  The self-similar momentum ordinary differential equation (MODE) and the
self-similar partial differential equation (MPDE) have been derived and the
investigation of the integrability of the MODE and the MPDE has been done by
performing Painlev\'e test. A detailed discussion of the leading order behavior
of the MODE and the MPDE has been presented with the latter being analyzed for
the cases in which terms of increasing orders of Reynolds number have been
considered. We have provided a brief introduction to Lie point symmetries and
have found the Lie infinitesimal operator which when acts on the MPDE to order
$\mathcal{O}(R)$ satisfies the Lie symmetry condition. Explicit calculations
and expressions for the Lie prolongation terms have been presented. We have
also investigated the integrability of various self-similar equations that
arise from the generalized self-similar equation for different values of
constants $\alpha_{1,2,3}$. Foundational work on transitional boundary
solutions has been presented and transition solutions have been found via
application of a junction condition at the leading edge-trailing edge boundary
domain. A detailed discussion of semi-analytical solutions via the homotopy
perturbation method is presented. We find semi-analytical solutions to the
Falkner-Skan equation and the MODE by considering a Taylor series expansion as
the initial approximation. An algorithmic scheme that involves consideration of
a multi-dimensional Taylor expansion as the initial approximation to the MPDE
has been presented.
