# Forcing Interpolation to be smooth at its boundaries

I'm looking at a piecewise function of two lines, which has a kink, such as:

f1[p_] := 1.41 * p + 2*^8;
f2[p_] := 1.45 * p + 1.9584*^8;
f3[p_] := Piecewise[{{f1[p], p <= 1.04*^8}, {f2[p], p > 1.04*^8}}];


I want to work with a function which is linear on both sides of the kink like above, but is smooth in a small region around the kink as an approximation to the above function.

I'd like to use Interpolation[] to find a smooth function to approximate a region of width delta p around the kink, p = 1.04 * 10^8.

However Interpolation can give me a function which will just introduce more kinks out the boundaries with f1 and f2. Is there a way I can enforce boundary conditions on Interpolation[] so that it can give me a smooth function in this region?

Edit to clarify: The boundary I'm considering is arbitrary; it is just centered around 1.04*10^8. For example, I tried using a region of width .01*10^8 centered around this value, so my code was:

region = Interpolation[{{1.03*^8, f1[1.03*^8]}, {1.035*^8, f1[1.035*^8]}, {1.045*^8, f2[1.045*^8]}, {1.05*^8, f2[1.05*^8]}}];


But if you plot this, it will not be smooth with the rest of the function (i.e., it doesn't have a slope of 1.41 at p=1.03*^8, and 1.45 at p=1.05*^8. I want to enforce those slopes at those values, or any other arbitrary value).

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Could you add the code for Interpolation that you have tried? What are the boundaries of p? –  Jens Aug 2 '13 at 19:10
Edited my question above to address this –  psachdeva Aug 2 '13 at 19:17

Since your goal is to enforce the derivatives, a better way to use Interpolation here is to apply it to the derivative of your data, as follows:

points = {1.03, 1.035, 1.045, 1.05} 10^8;

derivs = Map[f3', points]

(* ==> {1.41, 1.41, 1.45, 1.45} *)

regionSlope = Interpolation[Transpose[{points, derivs}]];

Plot[
Evaluate[{D[f3[p], p], regionSlope[p]} /. p -> p1], {p1, 1.03 10^8,
1.05 10^8}]


The slopes are now matched at the boundaries. The original function can be interpolated by the integral of the InterpolatingFunction obtained above:

Clear[region]

region[t_?NumericQ] :=
f3[points[[1]]] + NIntegrate[regionSlope[p], {p, points[[1]], t}]

Plot[Evaluate[{f3[p], region[p]}], {p, 1.03 10^8, 1.05 10^8}]


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