**1. Introduction**

An important drawback of asymptotic methods is the local character of solutions obtained [1–4]. Since the constructed series are often asymptotic, a simple increase in the number of terms does not remove this drawback. Essence of the problem consists of divergence of obtained series. There exist a lot of approaches to these problems [5, 6]. The method of analytic continuation (e.g., the Euler transform or generalized Euler transform [7–12]) requires a priori information about the singularities of the searched function in the complex domain [4, 9]. These methods are useful if a large number of terms of the series are known. In this case, it is possible to use the Domb-Sykes plot [5, 8]. But usually only a few terms of asymptotic series are known, and to get information from them, the method of Padé approximations (PAs) is useful [1, 2, 5, 13–15]. PAs yield meromorphic continuations of functions defined by power series and can be used even in cases where analytic continuations are inapplicable. If a PAs converges to the given function, then roots of the denominator tend to points of singularities. One-point PAs give possibilities to improve convergence of series [16–20]. Two-point PAs (TPPAs) allow matching asymptotics in transition zones and are widely used in mechanics and physics [1, 2, 4, 14, 21–24]. Overcoming the mentioned limitations of asymptotic methods for practically important problem is the purpose of this chapter. We consider at the beginning (Section 2) the mathematical bases of asymptotic methods and the use of Padé approximants for the summations of the asymptotic series. Section 3 discusses the method of combining of internal and external asymptotics (matching method) by means of Padé approximants. In the Section 4, the methods of solving specific problems of mathematical physics and mechanics of fluid and gas are demonstrated. Section 5 presents a discussion of the obtained.
