*3.1.1 m dependent and n independent variables*

*<sup>m</sup>* dependent variables *<sup>u</sup>* <sup>¼</sup> *<sup>u</sup>*1, *<sup>u</sup>*2, … , *um* ð Þ and *<sup>n</sup>* independent variables *<sup>x</sup>* <sup>¼</sup> *<sup>x</sup>*1, *<sup>x</sup>*2, … , *xn* ð Þ, *<sup>u</sup>* <sup>¼</sup> *u x*ð Þ with *<sup>m</sup>* <sup>≥</sup> 2, arise in studying systems of differential equations. We consider extended transformations from ð Þ� *x*, *u* space to *x*, *u*, *u*ð Þ<sup>1</sup> , *u*ð Þ<sup>2</sup> , … , *u*ð Þ*<sup>k</sup>* � �� space. Here *<sup>u</sup>*ð Þ*<sup>k</sup>* denotes the components of all *<sup>k</sup>*th-order partial derivatives of *u* wrt *x:*.

**definition** *Total derivative.* The total differentiation operator wrt *xi* is defined by

$$D\_i = \frac{\partial}{\partial \mathbf{x}^i} + u^a\_i \frac{\partial}{\partial u^a} + u^a\_{\vec{\eta}} \frac{\partial}{\partial u^a\_i} + \dots + u^a\_{\vec{u}\_1 i\_2 \dots i\_k} \frac{\partial}{\partial u^a\_{i\_1 i\_2 \dots i\_u}} + \dots, \quad i = 1, 2, \dots, n.$$

where

$$u^a\_i = \frac{\partial u^a}{\partial \mathbf{x}^i}, \quad u^a\_{ij} = \frac{\partial^2 u^a}{\partial \mathbf{x}^i \partial \mathbf{x}^j}, \quad \text{etc.}$$

We seek the one-parameter Lie group of transformations

$$\begin{aligned} \overline{\boldsymbol{\mathfrak{x}}}^i &= \boldsymbol{\mathfrak{x}}^i + \epsilon \xi^i(\boldsymbol{\mathfrak{x}}, \boldsymbol{\mathfrak{u}}) + O\left(\epsilon^2\right), \\\\ \overline{\boldsymbol{\mathfrak{u}}}^a &= \boldsymbol{\mathfrak{u}}^a + \epsilon \eta^a(\boldsymbol{\mathfrak{x}}, \boldsymbol{\mathfrak{u}}) + O\left(\epsilon^2\right), \end{aligned} \tag{14}$$

*ζα*

*DOI: http://dx.doi.org/10.5772/intechopen.95490*

*ζα*

*ζα*

*ζα*

*ζα*

*ζα*

*∂ <sup>∂</sup>xi* <sup>þ</sup> *ηα*

differential invariant of a group *G* if

differential invariant of a group *G* if

**3.2 Approximate methods**

sional transform given by

� � <sup>¼</sup> <sup>1</sup>

*F k*1, *k*2, … , *kp*

**93**

where *X*½ � *<sup>p</sup>* is the *p*th prolongation of *X:*.

*3.2.1 p-dimensional differential transform methods*

For an analytic multivariable function *f x*1, *x*2, … , *xp*

*k*1!*k*2! … *kp*!

*: : : : : :*

*<sup>i</sup>* ð Þ *x*, *u*

Theorem 1.2 A differential function *F x*, *u*, *u*1, … , *u*ð Þ *<sup>p</sup>*

*F x*, *u*, *u*1, … , *u*ð Þ *<sup>p</sup>*

Theorem 1.3 A differential function *F x*, *u*, *u*1, … , *u*ð Þ *<sup>p</sup>*

then

*<sup>X</sup>*½ � *<sup>k</sup>* <sup>¼</sup> *<sup>ξ</sup><sup>i</sup>*

ð Þ *x*, *u*

*<sup>i</sup>* <sup>¼</sup> *Di ηα* ð Þ� *<sup>u</sup><sup>α</sup>*

*i* � � � *<sup>u</sup><sup>α</sup>*

*i*<sup>1</sup> … *ik*�<sup>1</sup> � � � *<sup>u</sup><sup>α</sup>*

*<sup>W</sup><sup>α</sup>* <sup>¼</sup> *<sup>η</sup><sup>α</sup>* � *<sup>ξ</sup> <sup>j</sup>*

*ij* <sup>¼</sup> *Dj <sup>ζ</sup><sup>α</sup>*

*<sup>i</sup>*1,*i*2, … ,*ik* <sup>¼</sup> *Dik <sup>ζ</sup><sup>α</sup>*

Introducing the *Lie Characteristic function* defined by

*<sup>i</sup>* <sup>¼</sup> *Di <sup>W</sup><sup>α</sup>* ð Þþ *<sup>ξ</sup> <sup>j</sup>*

*ij* <sup>¼</sup> *DiDj <sup>W</sup><sup>α</sup>* ð Þþ *<sup>ξ</sup>kukij*,

*<sup>i</sup>*1,*i*2, … ,*ik* <sup>¼</sup> *Di*<sup>1</sup> … *Dik <sup>W</sup><sup>α</sup>* ð Þþ *<sup>ξ</sup> <sup>j</sup>*

*∂ <sup>∂</sup>u<sup>α</sup>* <sup>þ</sup> *<sup>ζ</sup><sup>α</sup> i ∂ ∂ui*

The corresponding (*k*th extended) infinitesimal generator is given by

� � <sup>¼</sup> *<sup>F</sup> <sup>x</sup>*, *<sup>u</sup>*, *<sup>u</sup>*1, … , *<sup>u</sup>*ð Þ *<sup>p</sup>*

*<sup>X</sup>*½ � *<sup>p</sup> <sup>F</sup>* <sup>¼</sup> 0,

*∂<sup>k</sup>*1þ*k*2<sup>þ</sup> … <sup>þ</sup>*kp f x*1, *x*2, … , *xp*

" #�

*∂xk*<sup>1</sup> <sup>1</sup> *<sup>∂</sup>xk*<sup>2</sup> <sup>2</sup> … *<sup>∂</sup><sup>x</sup> kp p*

� �

*: : : : : :* *jDi <sup>ξ</sup> <sup>j</sup>* � �,

*Survey of Some Exact and Approximate Analytical Solutions for Heat Transfer in Extended…*

*ilDj <sup>ξ</sup><sup>l</sup>* � �,

*i*1,*i*<sup>2</sup> … *ik*�1*l*

*uα j*,

*uα ji*, *Dik ξ<sup>l</sup>* � �,

*uji*1,*i*<sup>2</sup> … *ik :*

*i*1,*i*2, … ,*ik*

� �*:*

� � *p*≥ 0, is a *p*th-order

� � *p*≥ 0, is a *p*th-order

� �, we have the *p*-dimen-

� � � �

ð Þ *<sup>x</sup>*1,*x*2, … ,*xp* ¼ð Þ 0,0, … ,0

*:*

(18)

<sup>þ</sup> … <sup>þ</sup> *<sup>ζ</sup><sup>α</sup>*

9

>>>>>>>>>>=

(16)

(17)

, *k*≥1*:*

>>>>>>>>>>;

9

>>>>>>>>>=

>>>>>>>>>;

*∂ ∂ui*1,*i*2, … ,*ik*

which leave the system of equation in question invariant. These transformations are generated by the base vector

$$X = \xi^i(\mathfrak{x}, \mathfrak{u}) \frac{\partial}{\partial \mathfrak{x}^i} + \eta^a(\mathfrak{x}, \mathfrak{u}) \frac{\partial}{\partial \mathfrak{u}^a}.$$

The *k*th-extended transformation of (14) are given by

$$\begin{array}{ll} \overline{u}^{a}\_{i} &= u^{a}\_{i} + \varepsilon^{\alpha}\_{\forall i} (\mathbf{x}, u, u\_{(1)}) + O(\varepsilon^{2}), \\\\ \overline{u}^{a}\_{\forall i} &= u^{a}\_{\forall} + \varepsilon^{\alpha}\_{\forall i} (\mathbf{x}, u, u\_{(1)}, u\_{(2)}) + O(\varepsilon^{2}), \\\\ \cdot & \cdot \\\\ \cdot & \cdot \\\\ \overline{u}^{a}\_{i,i,1,\ldots,i\_{k}} &= u^{a}\_{i,i,\ldots,i\_{k}} + \varepsilon^{\alpha}\_{\forall,i,i,\ldots,i\_{k}} (\mathbf{x}, u, u\_{(1)}, u\_{(2)}, \ldots, u\_{(k)}) + O(\varepsilon^{2}), \end{array} \tag{15}$$

Theorem 1.1 The extended infinitesimals satisfy the recursion relations

*Survey of Some Exact and Approximate Analytical Solutions for Heat Transfer in Extended… DOI: http://dx.doi.org/10.5772/intechopen.95490*

$$\begin{array}{lcl} \zeta\_{i}^{a} &= D\_{i}(\eta^{a}) - \boldsymbol{u}\_{j}^{a} D\_{i}(\xi^{j}), \\ \zeta\_{ij}^{a} &= D\_{j}(\zeta\_{i}^{a}) - \boldsymbol{u}\_{il}^{a} D\_{j}(\xi^{l}), \\ \vdots & \vdots \\ \cdot & \cdot \\ \zeta\_{i\_{1},i\_{2},\ldots,i\_{k}}^{a} &= D\_{i\_{k}}(\zeta\_{i\_{1}\ldots i\_{k-1}}^{a}) - \boldsymbol{u}\_{i\_{1},i\_{2}\ldots i\_{k-1}}^{a} D\_{i\_{k}}(\xi^{l}), \end{array} \tag{16}$$

Introducing the *Lie Characteristic function* defined by

$$\mathcal{W}^a = \eta^a - \xi^j \mathfrak{u}\_j^a,$$

then

**3. Methods of solutions**

*x*, *u*, *u*ð Þ<sup>1</sup> , *u*ð Þ<sup>2</sup> , … , *u*ð Þ*<sup>k</sup>*

partial derivatives of *u* wrt *x:*.

*<sup>∂</sup>xi* <sup>þ</sup> *<sup>u</sup><sup>α</sup> i ∂ <sup>∂</sup>u<sup>α</sup>* <sup>þ</sup> *<sup>u</sup><sup>α</sup> ij ∂ ∂u<sup>α</sup> i*

are generated by the base vector

Anco [31].

defined by

where

*uα*

*uα*

*uα*

**92**

*<sup>i</sup>* <sup>¼</sup> *<sup>u</sup><sup>α</sup>*

*ij* <sup>¼</sup> *<sup>u</sup><sup>α</sup>*

*: :*

*: :*

*: :*

*<sup>i</sup>*1,*i*2, … ,*ik* <sup>¼</sup> *<sup>u</sup><sup>α</sup>*

*<sup>i</sup>* <sup>þ</sup> *εζ<sup>α</sup>*

*ij* <sup>þ</sup> *εζ<sup>α</sup>*

*<sup>i</sup>*1,*i*2, … ,*ik* <sup>þ</sup> *εζ<sup>α</sup>*

*Di* <sup>¼</sup> *<sup>∂</sup>*

**3.1 Brief account on lie symmetry methods**

*Heat Transfer - Design, Experimentation and Applications*

*3.1.1 m dependent and n independent variables*

In this subsection we provide a brief theory of Lie point symmetries. This discussion and further account can be found in the book of Bluman and

*<sup>m</sup>* dependent variables *<sup>u</sup>* <sup>¼</sup> *<sup>u</sup>*1, *<sup>u</sup>*2, … , *um* ð Þ and *<sup>n</sup>* independent variables *<sup>x</sup>* <sup>¼</sup> *<sup>x</sup>*1, *<sup>x</sup>*2, … , *xn* ð Þ, *<sup>u</sup>* <sup>¼</sup> *u x*ð Þ with *<sup>m</sup>* <sup>≥</sup> 2, arise in studying systems of differential equations. We consider extended transformations from ð Þ� *x*, *u* space to

� �� space. Here *<sup>u</sup>*ð Þ*<sup>k</sup>* denotes the components of all *<sup>k</sup>*th-order

*ii*1*i*<sup>2</sup> … *ik*

*ij* <sup>¼</sup> *<sup>∂</sup>*<sup>2</sup>

*<sup>u</sup><sup>α</sup>* <sup>¼</sup> *<sup>u</sup><sup>α</sup>* <sup>þ</sup> *εη<sup>α</sup>*ð Þþ *<sup>x</sup>*, *<sup>u</sup> <sup>O</sup> <sup>ε</sup>*<sup>2</sup> � �,

*∂*

which leave the system of equation in question invariant. These transformations

*<sup>∂</sup>xi* <sup>þ</sup> *ηα*ð Þ *<sup>x</sup>*, *<sup>u</sup>*

*<sup>i</sup>*1,*i*2, … ,*ik x*, *u*, *u*ð Þ<sup>1</sup> , *u*ð Þ<sup>2</sup> , … , *u*ð Þ*<sup>k</sup>*

Theorem 1.1 The extended infinitesimals satisfy the recursion relations

� � <sup>þ</sup> *<sup>O</sup> <sup>ε</sup>*<sup>2</sup> ð Þ,

*uα ∂xi*

ð Þþ *<sup>x</sup>*, *<sup>u</sup> <sup>O</sup> <sup>ε</sup>*<sup>2</sup> � �,

*∂ ∂u<sup>α</sup> i*1*i*<sup>2</sup> … *in*

*<sup>∂</sup><sup>x</sup> <sup>j</sup>* , etc*:*

*∂ <sup>∂</sup>u<sup>α</sup> :*

þ … , *i* ¼ 1, 2, … , *n:*

(14)

9

>>>>>>>>>>>>>=

(15)

>>>>>>>>>>>>>;

**definition** *Total derivative.* The total differentiation operator wrt *xi* is

<sup>þ</sup> … <sup>þ</sup> *<sup>u</sup><sup>α</sup>*

*<sup>∂</sup>xi* , *<sup>u</sup><sup>α</sup>*

We seek the one-parameter Lie group of transformations

*<sup>X</sup>* <sup>¼</sup> *<sup>ξ</sup><sup>i</sup>*

The *k*th-extended transformation of (14) are given by

*ij x*, *u*, *u*ð Þ<sup>1</sup> , *u*ð Þ<sup>2</sup>

*<sup>i</sup> x*, *u*, *u*ð Þ<sup>1</sup>

ð Þ *x*, *u*

� � <sup>þ</sup> *<sup>O</sup> <sup>ε</sup>*<sup>2</sup> ð Þ,

� � <sup>þ</sup> *<sup>O</sup> <sup>ε</sup>*<sup>2</sup> ð Þ,

*<sup>x</sup><sup>i</sup>* <sup>¼</sup> *<sup>x</sup><sup>i</sup>* <sup>þ</sup> *εξ<sup>i</sup>*

*uα <sup>i</sup>* <sup>¼</sup> *<sup>∂</sup>u<sup>α</sup>*

$$\begin{array}{lcl} \zeta\_i^a &= D\_i(\mathcal{W}^a) + \xi^j u\_{ji}^a, \\ \zeta\_{ij}^a &= D\_i D\_j(\mathcal{W}^a) + \xi^k u\_{kij}, \\ \cdot & \cdot \\ \cdot & \cdot \\ \cdot & \cdot \\ \cdot & \cdot \\ \zeta\_{i\_1 i\_2 \dots i\_k}^a &= D\_{i1} \dots D\_{i\_k}(\mathcal{W}^a) + \xi^j u\_{j i\_1 i\_2 \dots i\_k}. \end{array} \tag{17}$$

The corresponding (*k*th extended) infinitesimal generator is given by

$$X^{[k]} = \xi^i(\mathbf{x}, u) \frac{\partial}{\partial \mathbf{x}^i} + \eta^a\_i(\mathbf{x}, u) \frac{\partial}{\partial u^a} + \zeta^a\_i \frac{\partial}{\partial u\_i} + \dots + \zeta^a\_{i\_1, i\_2, \dots, i\_k} \frac{\partial}{\partial u\_{i\_1, i\_2, \dots, i\_k}}, \qquad k \ge 1.$$

Theorem 1.2 A differential function *F x*, *u*, *u*1, … , *u*ð Þ *<sup>p</sup>* � � *p*≥ 0, is a *p*th-order differential invariant of a group *G* if

$$F(\mathfrak{x}, u, u\_1, \dots, u\_{(p)}) = F(\overline{\mathfrak{x}}, \overline{u}, \overline{u\_1}, \dots, \overline{u}\_{(\overline{p})}) \dots$$

Theorem 1.3 A differential function *F x*, *u*, *u*1, … , *u*ð Þ *<sup>p</sup>* � � *p*≥ 0, is a *p*th-order differential invariant of a group *G* if

$$\mathbf{X}^{[p]}F = \mathbf{0},$$

where *X*½ � *<sup>p</sup>* is the *p*th prolongation of *X:*.

#### **3.2 Approximate methods**

#### *3.2.1 p-dimensional differential transform methods*

For an analytic multivariable function *f x*1, *x*2, … , *xp* � �, we have the *p*-dimensional transform given by

$$F(k\_1, k\_2, \dots, k\_p) = \frac{1}{k\_1! k\_2! \dots k\_p!} \left[ \frac{\partial^{k\_1 + k\_2 + \dots + k\_p} f(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_p)}{\partial \mathbf{x}\_1^{k\_1} \partial \mathbf{x}\_2^{k\_2} \dots \partial \mathbf{x}\_p^{k\_p}} \right] \bigg|\_{\substack{\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_p \text{ \rightarrow} \ \mathbf{0} \text{ \rightarrow} \mathbf{0}}} \tag{18}$$

The upper and lower case letters are for the transformed and the original functions respectively. The transformed function is also referred to as the T-function, the differential inverse transform is given by

$$f(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_p) = \sum\_{k\_1=0}^{\infty} \sum\_{k\_2=0}^{\infty} \dots \sum\_{k\_p=0}^{\infty} F(k\_1, k\_2, \dots, k\_p) \prod\_{l=1}^p \mathbf{x}\_l^{k\_l}.\tag{19}$$

*3.2.2 Variational iteration methods*

equation,

functional for Eq. (22) can be written as

*DOI: http://dx.doi.org/10.5772/intechopen.95490*

*θ*00

*θ <sup>j</sup>*þ<sup>1</sup>ð Þ¼ *x θ <sup>j</sup>*ð Þþ *x*

¼ 1 þ

*δθ <sup>j</sup>*þ<sup>1</sup>ð Þ¼ *x δθ <sup>j</sup>*ð Þþ *x δ*

obtained upon using *<sup>δ</sup>*~*<sup>θ</sup> <sup>j</sup>* <sup>¼</sup> 0 and *<sup>δ</sup>*~*<sup>θ</sup>*

*δ δθ <sup>j</sup>* ð*x* 0 *λ*ð Þ*t θ*<sup>00</sup> *j* ðÞþ*<sup>t</sup> <sup>a</sup>*~*<sup>θ</sup>* 0 *j*

> ð*x* 0 *λ*ð Þ*t θ*<sup>00</sup> *j* ðÞþ*<sup>t</sup> <sup>a</sup>*~*<sup>θ</sup>* 0 *j*

*δθ <sup>j</sup>*þ<sup>1</sup>ð Þ¼ *x δθ <sup>j</sup>*ð Þþ *x δ*

*δθ <sup>j</sup>*þ<sup>1</sup> ¼ *δθ <sup>j</sup>* þ *δλθ*<sup>0</sup>

*δθ <sup>j</sup>*þ<sup>1</sup> ¼ *δ* 1 � *λ*<sup>0</sup>

for this equation as follows,

variable *θ <sup>j</sup>* gives,

*δθ <sup>j</sup>*þ<sup>1</sup> *δθ <sup>j</sup>*

or equivalently

which gives

parts gives,

**95**

or equivalently

ð Þþ *x aθ*<sup>0</sup>

ð*x* 0 *λ*ð Þ*t θ*<sup>00</sup> *j* ð Þþ*<sup>t</sup> <sup>a</sup>*~*<sup>θ</sup>* 0 *j*

*θ <sup>j</sup>*þ1ð Þ¼ *x θ <sup>j</sup>*ð Þþ *x*

*Lθ* þ *Nθ* ¼ *g x*ð Þ, (22)

*<sup>λ</sup>*ð Þ*<sup>t</sup> <sup>L</sup><sup>θ</sup> <sup>j</sup>*ð Þþ*<sup>t</sup> <sup>N</sup>*~*θθ*ðÞ�*<sup>t</sup> g t*ð Þ � �*dt*, (23)

ðÞþ*<sup>t</sup> <sup>b</sup>*~*<sup>θ</sup> <sup>j</sup>*ðÞ�*<sup>t</sup> g t*ð Þ � �*dt:* (25)

ðÞþ*<sup>t</sup> <sup>b</sup>*~*<sup>θ</sup> <sup>j</sup>*ð Þ�*<sup>t</sup> g t*ð Þ � �*dt* � �, (26)

ð Þþ*<sup>t</sup> <sup>b</sup>*~*<sup>θ</sup> <sup>j</sup>*ðÞ�*<sup>t</sup> g t*ð Þ � �*dt* � �, (27)

*<sup>j</sup>* ¼ 0. Evaluating the integral of Eq. (28) by

*θ jdt*, (29)

*θ jdt:* (30)

ð*x* 0 *λ*ð Þ*t θ*<sup>00</sup> *j* ð Þ*<sup>t</sup> dt* � �,

�

*<sup>j</sup>* � *δλ*<sup>0</sup>

*θ <sup>j</sup>* þ *δ* ð*x* 0 *λ*00

> *<sup>j</sup>* þ *δ* ð*x* 0 *λ*00

0

j *t*¼*x* � �*<sup>θ</sup> <sup>j</sup>* <sup>þ</sup> *δλθ*<sup>0</sup> ð Þ¼ 0 *β*, (24)

(28)

where *L* and *N* are linear and nonlinear operators, respectively, and *g x*ð Þ is the source inhomogeneous term. He [33], proposed the VIM where a correctional

*Survey of Some Exact and Approximate Analytical Solutions for Heat Transfer in Extended…*

ð Þþ *x bθ*ð Þ¼ *x g x*ð Þ, *θ*ð Þ¼ 0 *α*, *θ*<sup>0</sup>

where *a* and *b* are constants. The VIM admits the use of a correctional function

Taking the variation on both sides of Eq. (25) with respect to the independent

ð*x* 0

where *λ* is the general Lagrange multiplier, which can be be identified optimally via the variation theory, and ~*θ<sup>n</sup>* is a restricted variation, which means *<sup>δ</sup>*~*θ<sup>n</sup>* <sup>¼</sup> 0 [34]. The Lagrange multiplier can be a constant or a function depending on the order of the deferential equation under consideration. The VIM should be employed by following two essential steps. First we determine the Lagrange multiplier by considering the following second order differential

It can easily be deduced that the substitution of (18) into (19) gives the Taylor series expansion of the function *f x*1, *x*2, … , *xp* � � about the point *x*1, *x*2, … , *xp* � � <sup>¼</sup>.

ð Þ 0, 0, … , 0 . This is given by

$$f\left(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_p\right) = \sum\_{k\_1=0}^{\infty} \sum\_{k\_2=0}^{\infty} \dots \sum\_{k\_p=0}^{\infty} \frac{\prod\_{l=1}^p \mathbf{x}\_l^{k\_l}}{k\_1! k\_2! \dots k\_p!} \left[ \frac{\partial^{k\_1+k\_2+\dots+k\_p} f\left(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_p\right)}{\partial \mathbf{x}\_1^{k\_1} \partial \mathbf{x}\_2^{k\_2} \dots \partial \mathbf{x}\_p^{k\_p}} \right] \bigg|\_{\mathbf{x}\_1=0, \dots, \mathbf{x}\_p=0} \tag{20}$$

For real world applications the function *f x*1, *x*2, … , *xp* � � is given in terms of a finite series for some *q*,*r*, *s*∈ ℤ. Then (19)becomes

$$f\left(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_p\right) = \sum\_{k\_1=0}^q \sum\_{k\_2=0}^r \dots \sum\_{k\_p=0}^s F(k\_1, k\_2, \dots, k\_p) \prod\_{l=1}^p \mathbf{x}\_l^{k\_l}.\tag{21}$$

We now give some important operations and theorems performed in the *p*dimensional DTM in **Table 1**. Those have been derived using the definition in (18) together with previously obtained results [32].

In the table

$$\delta(k\_1 - e\_1, k\_2 - e\_2, \dots, k\_p - e\_p) = \begin{cases} 1 & \text{if } \ k\_i = e\_i \text{ for } i = 1, 2, \dots, p \\ 0 & \text{otherwise.} \end{cases}$$

We now provide one result of the *p*-dimensional DTM without proof. **Theorem**. Proof in [32].

If

$$f\left(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_p\right) = \mathbf{g}\left(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_p\right) h\left(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_p\right),$$

then

$$F(k\_1, k\_2, \dots, k\_p) = \sum\_{i\_1=0}^{k\_1} \sum\_{i\_2=0}^{k\_2} \dots \sum\_{i\_p=0}^{k\_p} G(k\_1, k\_2, \dots, k\_p + i\_p) H(k\_1 + i\_1, \dots, k\_{n-1} + i\_{p-1}, k\_p) \dots$$


**Table 1.**

*Theorems and operations performed in p-dimensional DTM.*

*Survey of Some Exact and Approximate Analytical Solutions for Heat Transfer in Extended… DOI: http://dx.doi.org/10.5772/intechopen.95490*

#### *3.2.2 Variational iteration methods*

The upper and lower case letters are for the transformed and the original functions respectively. The transformed function is also referred to as the T-function,

> … <sup>X</sup><sup>∞</sup> *kp*¼0

It can easily be deduced that the substitution of (18) into (19) gives the Taylor

*F k*1, *k*2, … , *kp* � �Y

� � about the point *x*1, *x*2, … , *xp*

*∂x<sup>k</sup>*<sup>1</sup> <sup>1</sup> *<sup>∂</sup>xk*<sup>2</sup> <sup>2</sup> … *<sup>∂</sup><sup>x</sup> kp p*

*F k*1, *k*2, … , *kp* � �Y

*∂<sup>k</sup>*1þ*k*2<sup>þ</sup> … <sup>þ</sup>*kp f x*1, *x*2, … , *xp*

" #�

*p*

*l*¼1 *x kl*

� �

� � is given in terms of a

*p*

*l*¼1 *x kl*

*<sup>l</sup> :* (19)

� � <sup>¼</sup>.

� � � �

*<sup>l</sup> :* (21)

*x*1¼0, … ,*xp*¼0

(20)

*:*

*k*1¼0

Q*<sup>p</sup> <sup>l</sup>*¼<sup>1</sup>*<sup>x</sup> kl l k*1!*k*2! … *kp*!

X∞ *k*2¼0

the differential inverse transform is given by

*Heat Transfer - Design, Experimentation and Applications*

series expansion of the function *f x*1, *x*2, … , *xp*

X∞ *k*2¼0

finite series for some *q*,*r*, *s*∈ ℤ. Then (19)becomes

… <sup>X</sup><sup>∞</sup> *kp*¼0

For real world applications the function *f x*1, *x*2, … , *xp*

*q*

X*r k*2¼0 … <sup>X</sup>*<sup>s</sup> kp*¼0

We now give some important operations and theorems performed in the *p*dimensional DTM in **Table 1**. Those have been derived using the definition in (18)

� � <sup>¼</sup> 1 if *ki* <sup>¼</sup> *ei* for *<sup>i</sup>* <sup>¼</sup> 1, 2, *::*, *<sup>p</sup>:*

*G k*1, *k*2, … , *kp* þ *ip*

� � **T-function** *F k***<sup>1</sup> ,** *k***2, … ,** *kp*

� � *F k*<sup>1</sup> , *k*2, … , *kp*

� � *F k*<sup>1</sup> , *k*2, … , *kp*

*<sup>l</sup> F k*<sup>1</sup> , *k*2, … , *kp*

We now provide one result of the *p*-dimensional DTM without proof.

� � <sup>¼</sup> *g x*1, *<sup>x</sup>*2, … , *xp*

�

0 otherwise*:*

� �*h x*1, *x*2, … , *xp*

� �,

� �*H k*<sup>1</sup> <sup>þ</sup> *<sup>i</sup>*1, … , *kn*�<sup>1</sup> <sup>þ</sup> *ip*�1, *kp*

� �

� � <sup>¼</sup> *<sup>δ</sup> <sup>k</sup>*<sup>1</sup> � *<sup>e</sup>*1, *<sup>k</sup>*<sup>2</sup> � *<sup>e</sup>*2, … , *kp* � *ep*

� �

� � <sup>¼</sup> *<sup>λ</sup>G k*1, *<sup>k</sup>*2, … , *kp*

� � <sup>¼</sup> *G k*<sup>1</sup> , *<sup>k</sup>*2, … , *kp*

� � <sup>¼</sup> ð Þ *<sup>k</sup>*1þ*r*<sup>1</sup> ! … ð Þ *kp*þ*rp* !

*F k*<sup>1</sup> , *k*2, … , *kp*

� �*:*

� � � *P k*1, *<sup>k</sup>*2, … , *kp*

� �

*<sup>k</sup>*<sup>1</sup> ! … *kp* ! *k*<sup>1</sup> þ *r*<sup>1</sup> , … , *kp* þ *rp* � �

� �

*k*1¼0

*f x*1, *x*2, … , *xp* � � <sup>¼</sup> <sup>X</sup><sup>∞</sup>

ð Þ 0, 0, … , 0 . This is given by

*k*1¼0

*f x*1, *x*2, … , *xp* � � <sup>¼</sup> <sup>X</sup>

together with previously obtained results [32].

*δ k*<sup>1</sup> � *e*1, *k*<sup>2</sup> � *e*2, … , *kp* � *ep*

*f x*1, *x*2, … , *xp*

X *k*2

… X *kp*

*ip*¼0

*i*2¼0

� � � *p x*<sup>1</sup> , *<sup>x</sup>*2, … , *xp*

*Theorems and operations performed in p-dimensional DTM.*

*k*1

*i*1¼0

**Theorem**. Proof in [32].

**Original function** *f x***1,** *x***2, … ,** *xp*

� � <sup>¼</sup> *<sup>λ</sup>g x*1, *<sup>x</sup>*2, … , *xp*

� � <sup>¼</sup> *g x*1, *<sup>x</sup>*2, … , *xp*

� � <sup>¼</sup> *<sup>∂</sup>r*<sup>1</sup> <sup>þ</sup>*r*2<sup>þ</sup> … <sup>þ</sup>*rp g x*ð Þ <sup>1</sup> , *<sup>x</sup>*<sup>2</sup> , … , *xp ∂x r*1 <sup>1</sup> *<sup>∂</sup><sup>x</sup> r*2 <sup>2</sup> … *<sup>∂</sup><sup>x</sup> rp p*

*<sup>l</sup>*¼<sup>1</sup>*xel*

*f x*1, *x*2, … , *xp* � � <sup>¼</sup> <sup>X</sup><sup>∞</sup>

In the table

If

then

*F k*1, *k*2, … , *kp* � � <sup>¼</sup> <sup>X</sup>

*f x*1, *x*<sup>2</sup> , … , *xp*

*f x*1, *x*<sup>2</sup> , … , *xp*

*f x*1, *x*<sup>2</sup> , … , *xp*

*f x*1, *x*<sup>2</sup> , … , *xp* � � <sup>¼</sup> <sup>Q</sup>*<sup>p</sup>*

**Table 1.**

**94**

$$L\theta + N\theta = \mathbf{g}(\mathbf{x}),\tag{22}$$

where *L* and *N* are linear and nonlinear operators, respectively, and *g x*ð Þ is the source inhomogeneous term. He [33], proposed the VIM where a correctional functional for Eq. (22) can be written as

$$\theta\_{j+1}(\mathbf{x}) = \theta\_j(\mathbf{x}) + \int\_0^\mathbf{x} \lambda(t) \left( L\theta\_j(t) + N\ddot{\theta}\_\theta(t) - \mathbf{g}(t) \right) dt,\tag{23}$$

where *λ* is the general Lagrange multiplier, which can be be identified optimally via the variation theory, and ~*θ<sup>n</sup>* is a restricted variation, which means *<sup>δ</sup>*~*θ<sup>n</sup>* <sup>¼</sup> 0 [34]. The Lagrange multiplier can be a constant or a function depending on the order of the deferential equation under consideration. The VIM should be employed by following two essential steps. First we determine the Lagrange multiplier by considering the following second order differential equation,

$$
\theta'(\mathbf{x}) + a\theta'(\mathbf{x}) + b\theta(\mathbf{x}) = \mathbf{g}(\mathbf{x}), \theta(\mathbf{0}) = a, \theta'(\mathbf{0}) = \beta,\tag{24}
$$

where *a* and *b* are constants. The VIM admits the use of a correctional function for this equation as follows,

$$\theta\_{j+1}(\mathbf{x}) = \theta\_j(\mathbf{x}) + \int\_0^\mathbf{x} \lambda(t) \left( \theta\_j'(t) + a\tilde{\theta}\_j'(t) + b\tilde{\theta}\_j(t) - \mathbf{g}(t) \right) dt. \tag{25}$$

Taking the variation on both sides of Eq. (25) with respect to the independent variable *θ <sup>j</sup>* gives,

$$\frac{\delta\theta\_{j+1}}{\delta\theta\_{j}} = \mathbf{1} + \frac{\delta}{\delta\theta\_{j}} \left( \int\_{0}^{\mathbf{x}} \lambda(\mathbf{t}) \Big(\boldsymbol{\theta}\_{j}^{\prime}(\mathbf{t}) + a\tilde{\boldsymbol{\theta}}\_{j}^{\prime}(\mathbf{t}) + b\tilde{\boldsymbol{\theta}}\_{j}(\mathbf{t}) - \mathbf{g}(\mathbf{t})\Big) d\mathbf{t} \right), \tag{26}$$

or equivalently

$$
\delta\theta\_{j+1}(\mathbf{x}) = \delta\theta\_j(\mathbf{x}) + \delta\left(\int\_0^\mathbf{x} \lambda(\mathbf{t}) \left(\boldsymbol{\theta}\_j^{\prime}(\mathbf{t}) + a\tilde{\theta}\_j^{\prime}(\mathbf{t}) + b\tilde{\theta}\_j(\mathbf{t}) - \mathbf{g}(\mathbf{t})\right) d\mathbf{t}\right), \tag{27}
$$

which gives

$$
\delta\theta \,\_{j+1}(\mathbf{x}) = \delta\theta \,\_{j}(\mathbf{x}) + \delta \left( \int\_{0}^{\mathbf{x}} \lambda(t) \left( \theta \,\_{j}^{\prime}(t) dt \right) , \tag{28}
$$

obtained upon using *<sup>δ</sup>*~*<sup>θ</sup> <sup>j</sup>* <sup>¼</sup> 0 and *<sup>δ</sup>*~*<sup>θ</sup>* 0 *<sup>j</sup>* ¼ 0. Evaluating the integral of Eq. (28) by parts gives,

$$
\delta\theta \,\,\dot{\theta}\_{j+1} = \delta\theta \,\_j + \delta\lambda \theta'\_j - \delta\lambda' \theta\_j + \delta\int\_0^\infty \lambda'^\prime \theta\_j dt,\tag{29}
$$

or equivalently

$$
\delta\theta \,\_{j+1} = \delta \left( \mathbf{1} - \lambda' \vert\_{t=\mathbf{x}} \right) \theta\_j + \delta\lambda \theta'\_j + \delta \int\_0^\mathbf{x} \lambda' \theta\_j dt. \tag{30}
$$

The extreme condition of *θ <sup>j</sup>*þ<sup>1</sup> requires that *δθ <sup>j</sup>*þ<sup>1</sup> ¼ 0. Equating both sides of Eq. (30) to 0, yields the following stationary conditions

$$\left.\mathbf{1} - \lambda'\right|\_{t=\mathbf{x}} = \mathbf{0},\tag{31}$$

and hence we start the double reduction first with *X*<sup>1</sup> which implies ð Þ *τ*,*r*, *θ* are

*∂ ∂r* þ 2*F n*

*n*2 � �*g*ð Þ*<sup>γ</sup> <sup>n</sup>*þ<sup>1</sup>

*n*2 � �*<sup>τ</sup>* <sup>þ</sup> *<sup>c</sup>*<sup>1</sup> � ��1*=<sup>n</sup>*

The difficulty for group-invariant solutions is the satisfaction of the imposed or prescribed boundary conditions. This has been seen in two dimensional steady state problems [7, 16], and 1 + 1 D transient problems [25]. Perhaps the most successful attempt in in [26]. For nonlinear steady state problems, some transformation such as Kirchoff [7, 16], may linearise the two dimensional problems which then becomes easier to solve using standard methods. Linearisation of nonlinear steady state one dimensional problems is possible when thermal conductivity is a differ-

In [5], preliminary group classification is invoked to determine the thermal conductivity which lead to exact solutions. It turned out that given a power law heat transfer coefficient, thermal conductivity also takes the power law form. Given Eq. (12) with both *<sup>k</sup>*ð Þ*<sup>θ</sup>* and *<sup>h</sup>*ð Þ*<sup>θ</sup>* given by *<sup>θ</sup><sup>n</sup>* then one obtains the solution

*cosh* ffiffiffiffiffiffiffiffiffiffiffi

*cosh* ffiffiffiffiffiffiffiffiffiffiffi

The expressions for fin efficiency and effectiveness can be explicit in this case. Furthermore, this solution led to the benchmarking of the approximate analytical solutions [35]. With established confidence in approximate methods, then one may

In this subsection we consider heat transfer in a cylindrical pin fin. We consider thermal conductivity given as a linear function of temperature 1 þ *βθ* and a power

*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> <sup>p</sup> *Mx* � �

*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> <sup>p</sup> *<sup>M</sup>* � � " #<sup>1</sup>*=*ð Þ *<sup>n</sup>*þ<sup>1</sup>

ð Þ¼ *<sup>γ</sup>* <sup>4</sup>*E*<sup>2</sup> *<sup>n</sup>* <sup>þ</sup> <sup>1</sup>

In terms of original variables one obtains the general exact solution

*<sup>n</sup>=*<sup>2</sup> �4*E*<sup>2</sup> *<sup>n</sup>* <sup>þ</sup> <sup>1</sup>

*rF<sup>n</sup>* ð Þ *Fr :*

*∂ ∂F :*

*:*

*:*

*:* (38)

invariants and leads to a steady state problem. Hence writing *θ* ¼ *F*ð Þ *τ*,*r* and

*Survey of Some Exact and Approximate Analytical Solutions for Heat Transfer in Extended…*

*<sup>F</sup><sup>τ</sup>* <sup>¼</sup> *<sup>E</sup>*<sup>2</sup> <sup>1</sup> *r ∂ ∂r*

*X*<sup>∗</sup> <sup>2</sup> ¼ *r*

This symmetry generator leads to the first order ODE

*g*0

*θ τ*ð Þ¼ ,*r r*

ential consequence of heat transfer coefficient [5].

*θ*ð Þ¼ *x*

solve other problems that are challenging to solve exactly.

**4.2 Some approximate solutions**

*4.2.1 Three dimensional DTM*

**97**

substitute in the original equation, one obtains

*DOI: http://dx.doi.org/10.5772/intechopen.95490*

*X*<sup>2</sup> becomes

*4.1.2 Example 2*

$$
\left.\lambda\right|\_{t=\mathbf{x}} = \mathbf{0},
\tag{32}
$$

$$\left.\lambda^{\prime\prime}\right|\_{t=\pm} = \mathbf{0}.\tag{33}$$

This in turn gives

$$
\lambda = t - \mathfrak{x}.\tag{34}
$$

In general, for the *nth* order ordinary differential equation, the Lagrange multiplier is given by,

$$
\lambda = \frac{(-1)^j}{(j-1)!}(t-x)^{j-1}.\tag{35}
$$

Having determined *λ* and substituting its value into (23) gives the iteration formula

$$
\theta\_{j+1}(\mathbf{x}) = \theta\_j(\mathbf{x}) + \int\_0^\mathbf{x} (t - \mathbf{x}) \left( \theta\_j'(t) + a\theta\_j'(t) + b\theta\_j(t) - \mathbf{g}(t) \right) dt,\tag{36}
$$

The iteration formula Eq. (36), without restricted variation, should be used for the determination of the successive approximations *θ <sup>j</sup>*þ<sup>1</sup>ð Þ *x* , *j*⩾0, of the solution *θ*ð Þ *x* . Consequently, the solution is given by

$$\theta(\mathbf{x}) = \lim\_{j \to \infty} \theta\_j(\mathbf{x}). \tag{37}$$

#### **4. Survey of some solutions**

In this section we demonstrate the challenge in the construction of exact solution for heat transfer in pin fin. Also, we consider the work in [5].

#### **4.1 Some exact solutions**

#### *4.1.1 Example 1*

Given the power law thermal conductivity in heat transfer through pin fins, that is in Eq. (3) *<sup>k</sup>*ð Þ¼ *<sup>θ</sup> <sup>θ</sup><sup>n</sup>* . The model admits four finite symmetry generators. Amongst the others, the two dimensional Lie subalgebra is given by

$$X\_1 = \frac{\partial}{\partial \mathbf{z}} , \quad X\_2 = z \frac{\partial}{\partial \mathbf{z}} + r \frac{\partial}{\partial r} + \frac{2\theta}{n} \frac{\partial}{\partial \theta}$$

Notice that

$$[X\_1, X\_2] = X\_1, \dots$$

*Survey of Some Exact and Approximate Analytical Solutions for Heat Transfer in Extended… DOI: http://dx.doi.org/10.5772/intechopen.95490*

and hence we start the double reduction first with *X*<sup>1</sup> which implies ð Þ *τ*,*r*, *θ* are invariants and leads to a steady state problem. Hence writing *θ* ¼ *F*ð Þ *τ*,*r* and substitute in the original equation, one obtains

$$F\_{\tau} = E^2 \frac{1}{r} \frac{\partial}{\partial r} (r F^n F\_r).$$

*X*<sup>2</sup> becomes

The extreme condition of *θ <sup>j</sup>*þ<sup>1</sup> requires that *δθ <sup>j</sup>*þ<sup>1</sup> ¼ 0. Equating both sides of

In general, for the *nth* order ordinary differential equation, the Lagrange multi-

Having determined *λ* and substituting its value into (23) gives the iteration

The iteration formula Eq. (36), without restricted variation, should be used for the determination of the successive approximations *θ <sup>j</sup>*þ<sup>1</sup>ð Þ *x* , *j*⩾0, of the solution

In this section we demonstrate the challenge in the construction of exact solution

Given the power law thermal conductivity in heat transfer through pin fins, that

*∂ ∂z* þ *r ∂ ∂r* þ 2*θ n ∂ ∂θ*

½ �¼ *X*1, *X*<sup>2</sup> *X*1,

, *X*<sup>2</sup> ¼ *z*

. The model admits four finite symmetry generators. Amongst

*j* ðÞþ*t aθ*<sup>0</sup> *j*

*θ*ð Þ¼ *x* lim

ð Þ *<sup>t</sup>* � *<sup>x</sup> <sup>j</sup>*�<sup>1</sup>

� �

*<sup>t</sup>*¼*<sup>x</sup>* <sup>¼</sup> 0, (31)

*<sup>t</sup>*¼*<sup>x</sup>* ¼ 0, (32)

*<sup>t</sup>*¼*<sup>x</sup>* <sup>¼</sup> <sup>0</sup>*:* (33)

*λ* ¼ *t* � *x:* (34)

ð Þþ*t bθ <sup>j</sup>*ðÞ�*t g t*ð Þ

*<sup>j</sup>*!<sup>∞</sup> *<sup>θ</sup> <sup>j</sup>*ð Þ *<sup>x</sup> :* (37)

*:* (35)

*dt*, (36)

1 � *λ*<sup>0</sup> j

*λ*j

*λ*00 � �

*<sup>λ</sup>* <sup>¼</sup> ð Þ �<sup>1</sup> *<sup>j</sup>* ð Þ *j* � 1 !

ð Þ *t* � *x θ*<sup>00</sup>

for heat transfer in pin fin. Also, we consider the work in [5].

the others, the two dimensional Lie subalgebra is given by

*<sup>X</sup>*<sup>1</sup> <sup>¼</sup> *<sup>∂</sup> ∂z*

Eq. (30) to 0, yields the following stationary conditions

*Heat Transfer - Design, Experimentation and Applications*

This in turn gives

*θ <sup>j</sup>*þ<sup>1</sup>ð Þ¼ *x θ <sup>j</sup>*ð Þþ *x*

**4. Survey of some solutions**

**4.1 Some exact solutions**

*4.1.1 Example 1*

is in Eq. (3) *<sup>k</sup>*ð Þ¼ *<sup>θ</sup> <sup>θ</sup><sup>n</sup>*

Notice that

**96**

*θ*ð Þ *x* . Consequently, the solution is given by

ð*x* 0

plier is given by,

formula

$$X\_2^\* = r\frac{\partial}{\partial r} + \frac{2F}{n}\frac{\partial}{\partial F}.$$

This symmetry generator leads to the first order ODE

$$\mathbf{g}'(\boldsymbol{\gamma}) = 4E^2 \left( \frac{n+1}{n^2} \right) \mathbf{g}(\boldsymbol{\gamma})^{n+1}.$$

In terms of original variables one obtains the general exact solution

$$\theta(\mathbf{r}, r) = r^{n/2} \left[ -4E^2 \left( \frac{n+1}{n^2} \right) \mathbf{r} + c\_1 \right]^{-1/n}.$$

The difficulty for group-invariant solutions is the satisfaction of the imposed or prescribed boundary conditions. This has been seen in two dimensional steady state problems [7, 16], and 1 + 1 D transient problems [25]. Perhaps the most successful attempt in in [26]. For nonlinear steady state problems, some transformation such as Kirchoff [7, 16], may linearise the two dimensional problems which then becomes easier to solve using standard methods. Linearisation of nonlinear steady state one dimensional problems is possible when thermal conductivity is a differential consequence of heat transfer coefficient [5].

#### *4.1.2 Example 2*

In [5], preliminary group classification is invoked to determine the thermal conductivity which lead to exact solutions. It turned out that given a power law heat transfer coefficient, thermal conductivity also takes the power law form. Given Eq. (12) with both *<sup>k</sup>*ð Þ*<sup>θ</sup>* and *<sup>h</sup>*ð Þ*<sup>θ</sup>* given by *<sup>θ</sup><sup>n</sup>* then one obtains the solution

$$\theta(\mathbf{x}) = \left[ \frac{\cosh\left(\sqrt{n+1}\text{ }\mathbf{Mx}\right)}{\cosh\left(\sqrt{n+1}\text{ }\mathbf{M}\right)} \right]^{1/(n+1)}.\tag{38}$$

The expressions for fin efficiency and effectiveness can be explicit in this case. Furthermore, this solution led to the benchmarking of the approximate analytical solutions [35]. With established confidence in approximate methods, then one may solve other problems that are challenging to solve exactly.

#### **4.2 Some approximate solutions**

#### *4.2.1 Three dimensional DTM*

In this subsection we consider heat transfer in a cylindrical pin fin. We consider thermal conductivity given as a linear function of temperature 1 þ *βθ* and a power

law heat transfer coefficient. The three dimensional DTM solution of Eq. (2) is given by

$$\begin{aligned} \theta(\mathbf{r}, \mathbf{r}, \mathbf{z}) &= c\mathbf{r} + c\mathbf{r}\mathbf{r} + c\tau\mathbf{r}^2 + c\tau\mathbf{r}^3 + c\tau\mathbf{r}^4 + c\tau\mathbf{r}^5 + c\tau\mathbf{r}^6 + c\tau\mathbf{r}^7 + \dots \ \dots \\ &+ c\tau\mathbf{z}^2 - \frac{Bic^{m+1}}{(1+\beta c)}\tau\mathbf{r}\mathbf{z}^2 - \frac{5c}{2E}\tau\mathbf{r}^2\mathbf{z}^2 + \frac{10Bic^{m+1}}{9E^2(1+\beta c)}\tau r^3\mathbf{z}^2 + \dots \ \dots \\ &+ c\tau\mathbf{z}^3 - \frac{Bic^{m+1}}{(1+\beta c)}\tau\mathbf{r}\mathbf{z}^3 - \frac{9c}{2E^2}\tau\mathbf{r}^2\mathbf{z}^3 + \frac{2Bic^{m+1}}{E^2(1+\beta c)}\tau r^3\mathbf{z}^3 + \dots \ \dots \\ &\vdots \end{aligned} \tag{39}$$

and *c* is obtained from

3*c*2*M*<sup>2</sup> *x*2 <sup>2</sup> � <sup>3</sup>*c*5*M*<sup>2</sup>

*<sup>r</sup>* <sup>þ</sup> … <sup>þ</sup> *<sup>c</sup>* � *Bicm*þ<sup>1</sup>

This solution is plotted in **Figure 3**

*DOI: http://dx.doi.org/10.5772/intechopen.95490*

The DTM solution is given by (see [35])

These solutions are depicted in **Figure 4**.

3*c*<sup>2</sup>*M*<sup>2</sup> *x*2 <sup>2</sup> � *<sup>c</sup>*<sup>2</sup>*M*<sup>2</sup>

1 þ *βc*

*x*2 2 þ

> *x*2 6 þ

*r* �

*4.2.3 Comparison of one dimensional exact, DTM and VIM solutions*

namely the exact solution given in Eq. (38). The VIM solutions is given by

*c*7*M*<sup>2</sup> *x*2 2 þ

<sup>12</sup>*<sup>c</sup>* <sup>þ</sup> <sup>18</sup>*βc*<sup>2</sup> <sup>þ</sup> <sup>4</sup>*E*<sup>2</sup>

*Survey of Some Exact and Approximate Analytical Solutions for Heat Transfer in Extended…*

Here the solutions for the one dimensional heat transfer problems are compared,

The constant *c* may be obtained using the boundary condition at the fin base.

*<sup>c</sup>*<sup>2</sup>*M*<sup>2</sup> <sup>3</sup> � <sup>4</sup>*M*<sup>2</sup> *c x*<sup>4</sup> 48 þ

Likewise, the constant *c* is obtained using the boundary conditions.

*Approximate analytical solutions for a two-dimensional cylindrical spine fin with a constant thermal conductivity (β* ¼ 0*) for τ* ¼ 0*:*4*. The parameters are set such that E* ¼ 2*, Bi* ¼ 0*:*2*, and m* ¼ 3*. (see also, [8]).*

*c*5*M*4*x*<sup>4</sup>

4*E*<sup>2</sup>

ð Þ 1 þ *βc*

<sup>8</sup> � <sup>3</sup>*c*7*M*4*x*<sup>4</sup>

2 þ

*<sup>c</sup>*<sup>2</sup>*M*<sup>2</sup> <sup>1</sup> � <sup>16</sup>*M*<sup>2</sup> *c x*<sup>5</sup>

*<sup>β</sup>c*<sup>2</sup> <sup>þ</sup> <sup>4</sup>*E*2*βBi*2*c*2*m*þ<sup>2</sup> ð Þ <sup>1</sup>þ*β<sup>c</sup>* <sup>2</sup>

*r*

5*c*9*M*4*x*<sup>4</sup>

<sup>2</sup> <sup>þ</sup> … <sup>¼</sup> <sup>1</sup>*:*

<sup>4</sup> � *<sup>c</sup>*11*M*4*x*<sup>4</sup> 4

<sup>240</sup> � <sup>⋯</sup>

(41)

(42)

*<sup>c</sup>* � *Bicm*þ<sup>1</sup> 1 þ *βc*

*θ*ð Þ¼ *x c* þ

þ ⋯

*θ*ð Þ¼ *x c* þ

**Figure 3.**

**99**

One may determine the value of *c* by invoking the boundary at the base of the fin, as such

$$\begin{aligned} c\tau + c\tau r + c\tau r^2 + \dots + c\tau - \frac{Bic^{m+1}}{(1+\beta c)}\tau r - \frac{5c}{2E^2}\tau r^2 \dots + c\tau - \frac{Bic^{m+1}}{(1+\beta c)}\tau r - \frac{9c}{2E^2}\tau r^2 + \dots \\ = 1. \end{aligned}$$

To plot a three dimensional figure for this solution one may fix temperature, say at *τ* ¼ 0*:*4 The results are shown in **Figure 2**.

#### *4.2.2 Two dimensional DTM*

The two dimensional DTM solution for a steady heat transfer through the cylindrical fin is given by

$$\theta(r, z) = \left[ \begin{array}{c} -\operatorname{Bi}^{m+1} \\ \mathbf{1} + \beta \varepsilon \end{array} r + \dots + \operatorname{cz}^2 - \frac{\operatorname{Bi}^{m+1}}{\mathbf{1} + \beta \varepsilon} \mathbf{z}^2 r + \dots + \operatorname{cz}^3 - \frac{\operatorname{Bi}^{m+1}}{\mathbf{1} + \beta \varepsilon} \mathbf{z}^3 r + \dots \right] \tag{40}$$

#### **Figure 2.**

*Approximate analytical solutions for a two-dimensional cylindrical spine fin with a constant thermal conductivity (β* ¼ 0*) for τ* ¼ 0*:*4*. The parameters are set such that E* ¼ 2*, Bi* ¼ 0*:*2*, and m* ¼ 3*. (see also, [8]).*

*Survey of Some Exact and Approximate Analytical Solutions for Heat Transfer in Extended… DOI: http://dx.doi.org/10.5772/intechopen.95490*

and *c* is obtained from

law heat transfer coefficient. The three dimensional DTM solution of Eq. (2) is

<sup>3</sup> <sup>þ</sup> *<sup>c</sup>τ<sup>r</sup>*

ð Þ <sup>1</sup> <sup>þ</sup> *<sup>β</sup><sup>c</sup> <sup>τ</sup><sup>r</sup>* � <sup>5</sup>*<sup>c</sup>*

*<sup>r</sup>* <sup>þ</sup> … <sup>þ</sup> *cz*<sup>2</sup> � *Bic<sup>m</sup>*þ<sup>1</sup>

*Approximate analytical solutions for a two-dimensional cylindrical spine fin with a constant thermal conductivity (β* ¼ 0*) for τ* ¼ 0*:*4*. The parameters are set such that E* ¼ 2*, Bi* ¼ 0*:*2*, and m* ¼ 3*. (see also, [8]).*

<sup>2</sup>*E*<sup>2</sup> *<sup>τ</sup><sup>r</sup>* 2 *<sup>z</sup>*<sup>2</sup> <sup>þ</sup>

<sup>2</sup>*E*<sup>2</sup> *<sup>τ</sup><sup>r</sup>* 2 *<sup>z</sup>*<sup>3</sup> <sup>þ</sup>

One may determine the value of *c* by invoking the boundary at the base of the

<sup>2</sup>*E*<sup>2</sup> *<sup>τ</sup><sup>r</sup>*

To plot a three dimensional figure for this solution one may fix temperature, say

The two dimensional DTM solution for a steady heat transfer through the cylin-

1 þ *βc*

*z*2

<sup>4</sup> <sup>þ</sup> *<sup>c</sup>τ<sup>r</sup>*

<sup>5</sup> <sup>þ</sup> *<sup>c</sup>τ<sup>r</sup>*

9*E*<sup>2</sup>

*E*2

<sup>6</sup> <sup>þ</sup> *<sup>c</sup>τ<sup>r</sup>*

ð Þ <sup>1</sup> <sup>þ</sup> *<sup>β</sup><sup>c</sup> <sup>τ</sup><sup>r</sup>*

ð Þ <sup>1</sup> <sup>þ</sup> *<sup>β</sup><sup>c</sup> <sup>τ</sup><sup>r</sup>*

<sup>2</sup> … <sup>þ</sup> *<sup>c</sup><sup>τ</sup>* � *Bic<sup>m</sup>*þ<sup>1</sup>

*<sup>r</sup>* <sup>þ</sup> … <sup>þ</sup> *cz*<sup>3</sup> � *Bic<sup>m</sup>*þ<sup>1</sup>

1 þ *βc*

*z*3 *r* þ …

(40)

10*Bicm*þ<sup>1</sup>

2*Bicm*þ<sup>1</sup>

<sup>7</sup> <sup>þ</sup> *:* … …

*<sup>z</sup>*<sup>2</sup> <sup>þ</sup> … …

(39)

<sup>2</sup> <sup>þ</sup> …

*<sup>z</sup>*<sup>3</sup> <sup>þ</sup> … …

ð Þ <sup>1</sup> <sup>þ</sup> *<sup>β</sup><sup>c</sup> <sup>τ</sup><sup>r</sup>* � <sup>9</sup>*<sup>c</sup>*

<sup>2</sup>*E*<sup>2</sup> *<sup>τ</sup><sup>r</sup>*

3

3

<sup>2</sup> <sup>þ</sup> *<sup>c</sup>τ<sup>r</sup>*

ð Þ <sup>1</sup> <sup>þ</sup> *<sup>β</sup><sup>c</sup> <sup>τ</sup>rz*<sup>2</sup> � <sup>5</sup>*<sup>c</sup>*

ð Þ <sup>1</sup> <sup>þ</sup> *<sup>β</sup><sup>c</sup> <sup>τ</sup>rz*<sup>3</sup> � <sup>9</sup>*<sup>c</sup>*

given by

fin, as such

¼ 1*:*

**Figure 2.**

**98**

*cτ* þ *cτr* þ *cτr*

*θ τ*ð Þ¼ ,*r*, *z cτ* þ *cτr* þ *cτr*

⋮

*4.2.2 Two dimensional DTM*

*<sup>θ</sup>*ð Þ¼ *<sup>r</sup>*, *<sup>z</sup> <sup>c</sup>* � *Bic<sup>m</sup>*þ<sup>1</sup>

drical fin is given by

<sup>þ</sup>*cτz*<sup>2</sup> � *Bicm*þ<sup>1</sup>

*Heat Transfer - Design, Experimentation and Applications*

<sup>þ</sup>*cτz*<sup>3</sup> � *Bicm*þ<sup>1</sup>

<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> *<sup>c</sup><sup>τ</sup>* � *Bic<sup>m</sup>*þ<sup>1</sup>

at *τ* ¼ 0*:*4 The results are shown in **Figure 2**.

1 þ *βc*

$$c - \frac{Bic^{m+1}}{\mathbf{1} + \beta c}r + \dots + c - \frac{Bic^{m+1}}{\mathbf{1} + \beta c}r - \frac{\mathbf{1} \mathbf{2}c + \mathbf{1} \mathbf{8} \beta c^2 + 4E^2 \beta c^2 + \frac{4E^2 \beta \mathbf{B}^2 c^{2m+2}}{(1 + \beta c)}}{4E^2 (1 + \beta c)}r^2 + \dots = \mathbf{1}.$$

This solution is plotted in **Figure 3**

#### *4.2.3 Comparison of one dimensional exact, DTM and VIM solutions*

Here the solutions for the one dimensional heat transfer problems are compared, namely the exact solution given in Eq. (38). The VIM solutions is given by

$$\theta(\mathbf{x}) = \mathbf{c} + \frac{3\mathbf{c}^2 \mathbf{M}^2 \mathbf{x}^2}{2} - \frac{3\mathbf{c}^5 \mathbf{M}^2 \mathbf{x}^2}{2} + \frac{\mathbf{c}^7 \mathbf{M}^2 \mathbf{x}^2}{2} + \frac{\mathbf{c}^5 \mathbf{M}^4 \mathbf{x}^4}{8} - \frac{3\mathbf{c}^7 \mathbf{M}^4 \mathbf{x}^4}{2} + \frac{5\mathbf{c}^9 \mathbf{M}^4 \mathbf{x}^4}{4} - \frac{\mathbf{c}^{11} \mathbf{M}^4 \mathbf{x}^4}{4} \tag{41}$$

The constant *c* may be obtained using the boundary condition at the fin base. The DTM solution is given by (see [35])

$$\theta(\mathbf{x}) = \mathbf{c} + \frac{3c^2M^2\mathbf{x}^2}{2} - \frac{c^2M^2\mathbf{x}^2}{6} + \frac{c^2M^2(3 - 4M^2c)\mathbf{x}^4}{48} + \frac{c^2M^2(1 - 16M^2c)\mathbf{x}^5}{240} - \dots \tag{42}$$

Likewise, the constant *c* is obtained using the boundary conditions. These solutions are depicted in **Figure 4**.

#### **Figure 3.**

*Approximate analytical solutions for a two-dimensional cylindrical spine fin with a constant thermal conductivity (β* ¼ 0*) for τ* ¼ 0*:*4*. The parameters are set such that E* ¼ 2*, Bi* ¼ 0*:*2*, and m* ¼ 3*. (see also, [8]).*

*h* dimensionless heat transfer coefficient

*DOI: http://dx.doi.org/10.5772/intechopen.95490*

*h* Dimensionless thermal conductivity *Ka* Thermal conductivity of the fluid *K* Thermal conductivity of the fin

*Hb* Heat transfer coefficient at the base of the fin

*Survey of Some Exact and Approximate Analytical Solutions for Heat Transfer in Extended…*

*Z* Length of a cylindrical pin fin. Greek letters

\*, Partner Luyanda Ndlovu<sup>1</sup> and Basetsana Pauline Ntsime<sup>2</sup>

1 School of Computer Science and Applied Mathematics, University of the

2 Department of Mathematical Science, College of Science, Engineering and

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Technology, University of South Africa, Florida, South Africa

\*Address all correspondence to: raseelo.moitsheki@wits.ac.za

*H* Heat transfer coefficient

*L* Length of the fin

*Tb* Base temperature *Ts* Fluid temperature *x* Dimensionless fin length

*y* Dimensionless fin length

*θ* dimensionless temperature

*τ* Dimentionless time

*R* Radius *Ra* Radius *t* time

*X* Fin length

*Y* Fin length

**Author details**

**101**

Raseelo Joel Moitsheki<sup>1</sup>

Witwatersrand, Johannesburg, South Africa

provided the original work is properly cited.

**Figure 4.** *A temperature distribution in a rectangular fin for varying values of n, M* ¼ 1*:*7*.(see also [36]).*
