## Saturation CurvesErkki Hartikainen November 27, 2016## The hyperbolic saturationImplicit 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 saturationThe differential equation is:(y''/y') - (y'/y) = 1. Solution of the differential aquation: y = c _{2} exp(c_{1} exp(x)), where exp(x) = e ^{x} .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 saturationThe 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.) |