The main challenge in using the linearization method as described in Makukula et al. [

23,

24] is how to generalize the method so as to find solutions of partial differential equations of the form (8)-(10). It is certainly not clear how the method may be applied directly to the terms on the right hand side of equations (

8)-(

10). For this reason, equations (

8)-(

10) are first simplified and reduced to sets of ordinary differential equations by assuming regular perturbation expansions for

*f, θ*, and

*ϕ* in powers of

*ξ* (which is assumed to be small) as follows

$f={f}_{{M}_{1}}\left(\eta ,\xi \right)=\sum _{i=0}^{{M}_{1}}{\xi}^{i}{f}_{i}\left(\eta \right),\phantom{\rule{1em}{0ex}}\theta ={\theta}_{{M}_{1}}\left(\eta ,\xi \right)=\sum _{i=0}^{{M}_{1}}{\xi}^{i}{\theta}_{i}\left(\eta \right),\phantom{\rule{1em}{0ex}}\varphi ={\varphi}_{{M}_{1}}\left(\eta ,\xi \right)=\sum _{i=0}^{{M}_{1}}{\xi}^{i}{\varphi}_{i}\left(\eta \right)$

(17)

where

*M*_{1} is the order of the approximate solution. Substituting (17) into equations (

8)-(

10) and equating the coefficients of like powers of

*ξ*, we obtain the zeroth order set of ordinary differential equations

${f}_{0}^{\u2034}+\frac{1}{2}{f}_{0}{f}_{0}^{\u2033}-H{a}_{x}{f}_{0}^{\prime}+G{r}_{x}{\theta}_{0}+G{c}_{x}{\varphi}_{0}=0,$

(18)

$\frac{1}{Pr}{\theta}_{0}^{\u2033}+\frac{1}{2}{f}_{0}{\theta}_{0}^{\prime}+{D}_{f}{\varphi}_{0}^{\u2033}=0,$

(19)

$\frac{1}{Sc}{\varphi}_{0}^{\u2033}+\frac{1}{2}{f}_{0}{\varphi}_{0}^{\prime}+{S}_{r}{\theta}_{0}^{\u2033}=0,$

(20)

with corresponding boundary conditions

$\left.\begin{array}{cc}{f}_{0}=0,{f}_{0}^{\prime}=0,\phantom{\rule{2.77695pt}{0ex}}{\theta}_{0}=1,\phantom{\rule{2.77695pt}{0ex}}{\varphi}_{0}=1,\hfill & \hfill \mathsf{\text{at}}\phantom{\rule{2.77695pt}{0ex}}\eta =0\hfill \\ {f}_{0}^{\prime}=1,\phantom{\rule{2.77695pt}{0ex}}{\theta}_{0}=0,\phantom{\rule{2.77695pt}{0ex}}{\varphi}_{0}=0,\hfill & \hfill \mathsf{\text{as}}\phantom{\rule{2.77695pt}{0ex}}\eta =\infty .\hfill \end{array}\right\}$

(21)

The

*O*(

*ξ*^{1}) equations are

${f}_{1}^{\u2034}+\frac{1}{2}{f}_{0}{f}_{1}^{\u2033}-\left(H{a}_{x}+{f}_{0}^{\prime}\right){f}_{1}^{\prime}+\frac{3}{2}{f}_{0}^{\u2033}{f}_{1}+G{r}_{x}{\theta}_{1}+G{c}_{x}{\varphi}_{1}=0,$

(22)

$\frac{1}{Pr}{\theta}_{1}^{\u2033}+\frac{1}{2}{f}_{0}{\theta}_{1}^{\prime}-{f}_{0}^{\prime}{\theta}_{1}+\frac{3}{2}{\theta}_{0}^{\prime}{f}_{1}+{D}_{f}{\varphi}_{1}^{\u2033}=0,$

(23)

$\frac{1}{Sc}{\varphi}_{1}^{\u2033}+\frac{1}{2}{f}_{0}{\varphi}_{1}^{\prime}-{f}_{0}^{\prime}{\varphi}_{1}+\frac{3}{2}{\varphi}_{0}^{\prime}{f}_{1}+{S}_{r}{\theta}_{1}^{\u2033}=0,$

(24)

with boundary conditions

$\left.\begin{array}{cc}{f}_{1}=0,{f}_{0}^{\prime}=0,\phantom{\rule{2.77695pt}{0ex}}{\theta}_{1}=0,\phantom{\rule{2.77695pt}{0ex}}{\varphi}_{1}=0,\hfill & \hfill \mathsf{\text{at}}\phantom{\rule{2.77695pt}{0ex}}\eta =0\hfill \\ {f}_{1}^{\prime}=0,\phantom{\rule{2.77695pt}{0ex}}{\theta}_{1}=0,\phantom{\rule{2.77695pt}{0ex}}{\varphi}_{1}=0,\hfill & \hfill \mathsf{\text{as}}\phantom{\rule{2.77695pt}{0ex}}\eta =\infty .\hfill \end{array}\right\}$

(25)

Finally, the

*O*(

*ξ*^{2}) equations are

${f}_{2}^{\u2034}+\frac{1}{2}{f}_{0}{f}_{2}^{\u2033}-\left(H{a}_{x}+2{f}_{0}^{\prime}\right){f}_{2}^{\prime}+\frac{5}{2}{f}_{0}^{\u2033}{f}_{2}+G{r}_{x}{\theta}_{2}+G{c}_{x}{\varphi}_{2}={f}_{1}^{\prime}{f}_{1}^{\prime}-\frac{3}{2}{f}_{1}{f}_{1}^{\u2033},$

(26)

$\frac{1}{Pr}{\theta}_{2}^{\u2033}+\frac{1}{2}{f}_{0}{\theta}_{2}^{\prime}-2{f}_{0}^{\prime}{\theta}_{2}+\frac{5}{2}{\theta}_{0}^{\prime}{f}_{2}+{D}_{f}{\varphi}_{2}^{\u2033}={f}_{0}^{\prime}{\theta}_{1}-\frac{3}{2}{f}_{1}{\theta}_{1}^{\prime},$

(27)

$\frac{1}{Sc}{\varphi}_{2}^{\u2033}+\frac{1}{2}{f}_{0}{\varphi}_{2}^{\prime}-2{f}_{0}^{\prime}{\varphi}_{2}+\frac{5}{2}{\varphi}_{0}^{\prime}{f}_{2}+{S}_{r}{\theta}_{2}^{\u2033}={f}_{1}^{\prime}{\varphi}_{1}-\frac{3}{2}{f}_{1}{\varphi}_{1}^{\prime}.$

(28)

These equations have to be solved subject to boundary conditions

$\left.\begin{array}{cc}{f}_{2}=0,{f}_{2}^{\prime}=0,\phantom{\rule{2.77695pt}{0ex}}{\theta}_{2}=0,\phantom{\rule{2.77695pt}{0ex}}{\varphi}_{2}=0,\hfill & \hfill \mathsf{\text{at}}\phantom{\rule{2.77695pt}{0ex}}\eta =0\hfill \\ {f}_{2}^{\prime}=0,\phantom{\rule{2.77695pt}{0ex}}{\theta}_{2}=0,\phantom{\rule{2.77695pt}{0ex}}{\varphi}_{2}=0,\hfill & \hfill \mathsf{\text{as}}\phantom{\rule{2.77695pt}{0ex}}\eta =\infty .\hfill \end{array}\right\}$

(29)

The coupled system of equations (18)-(20), (22)-(24), and (26)-(28) together with the associated boundary conditions (21), (25), and (29) may be solved independently pairwise one after another. These equations may now be solved using the successive linearization method in the manner described in Makukula et al. [23, 24]). We begin by solving equations (18)-(20) with boundary conditions (21).

The successive linearization method is a non-perturbation method requiring neither the presence of an embedded perturbation parameter nor the addition of an artificial parameter. The method is therefore free of the major limitations associated with other perturbation methods. In the SLM algorithm assumption is made that the functions

*f*_{0}(

*η*),

*θ*_{0}(

*η*), and

*ϕ*_{0}(

*η*) may be expressed as

${f}_{0}\left(\eta \right)={F}_{i}\left(\eta \right)+\sum _{m=0}^{i-1}{F}_{m}\left(\eta \right),\phantom{\rule{1em}{0ex}}{\theta}_{0}\left(\eta \right)={\Theta}_{i}\left(\eta \right)+\sum _{m=0}^{i-1}{\Theta}_{m}\left(\eta \right),\phantom{\rule{1em}{0ex}}\varphi \left(\eta \right)={\Phi}_{i}\left(\eta \right)+\sum _{m=0}^{i-1}{\Phi}_{m}\left(\eta \right),$

(30)

where *F*_{
i
}, *Θ*_{
i
}, and *Φ*_{
i
}(*i* ≥ 1) are unknown functions and *F*_{
m
}, *Θ*_{
m
}, and *Φ*_{
m
}are successive approximations which are obtained by recursively solving the linear part of the system that is obtained from substituting equations (30) in (18)-(20). In choosing the form of the expansions (30), prior knowledge of the general nature of the solutions, as is often the case with perturbation methods, is not necessary.

Suitable initial guesses

*F*_{0}(

*η*),

*Θ*_{0}(

*η*), and

*Φ*_{0}(

*η*) which are selected to satisfy the boundary conditions (21) are

${F}_{0}\left(\eta \right)=\eta +{e}^{-\eta}-1,\phantom{\rule{1em}{0ex}}{\Theta}_{0}\left(\eta \right)={e}^{-\eta},\phantom{\rule{1em}{0ex}}{\Phi}_{0}\left(\eta \right)={e}^{-\eta},$

(31)

The subsequent solutions

*F*_{
i
},

*Θ*_{
i
}, and

*Φ*_{
i
}are obtained by iteratively solving the linearized form of the equations that are obtained by substituting equation (

30) in the governing equations (

18)-(

20). The linearized equations to be solved are

${F}_{i}^{\u2034}+{a}_{1,i-1}{F}_{i}^{\u2033}-H{a}_{x}{F}_{i}^{\prime}+{a}_{2,i-1}{F}_{i}+G{r}_{x}{\Theta}_{i}+G{c}_{x}{\Phi}_{i}={r}_{1,i-1},$

(32)

$\frac{1}{Pr}{\Theta}_{i}^{\u2033}+{b}_{1,i-1}{\Theta}_{i}^{\prime}+{b}_{2,i-1}{F}_{i}+{D}_{f}{\Phi}_{i}^{\u2033}={r}_{2,i-1},$

(33)

$\frac{1}{Sc}{\Phi}_{i}^{\u2033}+{c}_{1,i-1}{\Phi}_{i}^{\prime}+{c}_{2,i-1}{F}_{i}+Sr{\Theta}_{i}^{\u2033}={r}_{3,i-1},$

(34)

subject to the boundary conditions

${F}_{i}\left(0\right)={F}_{i}^{\prime}\left(0\right)={F}_{i}^{\prime}\left(\infty \right)={\Theta}_{i}\left(0\right)={\Theta}_{i}\left(\infty \right)={\Phi}_{i}\left(0\right)={\Phi}_{i}\left(\infty \right),$

(35)

where the coefficients parameters

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

*b*_{k,i-1},

*c*_{k,i-1},

*d*_{k,i-1}, and

*r*_{k,i- 1}are defined by

$\begin{array}{ll}\hfill {a}_{1,i-1}& =\frac{1}{2}\sum _{m=0}^{i-1}{F}_{m},\phantom{\rule{1em}{0ex}}{a}_{2,i-1}=\frac{1}{2}\sum _{m=0}^{i-1}{F}_{m}^{\prime \prime},\phantom{\rule{1em}{0ex}}{b}_{1,i-1}=\frac{1}{2}\sum _{m=0}^{i-1}{F}_{m}\phantom{\rule{2em}{0ex}}\\ \hfill {b}_{2,i-1}& =\frac{1}{2}\sum _{m=0}^{i-1}{\Theta}_{m}^{\prime},\phantom{\rule{1em}{0ex}}{c}_{1,i-1}=\frac{1}{2}\sum _{m=0}^{i-1}{F}_{m}\phantom{\rule{1em}{0ex}}{c}_{2,i-1}=\frac{1}{2}\sum _{m=0}^{i-1}{\Phi}_{m}^{\prime},\phantom{\rule{2em}{0ex}}\\ \hfill {r}_{1,i-1}& =H{a}_{x}\sum _{m=0}^{i-1}{F}_{m}^{\prime}-\frac{1}{2}\sum _{m=0}^{i-1}{F}_{m}\sum _{m=0}^{i-1}{F}_{m}^{\prime \prime}-\sum _{m=0}^{i-1}{F}_{m}^{\prime \prime \prime}-G{r}_{x}\sum _{m=0}^{i-1}{\Theta}_{m}-G{c}_{x}\sum _{m=0}^{i-1}{\Phi}_{m},\phantom{\rule{2em}{0ex}}\\ \hfill {r}_{2,i-1}& =-\frac{1}{Pr}\sum _{m=0}^{i-1}{\Theta}_{m}^{\prime \prime}-\frac{1}{2}\sum _{m=0}^{i-1}{F}_{m}\sum _{m=0}^{i-1}{\Phi}_{m}^{\prime}-{D}_{f}\sum _{m=0}^{i-1}{\Phi}_{m}^{\prime \prime},\phantom{\rule{2em}{0ex}}\\ \hfill {r}_{3,i-1}& =-\frac{1}{Sc}\sum _{m=0}^{i-1}{\Phi}_{m}^{\prime \prime}-\frac{1}{2}\sum _{m=0}^{i-1}{F}_{m}\sum _{m=0}^{i-1}{\Phi}_{m}^{\prime}-Sr\sum _{m=0}^{i-1}{\Phi}_{m}^{\prime \prime},\phantom{\rule{2em}{0ex}}\end{array}$

Once each solution

*F*_{
i
},

*Θ*_{
i
}, and

*Φ*_{
i
}(

*i* ≥ 1), has been found from iteratively solving equations (

32)-(

34), the approximate solutions for the system (18)-(20) are obtained as

${f}_{0}\left(\eta \right)\approx \sum _{i=0}^{{M}_{2}}{F}_{i}\left(\eta \right),\phantom{\rule{1em}{0ex}}{\theta}_{0}\left(\eta \right)\approx \sum _{i=0}^{{M}_{2}}{\Theta}_{i}\left(\eta \right),\phantom{\rule{1em}{0ex}}{\varphi}_{0}\left(\eta \right)\approx \sum _{i=0}^{{M}_{2}}{\Phi}_{i}\left(\eta \right),$

(36)

where

*M*_{2} is the order of the SLM approximations. In this study, we used the Chebyshev spectral collocation method to solve equations (

32)-(

34). The physical region [0, ∞) is first transformed into the spectral domain [-1,1] using the domain truncation technique in which the problem is solved on the interval [0,

*L*], where

*L* is a scaling parameter used to invoke the boundary condition at infinity. This is achieved by using the mapping

$\frac{\eta}{L}=\frac{\varsigma +1}{2},\phantom{\rule{1em}{0ex}}-1\le \varsigma \le 1,$

(37)

We discretise the spectral domain [-1,1] using the Gauss-Lobatto collocation points given by

${\varsigma}_{j}=\text{cos}\frac{\pi j}{N},\phantom{\rule{1em}{0ex}}j=0,1,\dots ,N,$

(38)

where

*N* is the number of collocation points used. The unknown functions

*F*_{
i
},

*Θ*_{
i
}, and

*Φ*_{
i
}are approximated at the collocation points by

${F}_{i}\left(\varsigma \right)\approx \sum _{k=0}^{N}{F}_{i}\left({\varsigma}_{k}\right){T}_{k}\left({\varsigma}_{j}\right),\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{\Theta}_{i}\left(\varsigma \right)\approx \sum _{k=0}^{N}{\Theta}_{i}\left({\varsigma}_{k}\right){T}_{k}\left({\varsigma}_{j}\right)\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{\Phi}_{i}\left(\varsigma \right)\approx \sum _{k=0}^{N}{\Phi}_{i}\left({\varsigma}_{k}\right){T}_{k}\left({\varsigma}_{j}\right),\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}j=0,1,\dots ,N$

(39)

where

*T*_{
k
}is the

*k* th Chebyshev polynomial defined as

${T}_{k}\left(\varsigma \right)=\text{cos}\left[k{\text{cos}}^{-1}\left(\varsigma \right)\right].$

(40)

The derivatives of the variables at the collocation points are represented as

${F}_{i}^{\left(r\right)}=\sum _{k=0}^{N}{\mathbf{D}}_{kj}^{r}{F}_{i}\left({\varsigma}_{k}\right),\phantom{\rule{1em}{0ex}}{\Theta}_{i}^{\left(r\right)}=\sum _{k=0}^{N}{\mathbf{D}}_{kj}^{r}{\Theta}_{i}\left({\varsigma}_{k}\right),\phantom{\rule{1em}{0ex}}{\Phi}_{i}^{\left(r\right)}=\sum _{k=0}^{N}{\mathbf{D}}_{kj}^{r}{\Phi}_{i}\left({\varsigma}_{k}\right),\phantom{\rule{1em}{0ex}}j=0,1,\dots ,N$

(41)

where

*r* is the order of differentiation and

$\mathbf{D}=\frac{2}{L}\mathcal{D}$ with

$\mathcal{D}$ being the Chebyshev spectral differentiation matrix whose entries are defined as;

$\left\{\begin{array}{c}{\mathcal{D}}_{jk}=\frac{{c}_{j}{\left(-1\right)}^{j+1}}{{c}_{k}{\varsigma}_{j}-{\varsigma}_{k}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}j\ne k;j,k=0,1,\dots ,N,\hfill \\ {\mathcal{D}}_{kk}=-\frac{{\varsigma}_{k}}{2\left(1-{\varsigma}_{k}^{2}\right)}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}k=1,2,\dots ,N-1,\hfill \\ {\mathcal{D}}_{00}=\frac{2{N}^{2}+1}{6}=-{\mathcal{D}}_{NN}.\hfill \end{array}\right.$

(42)

Substituting equations (

37)-(

41) into (32)-(34) gives the following linear system of equations

${\mathbf{A}}_{i-1}{\mathbf{X}}_{i}={\mathbf{R}}_{i-1}$

(43)

subject to the boundary conditions

${F}_{i}\left({\varsigma}_{0}\right)=\sum _{k=0}^{N}{\mathcal{D}}_{0k}{F}_{i}\left({\varsigma}_{k}\right)={g}_{i}\left({\varsigma}_{0}\right)={\Theta}_{i}\left({\varsigma}_{0}\right)={\Phi}_{i}\left({\varsigma}_{0}\right)={F}_{i}\left({\varsigma}_{N}\right)={g}_{i}\left({\varsigma}_{N}\right)={\Theta}_{i}\left({\varsigma}_{N}\right)={\Phi}_{i}\left({\varsigma}_{N}\right)=0.$

(44)

Here

**A**_{i- 1}is a 3(

*N* + 1) × 3(

*N* + 1) square matrix, while

**X**_{
i
}and

**R**_{i- 1}are 3(

*N* + 1) × 1 column vectors defined by

${\mathbf{A}}_{i-1}=\left[\begin{array}{c}{A}_{11}\phantom{\rule{2.77695pt}{0ex}}{A}_{12}\phantom{\rule{2.77695pt}{0ex}}{A}_{13}\hfill \\ {A}_{21}\phantom{\rule{2.77695pt}{0ex}}{A}_{22}\phantom{\rule{2.77695pt}{0ex}}{A}_{23}\hfill \\ {A}_{31}\phantom{\rule{2.77695pt}{0ex}}{A}_{32}\phantom{\rule{2.77695pt}{0ex}}{A}_{33}\hfill \end{array}\right],\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{\mathbf{X}}_{i}=\left[\begin{array}{c}{\stackrel{\u0303}{\mathbf{F}}}_{i}\hfill \\ {\stackrel{\u0303}{\Theta}}_{i}\hfill \\ {\stackrel{\u0303}{\Phi}}_{i}\hfill \end{array}\right],\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{\mathbf{R}}_{i-1}=\left[\begin{array}{c}{\mathbf{r}}_{1,i-1}\hfill \\ {\mathbf{r}}_{2,i-1}\hfill \\ {\mathbf{r}}_{3,i-1}\hfill \end{array}\right]$

(45)

where

${\stackrel{\u0303}{\mathbf{F}}}_{i}={\left[{F}_{i}\left({\varsigma}_{0}\right),{F}_{i}\left({\varsigma}_{1}\right),\dots ,{F}_{i}\left({\varsigma}_{N-1}\right),{F}_{i}\left({\varsigma}_{N}\right)\right]}^{T},$

(46)

${\stackrel{\u0303}{\Theta}}_{i}={\left[{\Theta}_{i}\left({\varsigma}_{0}\right),{\Theta}_{i}\left({\varsigma}_{1}\right),\dots ,{\Theta}_{i}\left({\varsigma}_{N-1}\right),{\Theta}_{i}\left({\varsigma}_{N}\right)\right]}^{T},$

(47)

${\stackrel{\u0303}{\Phi}}_{i}={\left[{\Phi}_{i}\left({\varsigma}_{0}\right),{\Phi}_{i}\left({\varsigma}_{1}\right),\dots ,{\Phi}_{i}\left({\varsigma}_{N-1}\right),{\Phi}_{i}\left({\varsigma}_{N}\right)\right]}^{T},$

(48)

${\mathbf{r}}_{1,i-1}={\left[{r}_{1,i-1}\left({\varsigma}_{0}\right),{r}_{1,i-1}\left({\varsigma}_{1}\right),\dots ,{r}_{1,i-1}\left({\varsigma}_{N-1}\right),{r}_{1,i-1}\left({\varsigma}_{N}\right)\right]}^{T}$

(49)

${\mathbf{r}}_{2,i-1}={\left[{r}_{2,i-1}\left({\varsigma}_{0}\right),{r}_{2,i-1}\left({\varsigma}_{1}\right),\dots ,{r}_{2,i-1}\left({\varsigma}_{N-1}\right),{r}_{2,i-1}\left({\varsigma}_{N}\right)\right]}^{T}$

(50)

${\mathbf{r}}_{3,i-1}={\left[{r}_{3,i-1}\left({\varsigma}_{0}\right),{r}_{3,i-1}\left({\varsigma}_{1}\right),\dots ,{r}_{3,i-1}\left({\varsigma}_{N-1}\right),{r}_{3,i-1}\left({\varsigma}_{N}\right)\right]}^{T}$

(51)

and

$\begin{array}{ll}\hfill {A}_{11}& ={\mathcal{D}}^{3}+{a}_{1,i-1}{\mathcal{D}}^{2}-H{a}_{x}\mathcal{D}+\left[{a}_{3,i-1}\right],\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{A}_{12}=\left[G{r}_{x}\right],\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{A}_{13}=\left[G{c}_{x}\right],\phantom{\rule{2em}{0ex}}\\ \hfill {A}_{21}& =\left[{b}_{2,i-1}\right],\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{A}_{22}=P{r}^{-1}{\mathcal{D}}^{2}+{b}_{1,i-1}\mathcal{D},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{A}_{23}={D}_{f}{\mathcal{D}}^{2}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{A}_{31}=\left[{c}_{2,i-1}\right],\phantom{\rule{2em}{0ex}}\\ \hfill {A}_{32}& =Sr{\mathcal{D}}^{2},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}{A}_{33}=S{c}^{-1}{\mathcal{D}}^{2}+{c}_{1,i-1}\mathcal{D},\phantom{\rule{2em}{0ex}}\end{array}$

(52)

where [

*·*] is a diagonal matrix of size (

*N* + 1) × (

*N +* 1) and

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

*b*_{k,i- 1},

*c*_{k,i- 1}are diagonal matrices of size (

*N* + 1) × (

*N* + 1) and

*T* is the transpose. After modifying the matrix system (43) to incorporate the boundary conditions (44), the solution is obtained as

${\mathbf{X}}_{i}={\mathbf{A}}_{i-1}^{-1}{\mathbf{R}}_{i-1}$

(53)

Equation (53) gives a solution of (18)-(20) for *f*_{0}, *θ*_{0} and *ϕ*_{0}. The procedure is repeated to obtain the *O*(*ξ*^{1}) and *O*(*ξ*^{2}) solutions using equations (22)-(24) and (26)-(28), respectively.