages and disadvantages $of$ the two types $of$ methods?

$(e)[3pts]$ Explain the difference between single$-$step and multi$-$step integration methods.

List examples $of$ explicit and implicit multistep methods.