AssumeConditions and Parameters

$T=T(r,t)$

$T(r,0)=22C$

$T_{\text{m}\text{a}\text{x}}=160C($melting point of PMMA$)$

Absorptivity of PMMA $=91\%$

Radial thermal conductivity of PMMA $=0\mathrm{.}19W{m}^{-1}{K}^{-1}$

Governing Equations

$(1)q=lon{A}_s(T_s^4-T_{\text{s}\text{u}\text{r}\text{r}}^4)$

$(2)q=hAT$

$(3)\frac{T}{t}=\frac{1}{r^2}\frac{}{r}(r^2\frac{T}{r})$

$(4)|\frac{dT}{dt}=**T$

create a temperature profile and determine the time it would take for the system to reach melting temperature. Assume one$-$dimensional radiation is the only source of heat. The radius of the laser is $\mathrm{.}01$m$.$ Use matlab.