International Journal of Modern Engineering Sciences, 2012, 1(2): 97-104
International Journal of Modern Engineering Sciences
ISSN: 2167-1133
Florida, USA
Journal homepage:www.ModernScientificPress.com/Journals/IJMES.aspx
Article
Fractional Model for Heat Conduction in Polar Bear Hairs
Afshan Kanwal, Muhammad Sohail and Syed Tauseef Mohyud-Din*
Department of Mathematics, HITEC University Taxila Cantt, Pakistan
* To whom correspondence should be addressed: E-Mail:[email protected]
Article history: Received 15 August 2012, Received in revised form 3 October 2012, Accepted 5
October 2012, Published 8 October 2012.
Abstract: In this paper, we apply Reduced Differential Transform Method (RDTM) to
solve time fractional heat equation, which can accurately describe heat conduction in
fractal media, such as wool fibers, goose down and polar bear hair. It is observed that the
proposed technique (RDTM) is highly suitable for such physical problems and the
numerical results re-confirm the efficiency of the suggested algorithm.
Keywords: Time fractional heat conduction equation, reduced differential transform
method.
1. Introduction
Fractional differential equations arise in almost all areas of physics, applied and engineering
sciences. In order to better understand these physical phenomena as well as further apply these
physical phenomena in practical scientific research, it is important to find their exact solutions. The
investigation of exact solutions of these equations is interesting and important. In the past several
decades, many authors mainly had paid attention to study the solutions of such equations by using
various developed methods. Recently, the Variational Iteration Method (VIM) [1-3] has been applied
to handle various kinds of nonlinear problems, for example, fractional differential equations [4],
nonlinear differential equations [5], nonlinear thermo elasticity [6], nonlinear wave equations [7]. In
Refs. [8-13] Adomian’s Decomposition Method (ADM) [8, 9], Homotopy Perturbation Method (HPM)
[10], Homotopy Analysis Method (HAM) [11] and Variation of Parameter Method (VPM) [12, 13] are
successfully applied to obtain the exact solutions of differential equations. In the present article, we use
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Int. J. Modern Eng. Sci. 2012, 1(2): 97-104
98
Reduced Differential Transform Method (RDTM) [14-18], to construct an appropriate solution to heat
conduction model of fractional order.
The reduced differential transform technique is an iterative procedure for obtaining Taylor
series solutions of differential equations. This method reduces the size of computational work and
easily applicable to many physical problems.
1.1. Fractional Model for Heat Conduction in Polar Bear Hairs [19-21]
The polar bear (Ursus Maritimus) has superior ability to survive in the harsh Arctic regions, its
hollow and membrane structure of the polar bear hair. The transparent part of the polar bear hair
enables its excellent optical character to absorb solar energy and relatively poor heat conduction to
prevent from body temperature lose and to absorb environment temperature from ice water or air;
while the hollow and membrane structure behaves as a good thermal insulation. If the hair is
considered as a continuous media, we can write down the following 1-D heat conduction equation:
̃
Where ̃ (
̃
̃
(
)
(1.1)
) is the temperature at the point x and time t and ̃ - the thermal diffusivity. The polar
bear’s body temperature is about 37
and the environment temperature cab be as low as -50
, we,
therefore, have the following boundary conditions:
̃(
̃(
)
)
(1.2)
Where L is the length of the hair.
Equation (1.1) is valid only for continuous media. For discontinuous polar bear hairs, equation
(1.1) can be modified as:
̃
̃⁄
Where
(̃
̃
)
(1.3)
is the modified Riemann-Liouville fractional derivative of order
with respect to .
2. Basic Definitions of Fractional Calculus
We present some basic definitions and properties of the fractional calculus [6-10].
Definition 2.1 A real function ( )
number (
) such that ( )
is said to be in the space
( ) where
( )
[
if there exists a real
) and it said to be in the space
iff
(2.1)
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Int. J. Modern Eng. Sci. 2012, 1(2): 97-104
99
Definition 2.2 The Riemann–Liouville fractional integral operator of order
, of a function
, is defined as
( )
( )
∫ (
)
( )
(2.2)
It has the following properties:
For
and
(i)
( )
( )
(ii)
( )
( )
(
(iii)
)
(
)
The Riemann–Liouville fractional derivative is mostly used by mathematicians but this
approach is not suitable for the physical problems of the real world since it requires the definition of
fractional order initial conditions, which have no physically meaningful explanation yet. Caputo
introduced an alternative definition which has the advantage of defining integer order initial conditions
for fractional order differential equations.
( )the Caputo sense is defined as
Definition 2.3 The fractional derivative of
( )
( )
(
)
∫ (
)
( )
(2.3)
For
Definition 2.4 For m to be the smallest integer that exceeds, the Caputo time –fractional derivative
operator of order
(
)
is defined as,
(
)
{
(
)
∫(
(
(
)
(2.4)
)
and the space-fractional derivative operator of order
(
)
(
)
)
(
)
(
∫(
)
is defined as
(
)
(
)
{
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)
Int. J. Modern Eng. Sci. 2012, 1(2): 97-104
100
3. Reduced Differential Transform Method
The Reduced Differential Transform Method (RDTM) is applicable to a large class of linear
and nonlinear physical problems with approximations that converge rapidly to the actual solutions. A
brief overview of this method is given below:
Definition 3.1 Let (
) be an analytic function that continuously differentiable with respect to t and
space x in domain of interest. Define
( )=
where
(
)
(
[
)]
(3.1)
is a parameter describing the order of time-fractional derivative in the Caputo sense and t( ) is defined as
dimensional spectrum function
(
)=∑
( )
(3.2)
Combining equations (3.1) and (3.2), we have
(
)=∑
(
)
(
[
)]
(3.3)
Table 1. Reduced Differential Transformations
Functional Form
(
(
)
)
(
(
( )
)
)
(
Transformed Form
(
(
)
( )=
( )=
)
( )
)
(
)
(
)
) (
)
(
)
(
)
(
)
(
)
(
( )
( )
(
[
)]
( )
( )
(
)
( ) (c is constant)
)
(
(
∑
( )
)
(
( )
)
( )
∑
(
)
(
( )
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)
( )
( )
( )
( )
Int. J. Modern Eng. Sci. 2012, 1(2): 97-104
101
4. Numerical Application
In this section, we apply Reduced Differential Transform method (RDTM) to solve fractional
heat conduction model.
Example 4.1 Consider the following time fractional heat conduction model subject to the following
initial condition [21]
(
)
(4.1)
subject to the following initial condition with assumption that the temperature decreases along the hair
exponentially
T(
)
(
)
(
( )
)
(
)
Where a is the body temperature, b is the environment temperature, and k is a constant.
Taking the reduced differential transform of equation (4.1), we have the following iterative relation
(
)
(
( )
)
( )
(
From the initial condition, we have,
(
)
Now using iterative relation (4.3), we obtain the following values of
(
( )
(
)
)
(
)
)
(
)
)
)
Finally the reduced differential inverse transform of
(
)
)
)
(
)
(
( )
(
)
)
(
(
)
)
(
(
( )
(
(
( )
(
( )
(
(
( )
( ) successively,
)
)
(
(x) gives,
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)
)
Int. J. Modern Eng. Sci. 2012, 1(2): 97-104
102
(
(
(
)
)
)
(
(
(
(
)
(
(
)
Fig. 1(a): 3D plot of (
)
)
(
( )
∑
)
(
(
)
(
)
(
)
(
)
)
(
)
(
)
(
)
(
)
)
(
)
(
) for
)
for example 4.1
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)
Int. J. Modern Eng. Sci. 2012, 1(2): 97-104
103
Fig. 1(b): 2D plot of ( ) for
for example 4.1
5. Conclusion
In this article, Reduced Differential Transform Method (RDTM) was successfully applied to
find appropriate solution of time fractional heat conduction model. It is observed that the proposed
algorithm (RDTM) is fully compatible with the complexity of such problem. Numerical results
explicitly reveal the complete reliability and efficiency of the proposed algorithm. In our work we use
the Maple package to calculate the series obtained from reduced differential transform method and to
show solution graphically.
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