Combination of inner and external asymptotics for mathematical physics problems

In this work, we present in some detail new trends in application of asymptotic to mathematical physics and mechanics problems. First we consider the various methods which give a possibility to extend a space of application of perturbation series and hence to omit local character. In this work we also are going to emphasize the role of Pade-Approximants (PA), which seems to play more important role in today's non-linear mechanics and mathematical physics. The method of combination of an asymptotics by means of Two-Points Pade Approximants (TPPA) is alternative to the known Matching method and is applied in local area of the transition layer where an asymptotics is uneven. The method is tested on solutions of the known problems of mathematical physics and showed the effectiveness at the solution of original problems of fluid mechanics and gas dynamics.