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I am working with System Dynamics Modeling for Sustainability. However, I would like to know details of the mathematical formulation of the functions of Vensim Delay Fixed and Delay material. Because t is not clear in the manual Vensim and I need to know what is the programming of these functions, to build the differential equations systems and try to solve in the analytical way

Maybe the following will help.
http://www.public.asu.edu/~kirkwood/sys ... o/ch-6.pdf

These are pure pipeline delays, so

y = DELAY FIXED( x, duration, x )

is essentially the same as

y = x(time-duration)

DELAY MATERIAL is the same, except for special treatment to conserve material if the duration increases.