Smoothing a single rectangular pulse in NDSolve

Question

Hello I am interested in smoothing out the sharp edges of a single rectangular wave centered at the origin.

I am looking for a smooth function that looks almost exactly the same as the square wave except it is differentiable everywhere.

I looked at Fourier Series but this creates a periodic function which I do not want. I want to create a rectangular wave of finite height and zero everywhere else.

Ultimately, I want to plug this approximating function into NDSolve in attempt to get rid of "spiky" effects present in the solution of the wave equation.

Thank You!

Edit::

This is the square pulse in question:

Answer 1

For NDSolve, an InterpolatingFunction might be sufficient. Here are smooth versions of UnitStep and UnitBox, with a smoothing "radius" of epsilon.

us[epsilon_] := 
  Interpolation[{{{-epsilon}, 0, 0}, {{epsilon}, 1, 0}}, 
   "ExtrapolationHandler" -> {If[# < -epsilon, 0, 1] &, "WarningMessage" -> False}];
ub[epsilon_] := 
 Interpolation[   
  {{{-1/2 - epsilon}, 0, 0}, {{-1/2 + epsilon}, 1, 0},
   {{1/2 - epsilon}, 1, 0}, {{1/2 + epsilon}, 0, 0}}, 
  "ExtrapolationHandler" -> {0 &, "WarningMessage" -> False}]

Example

sol = NDSolve[{y''[x] + ub[0.1][(x - 6)/6.5] y[x] == 0, y[0] == 1, y'[0] == 0}, y, {x, 0, 15}];
Plot[y[x] /. First[sol], {x, 0, 15}]

Response to updated question

You could construct an explicit interpolation for your particular use case instead of using the generic functions us and ub. For example, for the updated question, one could replace the ± 1/2 by ± 8.5 in ub, or by arbitrary cutoffs a, b, if a general function is needed, as follows.

smoothbox[a_, b_, epsilon_] /; a < b && epsilon > 0 := Interpolation[{
    {{a - epsilon}, 0, 0}, {{a + epsilon}, 1, 0},
    {{b - epsilon}, 1, 0}, {{b + epsilon}, 0, 0}},
  "ExtrapolationHandler" -> {0 &, "WarningMessage" -> False}]

GraphicsRow@{
  Plot[smoothbox[-8.5, 8.5, 2.5][x], {x, -15, 15}],
  Plot[smoothbox[-8.5, 8.5, 2.5][x], {x, 5, 12}]}