Saturation Curves

Erkki Hartikainen November 27, 2016

The hyperbolic saturation



uusi-hyperbeli

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:

gompertz

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:

logistic

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:

genralized-logistic

y = 1 - exp(-(exp(2x-4))

(1 is an asymptote, 2x-4 is a translation to the correct place.)