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,

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

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

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).