Following Boyd [19], we begin by transforming the domain of the problem from
to
using the domain truncation method. This approximates
by the computational domain
where
is a fixed length that is taken to be larger than the thickness of the boundary layer. The interval
is then transformed to the domain
using the algebraic mapping
For convenience we make the boundary conditions homogeneous by applying the transformations
where
and
are chosen so as to satisfy boundary conditions (2.10). The chain rule gives
Substituting (3.2) and (3.3)-(3.4) in the governing equations gives
where prime denotes derivative with respect to
and
As initial guesses we employ the exponentially decaying functions used by Yang and Liao [12], namely,
The initial solution is obtained by solving the linear parts of (3.5), namely,
subject to
The system (3.8)-(3.9) is solved using the Chebyshev pseudospectral method where the unknown functions
and
are approximated as truncated series of Chebyshev polynomials of the form
where
and
are the
th Chebyshev polynomials with coefficients
and
, respectively,
are Gauss-Lobatto collocation points defined by
and
is the number of collocation points. Derivatives of the functions
and
at the collocation points are represented as
where
is the order of differentiation and
is the Chebyshev spectral differentiation matrix (see, e.g., [20, 21]). Substituting (3.10)–(3.12) in (3.8)-(3.9) yields
subject to the boundary conditions
where
The superscript
denotes the transpose,
is a diagonal matrix, and
is an identity matrix of size
. We implement boundary conditions (3.14) in rows 1,
, and
of
in columns 1 through to
by setting all entries in the remaining columns to be zero. The second set (3.15) is implemented in rows
and
, respectively, by setting
,
and setting all other columns to be zero. We further set entries of
in rows
,
,
,
, and
to zero.
The values of
are determined from the equation
which provides the initial approximation for the solution of (3.5).
We now seek the approximate solutions of (3.5) by first defining the following linear operators:
where
is the embedding parameter and
and
are unknown functions. The zero th-order deformation equations are given by
where
is the nonzero convergence controlling auxiliary parameter and
and
are nonlinear operators given by
The
th-order deformation equations are given by
subject to the boundary conditions
where
Applying the Chebyshev pseudospectral transformation to (3.21)–(3.23) gives
subject to the boundary conditions
where
and
are as defined in (3.16) and
Boundary conditions (3.26) are implemented in matrix
on the left-hand side of (3.25) in rows
,
,
,
, and
, respectively, as before with the initial solution above. The corresponding rows, all columns, of
on the right-hand side of (3.25),
and
are all set to be zero. This results in the following recursive formula for
:
The matrix
is the matrix
on the right-hand side of (3.25) but with the boundary conditions incorporated by setting the first,
,
,
, and
, rows and columns to zero. Thus, starting from the initial approximation, which is obtained from (3.17), higher-order approximations
for
can be obtained through recursive formula (3.28).