Let the heat equation

$u(0,t)=u(L,t)=0.q(x,t)xtu{u}_{a}x{u}_{ex}{u}_{t}u(x,t)={\displaystyle \underset{n=1}{\overset{}{}}}{b}_n(t)sin\frac{n}{L}x.u_t,u_{az}q(x,t)b_n(t)b_n(t)b_n(t{)}^H{b}_n(t{)}^P{A}_n{b}_n(t)u(x,0)=f(x).u_t=u_{zz}+q(x,t),$

$0$

subject $to$ the boundary conditions

$u(0,t)=u(L,t)=0.$

Assuming that $q(x,t)is$ a piecewise smooth function $ofx$ for each positive $t.$ Further, assume that $u$ and $u_a$ are continuous functions $ofx$ and $u_{ex}$ and $u_t$ are piecewise smooth. Then

$u(x,t)={\displaystyle \underset{n=1}{\overset{}{}}}{b}_n(t)sin\frac{n}{L}x.$

$(i)$ Using $u_t,u_{az}$ and $q(x,t),$ obtain the ordinary differential equations satisfied $by{b}_n(t).$

$(ii)$ Solve for $b_n(t)-$ the homogenous and particular solutions, $b_n(t{)}^H$ and $b_n(t{)}^P.$

$(iii)$ Apply initial conditions, $to$ find coefficient $A_{n}in$ the $b_n(t)$ solution assuming

$u(x,0)=f(x).$

$[TURN$ OVER$]$