This paper is devoted to an overview of the basic properties of the Padé transformation and its generalizations. The merits and limitations of the described approaches are discussed. Particular attention is paid to the application of Padé approximants in the mechanics of liquids and gases. One of the disadvantages of asymptotic methods is that the standard ansatz in the form of a power series in some parameter usually does not reflect the symmetry of the original problem. The search for asymptotic ansatzes that adequately take into account this symmetry has become one of the most important problems of asymptotic analysis. The most developed technique from this point of view is the Padé approximation.

In this chapter, we describe the applications of asymptotic methods to the problems of mathematical physics and mechanics, primarily, to the solution of nonlinear singular perturbed problems. We also discuss the applications of Padé approximations for the transformation of asymptotic expansions to rational or quasi-fractional functions. The applications of the method of matching of internal and external asymptotics in the problem of boundary layer of viscous gas by means of Padé approximation are considered.

The urgency of present problem is caused by necessity of operative control of biological sewage treatment processes. At present these processes are realizing in aeration tanks of sewage treatment plants both municipal and of large enterprises. A process of biological treatment is being checked by chemical analysis practically by hand. By the way a duration of checking procedure is commensurable with the time the sewage water is passing through an aeration tank. Thus any operative control of sewage treatment is practically impossible. In the present work it is proposed to use for operative control the hardware- software complex (HSC) AQUATEST-AT4M, that is able to produce an analysis of water quality in some minutes. By the way the measurement data give an opportunity to identify a mathematical model of these biological sewage treatment. In this work there are presented the results of experiments and of mathematical modelling of work of the Kirov City communal systems sewage treatment plants (KCS STP).