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I want to smooth a unit step function for use in NDSolve so this function is a smooth function of time (t). How can I do the smoothing? The unit step function is defined by:

α = π/6;
Vbusd[t_] := 5 (1 - Exp[-t]);
Vbusq[t_] := 3 Exp[-t];

n = UnitStep[Vbusd[t]*Cos[t - α - π/6] - Vbusq[t]*Sin[t - α - π/6]]  -
    UnitStep[Vbusd[t]*Cos[t - α - (5 π)/6] - Vbusq[t]*Sin[t - α - (5 π)/6]];
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  • $\begingroup$ After a second look I think your question is a little unclear. What do you really want? Smoothen n with NDSolve or find a smooth approximation of n which is to be used in NDSolve? $\endgroup$
    – xzczd ♦
    Commented Jun 21, 2015 at 9:17
  • $\begingroup$ Thanks alot for help. finding a smooth approximation of n which is to be used in NDSolve. $\endgroup$
    – mard
    Commented Jun 21, 2015 at 9:21
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    $\begingroup$ I normally use Tanh $\endgroup$
    – yohbs
    Commented Jun 21, 2015 at 10:48

1 Answer 1

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These are the transitions I usually use: (both are $C^{\infty}$)

f[x_] = Piecewise[{{(Erf[Sqrt[2 π] ArcTanh[x]] + 1)/2, -1 < x < 1}}, UnitStep[x]]
g[x_] = Piecewise[{{(Tanh[Sqrt[2] Tan[π/2 x]] + 1)/2, -1 < x < 1}}, UnitStep[x]]

Plot[{UnitStep[x], f[x], g[x]}, {x, -2, 2}]

Picture

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    $\begingroup$ Could you please post text (code-styled markdown) that can be cut and pasted rather than images of your code? This makes it easy for people to grab your solution and experiment with it. $\endgroup$
    – dionys
    Commented Jun 21, 2015 at 12:45
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    $\begingroup$ But I do not know time of rising and falling levels for difinition of unitstep and Tanh ! $\endgroup$
    – mard
    Commented Jun 21, 2015 at 13:24

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