91Ó°ÊÓ

In Brercises \(1-8\), show that the given set of functions is orthogonal with respect to the given weight on the prescribed interval. \(1,1-x,\left(2-4 x+x^{2}\right) / 2 ; w(x)=e^{-x}\) on \((0, \infty)\), (These are examples of Laguerre) polynomials.)

Project Problem: Heat problem on a disk with Robin conditions. Use the method of separation of variable to solve the heat equation on a disk of unit radius $$ \frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}, \quad 0 \leq r<1, t>0 $$ with initial temperature distribution \(u(r, 0)=100(0 \leq r<1)\), and Robin boundary condition $$ \left.\frac{\partial u}{\partial r}(r, t)\right|_{r=1}=-u(1, t) $$ The problem models the temperature diatribution in a plate with insulated lateral surfice, whose boundary is exchanging beat with the surrounding medium at a rate proportional to the temperature at the boundary. Here the heat transfer constant or convection constant \(\kappa\) is equal to 1 .