3 7.28, §3 Boundaries

• Get rid of the boundaries by imagining that the pipe extends from $-\infty < x <\infty$
• To do so, we must extend the initial conditions $f(x)$ and $g(x)$ to all $x$. Call the extended functions $\bar f(x)$ (or $F$) and $\bar g(x)$ (or $G$).
• The extended functions should make the given boundary conditions automatic.

To make the boundary condition $u_x=0$ at $x=0$ automatic, create symmetry around $x=0$ To make the boundary condition $u=0$ at $x=\ell$ automatic, create antisymmetry (odd symmetry) around $x=\ell$.

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Create the extended function $\bar g(x)$ or $G$ the same way.