1. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN
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FRACTIONAL GREEN FUNCTION APPROACH TO STUDY HEAT
TRANSFER THROUGH DIATHERMANOUS MATERIALS
JYOTINDRA C. PRAJAPATI1
, KRUNAL B. KACHHIA2
1, 2
(Department of Mathematical Sciences, Faculty of Applied Sciences, Charotar University of
Science and Technology (Charusat), Changa, Anand-388421, Gujarat, India
ABSTRACT
In recent years, there has been a considerable growth of interest in use of Fourier transform as
an efficient method for solving certain types of fractional differential equations. In addition, to such
applications Fourier Transforms also have a number of close connections with important part of
mathematics. In this paper, authors applied fractional Green function to discuss a solution of problem
regarding heating of a diathermanous materials. The solution of this problem obtained in the form of
generalized Wright function.
MSC2010 Classifications: 26A33, 34B27, 35A99
Keywords: Fractional derivatives, Fractional differential equation, Fractional Green function,
Fractional time derivative, Generalized Wright function.
I. INTRODUCTION
Considerable practical interest has recently arisen in situations where diathermanous
materials are exposed to a time-invariant thermal flux. At the heating rates and irradiation periods in
question, the samples may be assumed to behave as one-dimensional semi-infinite solids (i.e. the
depth of penetration for the heat wave may be assumed slight, relative to the dimensions of the
exposed area) and any cooling losses from the irradiated surface may be neglected. It is desired to
develop an expression for the temperature-time-space relation in these irradiated slabs subject to the
following assumptions Mickley, Sherwood and Reed [1]:
(1) The materials in questions are homogenous, isotropic and have physical properties (that is k:
thermal conductivity, ρ: density, cp: specific heat, etc.) that are independent of temperature.
(2) The absorption of radiant energy below the surface may be characterized by a Lambert’s-decay-
law expression of the form
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0
x
xI I e γ−
= (1.1)
where
(i) Ix = intensity at any depth ‫ݔ‬
(ii) I0 = intensity of un-reflected radiation at surface
(iii) ߛ = extinction coefficient
(3) The initial temperature of the material is uniform at ܶ଴.
The applicable differential equation is given by classical non-homogenous heat equation
defined in Keshvani [2]:
2
2
x
e
t x
γθ θ
α β −∂ ∂
− =
∂ ∂
(1.2)
where
k
c
α
ρ
= , 0I
c
γ
β
ρ
= and 0T Tθ = − (1.3)
The relevant boundary conditions are:
( ,0) 0xθ = for all ‫ݔ‬ (1.4)
| |
lim ( , ) 0
x
x tθ
→∞
= for all ‫ݐ‬ (1.5)
The Green function for the equation (1.2) is given by
2
4
1
( , )
2
x
t
G x t e
t
α
α
−
= (1.6)
It seems that the notion of Green’s function of a fractional differential equation appeared for
the first time in the book by Meshkov [3]. The definition of fractional Green’s function suggested
and used by Miller and Ross [4] applies to fractional differential equations containing only
derivatives of order ݇ߙ, where ݇ is integer.
Fractional derivatives provide an excellent instrument for the description of memory and
hereditary properties of various materials and processes. The advantages of fractional derivatives
become apparent in modeling mechanical and electrical properties of real materials as well as in the
description of rheological properties of rocks and in many other fields. The subject of fractional
calculus deals with the investigations of integrals and derivatives of any arbitrary real or complex
order, which unify and extend the notions of integer-order derivative and n-fold integral. It has
gained importance and popularity during the last four decades or so, mainly due to its vast potential
of demonstrated applications in various seemingly diversified fields of science and engineering, such
as fluid flow, rheology, diffusion, relaxation, oscillation, anomalous, reaction-diffusion, turbulence,
diffusive transport akin to diffusion, electric network, polymer physics, chemical physics,
electrochemistry of corrosion, relaxation processes in complex systems, propagation of seismic

3. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN
0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 5, July – August (2013), © IAEME
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waves, dynamical processes in self-similar and porous structures and others. The importance of this
subject further lies in the fact that during the last three decades, three international conferences
dedicated exclusively to fractional calculus and its application were held in the University of New
Haven in 1974, University of Glasgow, Scotland in 1984 and the third in Nihon University in Tokyo,
Japan in 1989 in which Various workers presented their investigations dealing with the theory and
applications of fractional calculus Free shear flows are inhomogeneous flows with mean velocity
gradients that develop in the absence of boundaries. Turbulence free shear flows are commonly
found in natural and engineering enviournments. The jet of air issuing from one’s nostrils or mouth
upon exhaling, the turbulent plume from a smoldering cigarette and the buoyant jet issuing from an
erupting volcano-all illustrate both the omnipresence of free turbulent shear flows and the range of
scales of such flows in the natural enviournment. Examples of the multitude of engineering free
shear flows are the walks behind moving bodies and the exhausts from jet engines. Most combustion
processes and many mixing processes involve turbulent free shear flows. Free shear flows in the real
world are most often turbulent. The tendency of free shear flows to become and remain turbulent can
be greatly modified by the presence of density gradients in the flow, especially if gravitational effects
are also important. Free share flows deals with incompressible constant-density flows away from
walls, which include shear layers, jets and wakes behind bodies. Hydrodynamic stability is of
fundamental importance in fluid dynamics and is a well-established subject of scientific investigation
that continuous to attract great interest of the fluid mechanics community. Hydrodynamic
instabilities of prototypical character are , for example the Rayleigh-Benard, the Taylor-Couette, the
Benard-Marangoni, the Rayleigh-Taylor, and the Kelvin-Helmholtz instabilities. Modeling of
various instability mechanisms in biological and biomedical systems is currently a very active and
rapidly developing area of research with important biotechnological and medical applications
(biofilm engineering, wound healing, etc.) The understanding of breaking symmetry in
hemodynamics could have important consequence for vascular biology and diseases and its
implication for vascular interventions (grafting, stenting, etc.). When in a porous medium filled with
one fluid and another fluid is injected which is immiscible in nature in ordinary condition then
instability occurs in the flow depending upon viscosity difference in two flowing phases. When a
fluid flow through porous medium displaced by another fluid of lesser viscocity then instead of
regular displacement of whole front protuberance take place which shoot through the porous medium
at a relatively high speed. This phenomenon is called fingering phenomenon (or instability
phenomenon). Many researchers have studied this phenomenon with different point of view.
The mathematical modeling and simulation of systems and processes based on the description
of their properties in terms of fractional derivatives naturally leads to differential equations of
fractional order and to the necessity to solve such equations.
The Riemann-Liouville fractional derivative of ݂ሺ‫ݐ‬ሻ order ‫ݒ‬ with ‫ݒ‬ ൒ 0 is defined in
Podulbny [5] as
11
( ( )) ( ) ( )
( )
tk
v k v
a t k
a
d
D f t t f d
k v dt
τ τ τ− −
= −
Γ − ∫ (1.7)
where ݇ െ 1 ൑ ‫ݒ‬ ൏ ݇, k is any natural number and 0
( ( )) ( )a tD f t f t=
The Caputo fractional derivative of ݂ሺ‫ݐ‬ሻ of order ‫ݒ‬ with ‫ݒ‬ ൒ 0 is defined in Podulbny [5] as
1 ( )1
( ( )) ( ) ( )
( )
t
C v k v k
a t
a
D f t t f d
k v
τ τ τ− −
= −
Γ − ∫ (1.8)
where ݇ െ 1 ൑ ‫ݒ‬ ൏ ݇, ݇ is any natural number.

4. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN
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The properties of fractional calculus associated with generalized Mittag-Leffler function also
studied by Shukla and Prajapati ([6], [7], [8] and [9]).
The Wright function of parameters ߙ and ߚ is defined as (in Podulbny [10])
( , ; )
( ) !
j
j o
z
W z
j j
α β
α β
∞
=
=
Γ +
∑ (1.9)
where ߙ ‫א‬ Թ and ߚ ‫א‬ ԧ.
The generalized Wright function is defined as (Luchko and Gorenflo [11])
( , ),( , ) ( )
( ) ( )
j
a b
j o
z
W z
j a j b
α β
α β
∞
=
=
Γ + Γ +
∑ (1.10)
where ߙ, ߚ ‫א‬ Թ and ܽ, ܾ ‫א‬ ԧ.
If ‫ܩ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ is a Green function related to the evolution equation
2
0 1 2 2
... ( , ), 0,
n
n n
a a a a q x t t x
t x x x
θ θ θ θ
θ
∂ ∂ ∂ ∂
+ + + + + = > ∈
∂ ∂ ∂ ∂
(1.11)
where ‫ݐ‬ ൐ 0, ‫ݔ‬ ‫א‬ Թ then the Fractional Green function ‫ܩ‬௩ሺ‫,ݔ‬ ‫ݐ‬ሻ for evaluation equation
2
0 1 2 2
... ( , ), 0,
v n
nv n
a a a a q x t t x
t x x x
θ θ θ θ
θ
∂ ∂ ∂ ∂
+ + + + + = > ∈
∂ ∂ ∂ ∂
(1.12)
where ‫ݐ‬ ൐ 0, ‫ݔ‬ ‫א‬ Թ and 0 ൏ ‫ݒ‬ ൑ 1 is given by relation defined as (Momani and Odibat [12])
1
0
( , ) ( , ) ( )v
vG x t t G x z t z dz
∞
− −
= Φ∫ (1.13)
where
( ) ( ,0; )t W v tΦ = − − (1.14)
II. FORMATION OF PROBLEM
Now, consider fractional partial differential equation in the form
2
2
v
x
v
e
t x
γθ θ
α β −∂ ∂
− =
∂ ∂
(2.1)
where
0, , 0 1t x v> ∈ < ≤ ,
k
c
α
ρ
= , 0I
c
γ
β
ρ
= and 0T Tθ = − (2.2)

5. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN
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The relevant boundary conditions are:
( ,0) 0xθ = for all ‫ݔ‬ (2.3)
| |
lim ( , ) 0
x
x tθ
→∞
= for all ‫ݐ‬ (2.4)
If we put ‫ݒ‬ ൌ 1 then equation (2.1) reduces in classical non-homogenous heat equation (1.2)
III. SOLUTION OF PROBLEM
Define,
( ) ( ,0; )t W v tΦ = − − (3.1)
where ( ,0; )W v t− − is Wright function defined as (1.9) and has the power series representation
0
( 1)
( )
( ) !
k k
k
t
t
vk k
∞
=
−
Φ =
Γ −
∑ (3.2)
The fractional Green function ‫ܩ‬௩ሺ‫,ݔ‬ ‫ݐ‬ሻ for equation (2.1) is given by using (1.13) as follows:
1
0
( , ) ( , ) ( )v
vG x t t G x z t z dz
∞
− −
= Φ∫
2
1 4
0
1
( ,0; )
2
x
vz
t e W v t z dz
z
α
α
∞ −
−
= − −∫ (3.3)
this can be written as,
2
1
4
00
1 ( 1) ( )
( , )
( ) !2
x k v k
z
v
k
t t z
G x t e dz
vk kz
α
α
∞ −− −∞
=
−
=
Γ −
∑∫ (3.4)
above equation reduces to
2
11
42
0 0
( 1)
( , )
( ) !2
xk vk
k
z
v
k
t t
G x t z e dz
vk k
α
α
∞ −− −∞ −
=
−
=
Γ −
∑ ∫ (3.5)
on substituting
2
4
x
z
y e α
= yields

6. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN
0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 5, July – August (2013), © IAEME
145
2 1 31
2
0 0
( 1)
( , )
( ) !2 2
kk vk
k
y
v
k
t t x
G x t y e dy
vk kα α
+ ∞− −∞ − −
−
=
−  
=  Γ −  
∑ ∫
simplification of above equation gives,
2 11
0
( 1) 1
( , )
( ) ! 22 2
kk vk
v
k
t t x
G x t k
vk kα α
+− −∞
=
−    
= Γ − −  Γ −   
∑ (3.6)
using the fact that
1 2 1
1 ( 1) 2 !
2 (2 1)!
k k
k
k
k
π
+ +
− 
− − 
+ 
Γ = (3.7)
we get
2 11 1 2 1
0
( 1) ( 1) 2 !
( , )
( ) ! (2 1)!2 2
kk vk k k
v
k
t t x k
G x t
vk k k
π
α α
+− − + +∞
=
− − 
=  Γ − + 
∑ (3.8)
this immediately gives
2 1
2 1
0
( , )
2 ( )( ) (2 1)!
vk k
v k
k
t x
G x t
t vk k
π
α α
− +∞
+
=
=−
Γ − +
∑
2
2
02 ( )( ) (2 1)!
vk k
k
k
x t x
t vk k
π
α α
−∞
=
=−
Γ − +
∑ (3.9)
this follows
2
( ,0),(2,2)( , )
2
v
v v
x n
G x t W
t
π τ
αα
−
−
 
=−  
 
(3.10)
where
2
( ,0),(2,2)
v
v
n
W
τ
α
−
−
 
 
 
is the generalized Wright function defined as (1.10).
In view of Momani and Odibat [12], the solution of the non-homogenous fractional heat
equation (2.1) is given by
2
( ,0),(2,2)
0
1
( , )
2
t v
inx n
v
n n
x t e W e d dnγτ
θ β τ
ατ α
∞ −
−
−
−∞
 
=−  
 
∫ ∫ (3.11)
this gives
2
( ,0),(2,2)
0
( , )
2
t inx v
n
v
ne n
x t W e dndγβ τ
θ τ
α τ α
∞ −
−
−
−∞
 
=−  
 
∫ ∫ (3.12)

7. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN
0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 5, July – August (2013), © IAEME
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as 0T Tθ = − and 0 0
2 2 2
I c I
c k k
β γ ρ γ
α ρ
= = , we finally arrived at
2
0
0 ( ,0),(2,2)
02
t inx v
n
v
I ne n
T T W e dn d
k
γγ τ
τ
τ α
∞ −
−
−
−∞
 
− =−  
 
∫ ∫ (3.13)
Equation (3.13) is apposite solution of the problem.
IV. CONCLUSION
Finally, the solution of fractional order non-homogenous heat equation for heat transfer
through diathermanous materials obtained in terms of generalized Wright function by using
fractional Green function approach. Usually, this method is very useful to study various problems
regarding fluid dynamics, control theory, aerodynamics, mathematical and applied sciences.
V. REFERENCES
[1] H. S. Mickley, T. S. Sherwood and C. E. Reed, Applied mathematics in chemical
engineering (Tata McGraw Hill Publishing Company Ltd).
[2] R. R. Keshvani, Solution of a problem regarding heating of a diathermanous solid and its
explicit representation using Gauss-Weirestrass Kernel, International Journal of Physical,
Chemical and Mathematical Sciences, 2(1) (2013), 153-163.
[3] S. I. Meshkov, Viscoelastic Properties of metals (Metallurgia, Moscow, 1974).
[4] K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional
differential equations ( Jhon Wiley and Sons Inc., New York, 1993).
[5] I. Podulbny, Fractional differential equations (Academic Press, New York, 1999).
[6] A. K. Shukla and J. C. Prajapati, Some properties of a class of polynomials suggested by
Mittal, Proyecciones Journal of Mathematics, Scielo Group, Latin America, 26 (2) (2007),
145-156.
[7] A. K. Shukla and J. C. Prajapati, A general class of polynomials associated with generalized
Mittag-Leffler function, Integral Transforms and Special Functions, Taylor and Francis, U.
K., 19 (1) (2008), 23-34.
[8] A. K. Shukla and J. C. Prajapati, On a generalized Mittag-Leffler type function and generated
integral operator, Mathematical Sciences Research Journal, U. S. A., 12 (12) (2008), 283-
290.
[9] A. K. Shukla and J. C. Prajapati, Some remarks on generalized Mittag-Leffler function,
Proyecciones Journal of Mathematics, Scielo Group, Latin America, 28(1) (2009), 27-34.
[10] I. Podulbny, The Laplace transform method for linear differential equations of fractional
order ( Slovac Academy of Science, Slovak Republic, 1994).
[11] Y. Luchko and R. Gorenflo, Scale-invarient solutions of a partial differential equation of
fractional order, Fract. Calc. Anal., 3(1) (1998), 63-78.
[12] S. Momani and Z. M. Odibat, Fractional Green function for linear time-fractional
inhomogenous partial differential equations in Fluid mechanics, J. Appl. Math and
Computing, Vol. 24 (2007), No. 1-2, pp. 167-178.
[13] Ajeet Kumar Rai, Shahbaz Ahmad and Sarfaraj Ahamad Idrisi, “Design, Fabrication and
Heat Transfer Study of Green House Dryer”, International Journal of Mechanical
Engineering & Technology (IJMET), Volume 4, Issue 4, 2013, pp. 1 - 7, ISSN Print:
0976 – 6340, ISSN Online: 0976 – 6359.