$$
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$$
Terminology: natural and essential boundary conditions
$$ (u',v') + (bu',v) = (f,v) + u'(L)v(L) - u'(0)v(0)$$
- Note: forgetting the boundary terms implies \( u'(L)=u'(0)=0 \)
(unless prescribe a Dirichlet condition)
- Conditions on \( u' \) are simply inserted in the variational form
and called natural conditions
- Conditions on \( u \) at \( x=0 \) requires modifying \( V \) (through \( \baspsi_i(0)=0 \))
and are known as essential conditions
Lesson learned.
It is easy to forget the boundary term when integrating by parts.
That mistake may prescribe a condition on \( u' \)!