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In this book, analytical solutions are constructed for two governing nonlinear boundary-value problems using perturbation techniques. We are concerned by equilibrium configurations of fluid drops or bubbles in a uniform flow. The computation of their shapes is reduced to the solution of a nonlinear equation involving a small parameter and a characteristic quantity of the state. As an example the Plateau problem is considered. It is treated as the determination of the shape of an axi-symmetric fluid drop rotating in weightlessness with constant angular velocity under the action of surface tension. The other similar example is the shape of a bubble or a liquid drop in a constant-velocity uniform flow in weightlessness. A subclass of Hermite-Padé approximants are applied to locate the singularities limiting the convergence of the series solutions and important properties of the shape functions including bifurcations are discussed. The proposed methodology is shown to work adequatelyfor solving single or multiple solutions of some nonlinear boundary value problems.