Saturation Curves
Erkki Hartikainen November 27, 2016The hyperbolic saturation
Implicit equation:
(v-c)(v-at)=δ,
v = velocity of the material body.
c = double-way velocity of the light.
t = time.
a = constant.
δ = constant.
The meaning of the last two constants is not my problem.
The last constant is empirical.
Next you must solve the differential equation (to get for example the acceleration)
(y'(t) - c)(y'(t)-at)-d=0.
The Gompertz saturation
The differential equation is:(y''/y') - (y'/y) = 1.
Solution of the differential aquation:
y = c2 exp(c1 exp(x)),
where
exp(x) = ex .
Example:
y = exp(-exp(2-x)).
(2-x is a transalation to the right place.)
The logistic saturation
The
differential equation is:
y'= y(1-y).
The solution is:
y(x) = -exp(x)/(c + exp(x)).
Example:
y = e(2x-5) / (1 + e(2x-5) .
(2x-1 is the translation to the correct place.)
Generalisted logistic saturation
The differential equation is:y'=r y ln(y/k).
The solution is:
y = a exp(b exp(-cx).
For example:
y = 1 - exp(-(exp(2x-4))
(1 is an asymptote, 2x-4 is a translation to the correct place.)