**4. Method of solution**

In this section, we give a brief overview of the Spectral Quasi-linearization Method (SQLM). The SQLM uses the Newton-Raphson-based quasi-linearization method (QLM) to linearize the governing non-linear equations. The QLM was developed by [41]. The SQLM then integrates the QLM using the Chebyshev Spectral collocation method.

On a New Numerical Approach on Micropolar Fluid, Heat and Mass Transfer Over an Unsteady Stretching Sheet Through Porous Media in the Presence of a Heat Source/Sink and Chemical Reaction http://dx.doi.org/10.5772/63800 295

#### **4.1. Spectral Quasi-Linearization Method (SQLM)**

#### *4.1.1. Main idea*

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The quantities of engineering interest in the present study are the skin-friction coefficient *Cfx*, the local wall couple stress *Mwx*, the local Nusselt number *Nux* and the local Sherwood number

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In this section, we give a brief overview of the Spectral Quasi-linearization Method (SQLM). The SQLM uses the Newton-Raphson-based quasi-linearization method (QLM) to linearize the governing non-linear equations. The QLM was developed by [41]. The SQLM then

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**3.2. Quantities of engineering interest**

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**4. Method of solution**

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294 Numerical Simulation - From Brain Imaging to Turbulent Flows

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Consider the problem of solving *n*th order non-linear differential equation

$$F\left(\mu, \mu', \mu'', \dots, \mu^{(n)}\right) = \mathbf{g}\left(\mathbf{x}\right) \text{ or } F\left(\mathbf{u}\right) = \mathbf{g}\left(\mathbf{x}\right),\tag{22}$$

subject to prescribed boundary conditions, where *a* ≤ *x* ≤*b*. The SQLM consists of two basic steps: quasi-linearization and Chebyshev differentiation in that order.

*Quasi-linearization*: If we expand left-hand side of Eq. (22) in Taylor series about **v**(*v*, *v* ′ , *v* ″ , ..., *v* (*n*) ) = *g*(*x*) and re-arrange the terms in the resulting equation we get:

$$\mathbf{u}\,\nabla F(\mathbf{v}) = \mathbf{v}\,\nabla F(\mathbf{v}) - F(\mathbf{v}) + \mathbf{g}\,(\mathbf{x}).\tag{23}$$

If **v** is given then the previous equation can be used to solve for u. Keeping this in mind, we replace v and u with approximations *ur* and *ur+1* of **u** at the end of *r* and *r*+1 iterations, respec‐ tively. This results in the *n*th order linear differential equation

$$a\_{0,r}u\_{r+1} + a\_{1,r}\frac{du\_{r+2}}{d\mathbf{x}} + a\_{2,r}\frac{d^2u\_{r+2}}{d\mathbf{x}^2} + ... + a\_{n,r}\frac{d^nu\_{r+2}}{d\mathbf{x}^n} = R\_r \tag{24}$$

Where *am*,*<sup>r</sup>* <sup>=</sup> *Fu* (*<sup>m</sup>*)(**u***r*), *Fu* (*<sup>m</sup>*)= <sup>∂</sup> *<sup>F</sup>* <sup>∂</sup> *<sup>u</sup>* (*<sup>m</sup>*) and

$$R\_r = a\_{0,r}\mu\_r + a\_{1,r}\mu\_r' + a\_{2,r}\mu\_r'' + ... + a\_{n,r}\mu\_r^{(m)} - F\left(\mathbf{u}\_r\right) + \mathbf{g}\left(\mathbf{x}\right) \tag{25}$$

*Chebyshev differentiation*: To solve the differential Eq. (24) we start by performing the following preliminary steps.

**1.** Using the linear mapping

$$\text{tr}\left(\boldsymbol{\xi}\right) = \frac{1-\boldsymbol{\xi}}{2}\boldsymbol{a} + \frac{1+\boldsymbol{\xi}}{2}\boldsymbol{b} \tag{26}$$

we transform Eq. (24) on the physical interval [*a*, *b*], say, on the *x* axis to its equivalent

$$a\_{0,r}u\_{r+1} + a\_{1,r} \beta \frac{du\_{r+1}}{d\,\xi} + a\_{2,r} \beta^2 \frac{d^2 u\_{r+1}}{d\,\xi^2} + ... + a\_{n,r} \beta^n \frac{d^n u\_{r+1}}{d\,\xi^n} = R\_r \tag{27}$$

on the computational interval [−1, 1] on the *ξ* axis, where *<sup>β</sup>* <sup>=</sup> <sup>2</sup> *<sup>b</sup>* <sup>−</sup> *<sup>a</sup>* .

**2.** Partition interval [−1, 1] using the collocation points *<sup>ξ</sup>* <sup>=</sup> *<sup>π</sup><sup>i</sup> <sup>N</sup>* where *i* = 0, 1, 2, ..., *N*.

Next we calculate differential Eq. (27) at each collocation point *ξ<sup>i</sup>* . This is followed by approximating each derivative using the formula:

$$\frac{d^{\boldsymbol{\rho}}\boldsymbol{u}\_{\boldsymbol{\rho}}}{d\boldsymbol{\xi}^{\boldsymbol{\rho}}}(\boldsymbol{\xi}\_{\boldsymbol{\iota}}) = \sum\_{\boldsymbol{\rho}=\boldsymbol{0}}^{N} \boldsymbol{D}^{\boldsymbol{\rho}} \int\_{\boldsymbol{\mathcal{Y}}} \boldsymbol{u}\_{\boldsymbol{\iota}} \left(\boldsymbol{\xi}\_{\boldsymbol{\iota}}\right)^{\boldsymbol{\epsilon}}$$

where *<sup>D</sup>* is the (*N*+1)×(*N*+1) Chebyshev differentiation matrix [1]. This process is called Chebyshev differentiation. The differential Eq. (27) is then evaluated at points *ξ*0, *ξ*1, .....*ξ<sup>N</sup>* with a linear system

$$A \begin{bmatrix} u\_r \left( \xi\_0 \right) \\ u\_r \left( \xi\_1 \right) \\ \cdot \\ \cdot \\ \cdot \\ \cdot \\ u\_r \left( \xi\_N \right) \end{bmatrix} = \begin{bmatrix} R\_r \left( \xi\_0 \right) \\ R\_r \left( \xi\_1 \right) \\ \cdot \\ \cdot \\ \cdot \\ \cdot \\ R\_r \left( \xi\_N \right) \end{bmatrix}$$

which upon including the boundary conditions and solving for each *r* generates a sequence {**u***r*} of approximation which we expect to converge.

#### **4.2. Application to current problem**

Eq. (14) is of the form

$$F\left(f, f', \phi, \phi', \phi''\right) = 0,\tag{28}$$

where

$$F\left(f, f', \phi, \phi', \phi''\right) = \frac{1}{Sc}\phi'' + f\phi' - f'\phi - K\phi - \frac{A}{2}(4\phi + \eta\phi'). \tag{29}$$

On a New Numerical Approach on Micropolar Fluid, Heat and Mass Transfer Over an Unsteady Stretching Sheet Through Porous Media in the Presence of a Heat Source/Sink and Chemical Reaction http://dx.doi.org/10.5772/63800 297

*Hence quasi-linearization* as directed in Section 4.2.1 replaces non-linear differential Eq. (28) with its linear counterpart

$$d\_{0r}f\_{r+1} + d\_{1r}f\_{r+1}' + d\_{2r}
\phi\_{r+1} + d\_{3r}
\phi\_{r+1}' + d\_{4r}
\phi\_{r+1}'' = R\_r^{(3)}\tag{30}$$

where

2

0, 1 1, 2, <sup>2</sup> n, ...

on the computational interval [−1, 1] on the *ξ* axis, where *<sup>β</sup>* <sup>=</sup> <sup>2</sup>

**2.** Partition interval [−1, 1] using the collocation points *<sup>ξ</sup>* <sup>=</sup> *<sup>π</sup><sup>i</sup>*

approximating each derivative using the formula:

with a linear system

**4.2. Application to current problem**

Eq. (14) is of the form

where

xx

Next we calculate differential Eq. (27) at each collocation point *ξ<sup>i</sup>*

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sequence {**u***r*} of approximation which we expect to converge.

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296 Numerical Simulation - From Brain Imaging to Turbulent Flows

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where *<sup>D</sup>* is the (*N*+1)×(*N*+1) Chebyshev differentiation matrix [1]. This process is called Chebyshev differentiation. The differential Eq. (27) is then evaluated at points *ξ*0, *ξ*1, .....*ξ<sup>N</sup>*

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<sup>+</sup> + + ++ = (27)

$$d\_{0r} = \phi\_r', \ d\_{1r} = -\phi\_r, \ d\_{2r} = f\_r' - K - 2A,\ d\_{3r} = f\_r - \frac{A}{2}\eta\_r, \ d\_{4r} = \frac{1}{\text{Sc}},\ R\_r^{(3)} = -f\_r'\phi\_r + f\_r\phi\_r'.$$

Chebyshev differentiation replaces differential Eq. (30) with a linear system

$$\begin{aligned} \boldsymbol{A}\_{41} &= \text{diag}\left\{\boldsymbol{\Phi}\_{r}^{\prime}\right\} - \text{diag}\left\{\boldsymbol{\Phi}\_{r}\right\} \hat{D}, \\\\ \boldsymbol{A}\_{44} &= \text{diag}\left\{\mathbf{F}\_{r}^{\prime}\right\} + \left(-K - 2\boldsymbol{A}\right)\boldsymbol{I} + \text{diag}\left\{\mathbf{F}\_{r} - \frac{\boldsymbol{A}}{2}\boldsymbol{\eta}\_{r}\right\} \hat{D} + \frac{1}{Sc} \hat{D}^{2}, \\\\ \mathbf{R}\_{r}^{(4)} &= -\mathbf{F}\_{r}^{\prime} \circ \mathbf{O}\_{r} + \mathbf{F}\_{r} \circ \boldsymbol{\Phi}\_{r}^{\prime} \\\\ \mathbf{F}\_{r+1} &= \left[f\_{r+1}\left(\boldsymbol{\xi}\_{0}\right) f\_{r+1}\left(\boldsymbol{\xi}\_{1}\right) \dots f\_{r+1}\left(\boldsymbol{\xi}\_{N}\right)\right]^{\top}, \\\\ \boldsymbol{\Phi}\_{r+1} &= \left[\boldsymbol{\phi}\_{r+1}\left(\boldsymbol{\xi}\_{0}\right) \boldsymbol{\phi}\_{r+1}\left(\boldsymbol{\xi}\_{1}\right) \dots \boldsymbol{\phi}\_{r+1}\left(\boldsymbol{\xi}\_{N}\right)\right]^{\top}. \end{aligned}$$

Therefore the use of Quasi-linearization followed by Chebyshev differentiation replaces differential Eq. (14) with a linear system (31). Similarly, differential Eqs. (11)–(13) are re‐ placed by linear systems, which if combined with a linear system (30) yield a larger linear system

$$
\begin{bmatrix} A\_{11} & A\_{12} & A\_{13} & A\_{14} \\ A\_{21} & A\_{22} & O & O \\ A\_{31} & O & A\_{33} & O \\ A\_{41} & O & O & A\_{44} \end{bmatrix} \begin{bmatrix} \mathbf{F}\_{r+1} \\ \mathbf{H}\_{r+1} \\ \boldsymbol{\Phi}\_{r+1} \\ \boldsymbol{\Phi}\_{r+1} \end{bmatrix} = \begin{bmatrix} \mathbf{R}\_{r}^{(1)} \\ \mathbf{R}\_{r}^{(2)} \\ \mathbf{R}\_{r}^{(3)} \\ \mathbf{R}\_{r}^{(4)} \end{bmatrix} \tag{32}
$$

subject to boundary conditions

$$f\_{r+1}\left(\boldsymbol{\xi}\right) = f\_{\boldsymbol{\nu}}, \ \sum\_{k=0}^{N} \hat{D}\_{\boldsymbol{\nu}k} f\left(\boldsymbol{\xi}\right) = 1, \ h\_{r+1}\left(\boldsymbol{\xi}\_{N}\right) = \boldsymbol{\xi}\_{N}, \ \theta\_{r+1}\left(\boldsymbol{\xi}\_{N}\right) = 1, \ \phi\_{r+1}\left(\boldsymbol{\xi}\_{N}\right) = 1,\tag{33}$$

$$\sum\_{k=0}^{N} D\_{0k} f\left(\xi\right) = 0, \ h\_{r+1}\left(\xi\_0\right) = 0, \ \mathcal{O}\_{r+1}\left(\xi\_0\right) = 0, \ \phi\_{r+1}\left(\xi\_0\right) = 0,\tag{34}$$

where

$$\begin{aligned} \mathbf{A}\_{11} &= \text{diag}\left\{\mathbf{F}\_r^{\boldsymbol{\sigma}}\right\} + \left(-2\text{diag}\left\{\mathbf{F}\_r^{\boldsymbol{\prime}}\right\} - \left(A + M + \frac{1}{K\_\rho}\right)I\right)\hat{D} + \text{diag}\left\{\mathbf{F}\_r - \frac{A}{2}\boldsymbol{\eta}\_r\right\}\hat{D}^2 + \left(\mathbb{I} + \Delta\right)\hat{D}^\dagger, \\\\ \mathbf{A}\_{12} &= \Delta\hat{D}, \ A\_{13} = \lambda\_1 I, \ A\_{14} = \lambda\_2 I, \ \mathbf{R}\_r^{(\mathbf{t})} = \mathbf{F} \circ \mathbf{F}\_r'' - \mathbf{F}\_r' \circ \mathbf{F}', \\ \mathbf{A}\_{21} &= \text{diag}\left\{\mathbf{H}\_r\right\} - \text{diag}\left\{\mathbf{H}\_r\right\}\hat{D} - \Delta B\hat{D}^2, \end{aligned}$$

$$\begin{split} \mathbf{A}\_{22} &= -\text{diag}\left(\mathbf{F}\_{r}^{\prime}\right) - \left(\frac{3}{2}A + 2\Delta B\right)I + \text{diag}\left\{\mathbf{F}\_{r} - \frac{A}{2}\eta\_{r}\right\}\hat{D} + \lambda\_{3}\hat{D}^{2}, \ \mathbf{R}\_{r}^{(2)} = \mathbf{F}\circ\mathbf{H}\_{r}^{\prime} - \mathbf{F}\_{r}^{\prime}\circ\mathbf{H}\_{r}, \\ \mathbf{A}\_{31} &= \text{diag}\left\{\boldsymbol{\Theta}\_{r}^{\prime}\right\} - \text{diag}\left\{\boldsymbol{\Theta}\_{r}\right\}\hat{D} + 2Ec\left(1+\Lambda\right)\text{diag}\left\{\mathbf{F}\_{r}^{\prime}\right\}\hat{D}^{2}, \end{split}$$

$$\begin{aligned} \mathbf{A}\_{32} &= -\operatorname{diag}\left(\mathbf{F}\_r\right) + \left(-2\mathcal{A} + \underline{Q}\_x\right)I + \operatorname{diag}\left\{\mathbf{F}\_r - \frac{\mathcal{A}}{2}\,\eta\_r\right\}\hat{D} + \frac{1}{\operatorname{Pr}}\hat{D}^2, \\ \mathbf{R}\_r^{(3)} &= Ec(\mathbf{l} + \Delta)\,\mathbf{F}\_r'' + \mathbf{F}\circ\boldsymbol{\Theta}\_r' - \mathbf{F}\_r'\circ\boldsymbol{\Theta}\_\mathbf{\hat{p}} \end{aligned}$$

$$\begin{aligned} \mathbf{A}\_{41} &= \text{diag}\left\{\boldsymbol{\Phi}\mathbf{P}'\_{r}\right\} - \text{diag}\left\{\boldsymbol{\Phi}\mathbf{P}\_{r}\right\}\hat{D}, \\ \mathbf{A}\_{42} &= -\text{diag}\left\{\mathbf{F}\_{r}\right\} - \left(K + 2\boldsymbol{A}\right)I + \text{diag}\left\{\mathbf{F}\_{r} - \frac{\boldsymbol{A}}{2}\boldsymbol{\eta}\_{r}\right\}\hat{D} + \frac{1}{Sc}\hat{D}^{2}, \\ \mathbf{R}\_{r}^{(4)} &= -\mathbf{F}'\circ\boldsymbol{\Phi}\_{r} + \mathbf{F}\_{r}\circ\boldsymbol{\Phi}', \end{aligned}$$

*A о B* denotes the Hadarmard product (element-wise multiplication) of matrices *A* and *B* of the same order, and *I* and *O* are the identity and zero matrices, respectively. Boundary conditions (33) and (34) of linear system (32) are in the same manner as done in [2]. This is followed by solution of linear system (32) to get approximations *f <sup>r</sup>*(*ξc*), *hr*(*ξc*), *θr*(*ξc*), *ϕr*(*ξc*) for each *r* =1, 2, ... and *c* =0, 1, 2, ..., *N* . However, this last step requires suitable initial approx‐ imation for which we choose

On a New Numerical Approach on Micropolar Fluid, Heat and Mass Transfer Over an Unsteady Stretching Sheet Through Porous Media in the Presence of a Heat Source/Sink and Chemical Reaction http://dx.doi.org/10.5772/63800 299

$$f\_0\left(\eta\right) = f\_\text{\(\eta\)} + 1 - e^{-\eta},\ h\_0\left(\eta\right) = \eta e^{-\eta},\ \theta\_0\left(\eta\right) = e^{-\eta},\ \phi\_0\left(\eta\right) = e^{-\eta},\ \phi\_0$$

so as to satisfy boundary conditions (33) and (34).
