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).