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- #1

$\begin{aligned}

{u_t} &= K{u_{xx}},{\text{ }}0 < x < L,{\text{ }}t > 0, \\

u(0,t) &= 0,{\text{ }}u(L,t) = 0,{\text{ for }}t > 0, \\

u(x,0) &= 6\sin \frac{{3\pi x}}{L}.

\end{aligned} $

2) Solve

$\begin{aligned}

{u_t} &= 4{u_{xx}},{\text{ }}0 < x < 1,{\text{ }}t > 0, \\

u(0,t) &= 0,{\text{ }}u(1,t) = 0,{\text{ for }}t > 0, \\

u(x,0) &= x^2(1-x),\text{ for }x\in[0,1].

\end{aligned}$

Is there a standard procedure to solve this?