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For the problems indicated in the title, a further development of a newapproach (different from those applied before) is given. A basic problem underconsideration arises in viscous incompressible fluid dynamics and describesself-similar solutions to the boundary layer equation for a stream functionwith zero pressure gradient (connected with the plane-parallel laminar flow ina mixing layer). Some previous results concerning singular nonlinear Cauchyproblems, smooth stable initial manifolds, and parametric exponential Lyapunovseries are used to state correctly and analyze the singular"initial-boundary-value" problem for a third-order nonlinear ordinarydifferential equation defined on the entire real axis. The detailed analysis ofthis singular nonlinear problem leads, in particular, to efficient methods forsolving it approximately and gives a possibility to obtain numerically theparticle trajectories in the plane of flow. Some results of the numericalexperiments are displayed and their physical interpretation is discussed. Aconnection of this basic problem with some known physical and mathematicalproblems, arising for self-similar solutions to the boundary layer equationswith zero pressure gradient, is described, namely the "flooded jet", the plane"semi-jet" and the "near-wall jet" problems are considered which are ofinterest by themselves.

Correct Statement, Analysis and Numerical Solution of Singular Nonlinear  Problems for Self-Similar Solutions to the Boundary Layer Equations with Zero  Pressure Gradient