  The anisotropic 3D equation describing the pointlike particles diffusion in
slender impermeable tubes of revolution with cross section smoothly depending
on the longitudial coordinate is the object of our study. We use singular
perturbations approach to find the rigorous asymptotic expression for the local
particles concentration as an expansion in the ratio of the characteristic
transversal and longitudial diffusion relaxation times. The corresponding
leading-term approximation is a generalization of well-known Fick-Jacobs
approximation. This result allowed us to delineate the conditions on temporal
and spatial scales under which the Fick-Jacobs approximation is valid. A
striking analogy between solution of our problem and the method of inner-outer
expansions for low Knudsen numbers gas kinetic theory is established. With the
aid of this analogy we clarify the physical and mathematical meaning of the
obtained results.
