Consider the Boundary$-$Initial Value problem

$u(x,t)u(x,t)=\frac{8}{{}^3}{\displaystyle \underset{n=1}{\overset{}{}}}{b}_{n}sin((2n-1)\frac{}{2}x)e^{-c_{n}t}{b}_n=\frac{24}{n^3{}^3}(1-(-1{)}^n)c_n=\frac{3}{2^2}{n}^2{}^{2}3\frac{de{l}^{2}u}{del{x}^2}=\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{t}},00$

$u(0,t)=0,u(2,t)=0,t>0$

$u(x,0)=x(2-x),0$

This models a heated wire, with zero endpoints temperatures. The solution $u(x,t)of$ the initial$-$boundary value problem $is$ given $by$ the series

$u(x,t)=\frac{8}{{}^3}{\displaystyle \underset{n=1}{\overset{}{}}}{b}_{n}sin((2n-1)\frac{}{2}x)e^{-c_{n}t}$

where

$b_n=\frac{24}{n^3{}^3}(1-(-1{)}^n)$

and

$c_n=\frac{3}{2^2}{n}^2{}^2$