How would you go about determining the actual values of the derivatives (i.e. $u_0, u_1$)?
If we have the first segment, by "natural" assumption, u0 is zero. Similarly if we have the last segment then u1 is zero. If the segment is in the middle then we want the derivative and curvature to agree with its predecessor and successor at each end points.
I'll echo the above question from simply_compicated. If we want the derivative and second derivative to match with the predecessor and succeessor, how do we determine what those values are?