We consider the general nonlinear third-order boundary value problem

subject to the boundary conditions

where *c*_{
i
} , *b*_{
j
} (*i* = 1, ..., 4 *j* = 1, 2, 3) are constants.

Equation 2.1 can be solved easily using methods such as the HAM and the SHAM. In each of these methods, an initial approximation *f*_{0}(*η*) is sought, which satisfies the boundary conditions. The speed of convergence of the method depends on whether *f*_{0}(*η*) is a good approximation of *f* (*η*) or not. The approach proposed here seeks to find an optimal initial approximation *f*_{0} that would lead to faster convergence of the method to the true solution. We thus first seek to improve the initial approximation that is used later in the SHAM to solve the governing nonlinear equation.

We assume that the solution

*f*(

*η*) may be expanded as an infinite sum:

where

*f*_{
i
} 's are unknown functions whose solutions are obtained using the SHAM at the

*i* th iteration and

*f*_{
n
} , (

*n* ≥ 1) are known from previous iterations. The algorithm starts with the initial approximation

*f*_{0}(

*η*) which is chosen to satisfy the boundary conditions (2.2). An appropriate initial guess is

Substituting (2.3) in the governing equation (

2.1-2.2) gives

subject to the boundary conditions

where the coefficient parameters

*a*_{k,i-1}, (

*k* = 1, ..., 3) and

*r*_{i-1}are defined as

Starting from the initial approximation (2.4), the subsequent solutions

*f*_{
i
} (

*i* ≥ 1) are obtained by recursively solving Equation

2.5 using the SHAM, [

24,

25]. To find the solutions of Equation

2.5, we begin by defining the following linear operator:

where *q* ∈ 0[1] is the embedding parameter, and *F*_{
i
} (*η*; *q*) is an unknown function.

The zeroth-order deformation equation is given by

where

*ħ* is the non-zero convergence controlling auxiliary parameter and

is a nonlinear operator given by

Differentiating (2.10)

*m* times with respect to

*q* and then setting

*q* = 0, and finally dividing the resulting equations by

*m*! yield the

*m* th-order deformation equations:

subject to the boundary conditions

The initial approximation

*f*_{i,0}that is used in the higher-order equations (

2.12) is obtained on solving the linear part of Equation

2.5 which is given by

subject to the boundary conditions:

Since the coefficient parameters and the right-hand side of Equation

2.15 for

*i* = 1, 2, 3, ... are known (from previous iterations), the equation can easily be solved using numerical methods such as finite differences, finite elements, Runge-Kutta-based shooting methods or collocation methods. In this article, Equation

2.15 are solved using the Chebyshev spectral collocation method. The method (see, for example, [

29–

31]), is based on the Chebyshev polynomials defined on the interval [-1, 1] by

To implement the method, the physical region [0, ∞) is transformed into the region [-1, 1] using the domain truncation technique whereby the problem is solved in the interval [0,

*L*] instead of [0, ∞). This leads to the mapping

where

*L* is the scaling parameter used to invoke the boundary condition at infinity. We use the popular Gauss-Lobatto collocation points [

29,

31] to define the Chebyshev nodes in [-1, 1], namely:

where

*N* is the number of collocation points. The variable

*f*_{i,0}is approximated by the interpolating polynomial in terms of its values at each of the collocation points by employing the truncated Chebyshev series of the form:

where

*T*_{
k
} is the

*k* th Chebyshev polynomial. Derivatives of the variables at the collocation points may be represented by

where

*s* is the order of differentiation and

, with

being the Chebyshev spectral differentiation matrix (see, for example [

29,

31]) whose entries are defined as

Substituting Equations

2.20-2.21 in 2.15-2.16 gives

In the above definitions,

*T* stands for transpose and

**a**_{k,i-1}(

*k* = 1, 2, 3) denotes a diagonal matrix of size (

*N* + 1) × (

*N* + 1). The boundary condition

*f*_{
i
} (

*ξ*_{
N
} ) = 0 is implemented by deleting last row and last column of

**A**_{i-1}, and deleting the last rows of

**F**_{i,0}and

**R**_{i-1}. The derivative boundary conditions in (2.24) are then imposed on the resulting first row and last row of

**A**_{i-1}and setting the first and last rows of

**F**_{i,0}and

**R**_{i-1}to be zero. The solutions for

*f*_{i.0}(

*ξ*) are then obtained from soloving

In a similar manner, applying the Chebyshev spectral transformation on the higher order deformation equations (

2.12)-(2.13) gives

subject to the boundary conditions

where

**A**_{i-1}and

**R**_{i-1}, are as defined in (2.25) and (2.27), respectively, and

To implement the boundary condition

*f*_{i,m}(

*ξ*_{
N
} ) = 0, we delete the last rows of

**P**_{i,m-1}and

**R**_{i-1}and delete the last row and the last column of

**A**_{i-1}in (2.29). The other boundary conditions in (2.30) are imposed on the first and the last rows of the modified

**A**_{i-1}matrix on the left side of the equal sign in (2.29). The first and the last rows of the modified

**A**_{i-1}matrix on the right side of the equal sign in (2.29) are then set to be zero. This results in the following recursive formula for

*m* ≥ 1:

where **Ã**_{i-1}is the modified matrix **A**_{i-1}after incorporating the boundary conditions (2.30). Thus, starting from the initial approximation, which is obtained from (2.28), higher-order approximations *f*_{i,m}(*ξ*) for *m* ≥ 1, can be obtained through the recursive formula (2.33).

The solutions for

*f*_{
i
} are then generated using the solutions for

*f*_{i, m}as follows:

The [*i*, *m*] approximate solution for *f* (*η*) is then obtained by substituting *f*_{
i
} (obtained from 2.34) in equation 2.3.