Heat Conduction – Basic Research
114
and it follows from (2.7) and (2.8), that
11
1
21
2
12
1
2
'''
,
'' '
(1) (21) ( 2)
''
(2 1) ( 1)
iii
m
i
i
ii i
m
i
ii
i
GGG
Ui
GGG
(2.10)
and so on. Here, the prime denotes the derivative with respective to
.
To determine u explicitly, we take the following four steps:
Step 1. Determine the integer m by substituting Eq. (2.7) along with Eq. (2.8) into Eq. (2.5) or
(2.6), and balancing the highest-order nonlinear term(s) and the highest-order partial
derivative.
Step 2. Substitute Eq. (2.7) with the value of m determined in Step 1, along with Eq. (2.8) into
, for
1, 2, ,im
, by use of Maple.
Step 4. Use the results obtained in the above steps to derive a series of fundamental
solutions
()u
of Eq. (2.5) or (2.6) depending on
'G
G
; since the solutions of Eq. (2.8) have
been well known for us, we can obtain exact solutions of Eqs. (2.1) and (2.2).
2.2 The Exp-function method
According to the classic Exp-function method, it is assumed that the solution of ODEs (2.5)
or (2.6) can be written as
exp( )
exp( ) exp( )
() ,
exp( ) exp( )
exp( )
g
n
nf f g
q
pq
are positive integers which are unknown, to be further determined, and
n
a and
m
b are unknown constants.
Exact Travelling Wave Solutions for
Generalized Forms of the Nonlinear Heat Conduction Equation
115
3. A generalized form of the nonlinear heat conduction equation
3.1 Application of the (G'/G)-expansion method
Introducing a complex variable
defined as Eq. (2.3), Eq. (1.1) becomes an ordinary
differential equation, which can be written as
2
() 0, 0
nn
kcU ak U U U a
(3.1)
or, equivalently,
22221
(1) 0,
nnn
kcU aknn U U aknU U U U
and
2
VV
in Eq.
(3.4) gives
2231,mm
(3.5)
so that
1.m
(3.6)
Suppose that the solutions of (3.4) can be expressed by a polynomial in
'G
G
as follows:
01 1
'
() , 0.
G
V
G
,
1
, ,,kc
and
. Solving the system of algebraic equations
with the aid of Maple 12, we obtain the following.
Heat Conduction – Basic Research
116
Case A:
When
2
40
Case A-1.
01
22 2
1111
,,.,
2
24 4 4
(3.9)
Substituting the general solution of (2.9) into Eq. (3.9), we get the generalized travelling
wave solution as follows:
22
12
22
12
44
sinh cosh
22
1
() 1 ,
2
44
cosh sinh
22
CC
V
CC
(3.10)
where
2
11
.
4
n
xat
na
.
inserting Eq. (3.10) into Eq. (3.3), it yields the following exact solution of Eq. (1.1):
1
1
12
(3.11)
in which
1
C
and
2
C
are arbitrary parameters that can be determined by the related initial
and boundary conditions.
Now, to obtain some special cases of the above general solution, we set
, (3.12)
Exact Travelling Wave Solutions for
Generalized Forms of the Nonlinear Heat Conduction Equation
117
and, when
1
0C
, the general solution (3.11) reduces to
1
1
11
(,) 1 coth
2
2
n
n
uxt x at
na
22
11'
() ,
2
24 4
G
V
G
(3.15)
Substituting the general solution of (2.9) into Eq. (3.15), we obtain
22
12
22
12
44
sinh cosh
22
1
() 1 ,
2
(3.16)
where
2
11
.
4
n
xat
na
na na
, (3.18)
Heat Conduction – Basic Research
118
and, when
1
0C
, the general solution (3.17) reduces to
1
1
11
(,) 1 coth
2
2
n
n
uxt x at
na
24 4 4
iini
kca
na
(3.20)
Inserting Eq. (3.20) into (3.7) results
22
1'
() ,
2
24 4
iiG
V
G
44
cosh cos .
22
i
i
(3.23)
in Eq. (3.22) and substituting the result into (3.3), we obtain the following exact solution of
Eq. (1.1):
Exact Travelling Wave Solutions for
Generalized Forms of the Nonlinear Heat Conduction Equation
(3.24)
We note that if we set
2
1'
() ,
2
24 4
iiG
V
G
(3.26)
Substituting the general solution of (2.9) for
2
40
into Eq. (3.26), we have
22
12
22
12
44
sin cos
22
1
(3.27)
in which
2
1
.
4
ni
xat
na
CxatCixat
na na
1
2
exp(3 )
,
exp(3 )
cf
v
cp
(3.29)
Heat Conduction – Basic Research
120
3
4
exp([2 3 ] )
,
exp(5 )
cfp
vv
cp
3
1
2
exp( 3 )
,
exp( 3 )
dg
v
dq
(3.33)
3
4
exp( [2 3 ] )
,
exp( 5 )
dgq
vv
dp
101
01
exp( ) exp( )
() ,
exp( ) exp( )
aaa
v
bb
(3.37)
Substituting Eq. (3.37) into Eq. (3.4), and making use of Maple, we arrive at
432101
234
1
[ exp(4 ) exp(3 ) exp(2 ) exp( ) exp( )
exp( 2 ) exp( 3 ) exp( 4 )] 0,
cccccc
A
ccc
001 1 1
,,, , ,abaa b k
, and
c
. Solving the
system of algebraic equations with the aid of Maple 12, we obtain:
Case 1.
001 1111
1
0, 0, 0, , , ,
2
n
aba abbbk ca
na
(3.40)
Substituting Eq. (3.40) into (3.37) and inserting the result into the transformation (3.3), we
get the generalized solitary wave solution of Eq. (1.1) as follows:
1
1
1
1
exp( )
(,) ,
exp( ) exp( )
is an arbitrary parameter which can be determined by
the initial and boundary conditions.
If we set
1
1b
and
1
1b
in (3.41), the solutions (3.18) and (3.19) can be recovered,
respectively.
Case 2.
001 1 11
1
0, 0, 1, 0, , ,
2
n
abaa bbk ca
na
(3.42)
By the same procedure as illustrated above, we obtain
and
1
b
is a free parameter.
If we set
1
1b
and
1
1b
in (3.43), then it can be easily converted to the same solutions
(3.12) and (3.13), respectively.
Case 3.
110000100
1
0, 0, , , , ,
n
ab aabbaabk cna
na
(3.44)
and consequently we get
na
and
00
,ab
, are arbitrary parameters; for example, if we put
0
0b , solution (3.45) reduces to
1
1
0
(,) cosh sinh ,
n
uxt a
(3.46)
Case 4.
1 0 0 0 0 1 00 0 1 00 0
0, , , ( ), ( ),
1
,
(3.48)
where
1n
xat
na
and
0
a ,
0
b are free parameters; for example, if we set
00
1, 0ab in Eq. (3.48), it can be easily converted to
1
1
1
(,) (1 coth csc ) ,
2
n
uxt h
(3.51)
in which
1n
xat
na
and
0
b is a free parameter.
Case B: 2, 1pf qg
Exact Travelling Wave Solutions for
Generalized Forms of the Nonlinear Heat Conduction Equation
123
Since the values of
g
and
f
can be freely chosen, we can put 2pf
na
(3.53)
Substituting Eq. (3.53) into (3.52), we have
0
() exp(2),va
(3.54)
Substituting Eq. (3.54) into Eq. (3.3), we get the generalized solitary wave solution of Eq.
(1.1) as
1
1
0
(,) [ exp(2)] ,
n
uxt a
(3.55)
where
1
()
na
(3.56)
Substituting Eq. (3.56) into (3.52), we have
0
0
() ,
exp(2 )
b
v
b
(3.57)
Inserting Eq. (3.57) into (3.3), it admits to the generalized solitary wave solution of Eq. (1.1)
as follows:
1
1
0
0
(,) ,
exp(2 )
n
b
ab
in Eq. (3.48), we can recover the solution (3.58).
Case 3.
1 0 0 11112 1
1
0, 0, 0, , , 0, 0, ,
3
n
aabbbababk ca
na
(3.59)
Substituting Eq. (3.59) into (3.52) we obtain
1
1
exp( )
() ,
exp(2 ) exp( )
b
v
b
(3.61)
in which
1
()
3
n
xat
na
and
1
b
is a free parameter that can be determined by the
initial and boundary conditions.
4. The generalized nonlinear heat conduction equation in
two dimensions
4.1 Application of the (G'/G)-expansion method
Using the wave variable (2.4) transforms Eq. (1.2) to the ODE
2
2() 0, 0
nn
kcU ak U U U a
01
22 2
1111
,,.,2
2
2
24 4 4
n
kca
na
(4.4)
Exact Travelling Wave Solutions for
Generalized Forms of the Nonlinear Heat Conduction Equation
125
By the same procedure as illustrated in Case A-1 of Section 3.1, Eqs. (3.9) and (3.10), we can
finally find the generalized solitary wave solution of Eq. (1.2) as
1
1
12
(4.5)
in which
1
C and
2
C are arbitrary parameters that can be determined by the related initial
and boundary conditions.
Now, to obtain some special cases of the above general solution, we set
2
0C
1
0C , the exact solution (4.5) reduces to
1
1
11
(,,) 1 coth 2
2
22
n
n
uxyt x y at
na
, (4.7)
Comparing the particular cases of our general solution, Eqs. (4.6) and (4.7), with Wazwaz’s
results (2005), Eqs. (87) and (88), it can be seen that the results are exactly the same.
22 22
(,,) 1
11
2
cosh 2 sinh 2
22 22
n
nn
CxyatC xyat
na na
uxyt
nn
CxyatCxyat
na na
22
n
n
uxyt x y at
na
, (4.10)
Heat Conduction – Basic Research
126
and, when
1
0C
, the general solution (4.9) reduces to
Case B-1.
01
22 2
11
,,.,2
2
2
24 4 4
iini
kca
na
(4.12)
By the same manipulation as illustrated in Case B-1 of Section 3.1, Eqs. (3.21)-(3.23), we can
finally obtain the following exact solution:
1
1
(4.13)
We note that, if we set
2
0C
and
1
0C
1
22 22
(,,) 1
11
2
cosh 2 sinh 2
22 22
n
nn
CxyatCixyat
na na
uxyt
nn
C xyatCi xyat
na na
4.2 Application of the Exp-function method
By the same manipulation as illustrated in Section 3.2, we obtain the following sets of
solutions.
Exact Travelling Wave Solutions for
Generalized Forms of the Nonlinear Heat Conduction Equation
127
Case 1.
111 0 0 11
1
,0,0,0, , , 2
22
n
aaa a b bak c a
na
(4.16)
Substituting Eq. (4.16) into (3.37) and inserting the result into the transformation (3.3), we
get the generalized solitary wave solution of Eq. (1.2) as follows:
1
1
1
1
exp( )
(,,) ,
exp( ) exp( )
a
is an arbitrary parameter which can be
determined by the initial and boundary conditions.
If we set
1
1a
and
1
1a
in (4.17), the solutions (4.10) and (4.11) can be recovered,
respectively.
Case 2.
001 1 11
1
0, 0, 1, 0, , , 2
22
n
abaa bbk c a
na
(4.18)
and
1
b
is a free parameter.
If we set
1
1b
and
1
1b
in (4.19), then it can be easily converted to the same solutions
(4.6) and (4.7), respectively.
Case 3.
111 0 1 0
1
,0,0, 0,0, , 2
22
n
aaa a b b k cna
na
x
y
nat
na
and
1
a
is an arbitrary parameter.
Heat Conduction – Basic Research
128
Case 4.
2
10
1001 110
0
1
1, , 0, , , , 2
2
ba
n
aaaa bbb k c a
a
na
(4.23)
where
1
2
2
n
xy at
na
and
0
a
,
1
b
are free parameters.
(,,)
exp( ) exp( )
n
aa
uxyt
aa
a
a
mathematical tools to solve the nonlinear partial differential equations (NPDEs) in the terms
Exact Travelling Wave Solutions for
Generalized Forms of the Nonlinear Heat Conduction Equation
129
of accuracy and efficiency. This is important, since systems of NPDEs have many
applications in engineering.
6. References
Abbasbandy, S. (2010). Homotopy analysis method for the Kawahara equation. Nonlinear
Analysis: Real World Applications, 11, 1, 307-312.
Bekir, A., Cevikel, C. (2009). New exact travelling wave solutions of nonlinear physical
models. Chaos, Solitons and Fractals, 41, 1733–1739.
Borhanifar, A., Kabir, MM. (2009). New periodic and soliton solutions by application of Exp-
function method for nonlinear evolution equations. Journal of Computational &
Applied Mathematics, 229, 158-167.
Borhanifar, A., Kabir, MM., Vahdat Lasemi, M. (2009). New periodic and soliton wave
solutions for the generalized Zakharov system and (2+1)-dimensional Nizhnik–
Novikov–Veselov system. Chaos, Solitons & Fractals, 42, 1646–1654.
Borhanifar, A., Kabir, MM. (2010). Soliton and Periodic solutions for (3+1)-dimensional
nonlinear evolution equations by Exp-function method. Applications and Applied
Mathematics: International Journal (AAM), 5, 1, 59-69.
Fan, E. (2002). Traveling wave solutions for nonlinear equations using symbolic
computation. Comput. Math. Appl., 43, 671–680.
He, JH. (1998). Approximate analytical solution for seepage flow with fractional derivatives
in porous media. Comput. Methods Appl. Mech. Eng., 167, 57-68.
He, JH. (2000). A coupling method of a homotopy technique and a perturbation technique
for non-linear problems. Int. J. Non-Linear Mechanics, 35, 37-43.
He, JH. (2006). New interpretation of homotopy perturbation method. Int. J. Mod. Phys. B,
20, 18, 2561–2568.
He, JH., Wu, XH. (2006). Exp-function method for nonlinear wave equations. Chaos, Solitons
Wazwaz, AM. (2006). New solitary wave solutions to the Kuramoto–Sivashinsky and the
Kawahara equations. Appl Math. Comput., 182, 1642-1650.
Zedan, HA. (2010). New classes of solutions for a system of partial differential equations by
G'/G)-expansion method. Nonlinear Science Letters A, 1, 3, 219–238.
Zhang, S., Wang, W., Tong, J. (2009). A generalized (G'/G)-expansion method and its
application to the (2+1)-dimensional Broer-Kaup equations. Appl. Math. Comput.,
209, 399-404.
6
Heat Conduction Problems of Thermosensitive
Solids under Complex Heat Exchange
Roman M. Kushnir and Vasyl S. Popovych
Pidstryhach Institute for Applied Problems of Mechanics and Mathematics,
Ukrainian National Academy of Sciences
Ukraine
1. Introduction
To provide efficient investigations for engineering problems related to heating/cooling
process in solids, the effect of thermosensitivity (the material characteristics depend on the
temperature) should be taken into consideration when solving the heat conductivity
problems (Carslaw & Jaeger, 1959; Noda, 1986; Nowinski, 1962; Podstrihach & Kolyano,
1972). It is important to construct the solutions to the aforementioned heat conduction
problems in analytical form. This requirement is motivated, for instance, by the need to
solve the thermoelasticity problems for thermosensitive bodies, for which the determined
temperature is a kind of input data, and thus, is desired in analytical form.
In general, the model of a thermosensitive body leads to a nonlinear heat conductivity
problem. It is mentioned in (Carslaw & Jaeger, 1959) that the exact solutions of such
problems can be determined when the temperature or heat flux is given on the surface by
assuming the material to be “simply nonlinear” (thermal conductivity
t
and volumetric
1959).
2. The step-by-step linearization method for solving the one-dimensional
transient heat conductivity problems with simple thermal non-linearity
Let us consider the step-by-step method for determining one-dimensional transient
temperature field
(,)tx
, which can be found from the following non-linear heat conduction
equation:
1
() ()
m
tv
m
tt
xt ct W
xx
x
, (1)
where
()
(2)
with the environment of constant temperature
a
t , where ()
a
t
is the temperature
dependent coefficient of heat exchange between the surface and the environment;
()
a
t
is
the temperature dependent emittance;
is the Stefan-Boltzmann constant. The surface
xb
is heated with constant temperature
b
t
or constant heat flux
b
q
:
t
or ( ) .
, temperature
0
Ttt
, and time
2
0
Fo al
(the Fourier number), we can present the functional parameters ()
t
t
, ()
v
ct,
()
a
t
, and ()
a
t
in the form
0
() ( )tT
tv
m
TT
xT cT qx
xx
x
, (5)
44
() Bi ()( )Sk ()( ) 0
taaaaaa
xa
T
TTTTTTT
x
. (8)
Here
2
000 00
/( )
t
Pql t
(the Pomerantsev number),
(0)
00
Bi
aa t
r
(the Biot number),
(0)
3
00 0
Sk
aa t
lt
(the Starc number),
000
Ki /( )
) and expressions
() ,
t
T
T
xx
() ()
Fo Fo Fo
tv
TT
TcT
, (11)
where
44
() Bi () () Sk ()(() )
aaa aaa a
QT T T T T T T
. The boundary condi-
tions (7) and initial condition (8) yield
b
xb
, ()T
denotes the temperature expressed through the Kirchhoff’s
variable and determined for certain ( )
t
T
by means of the integral equation (9).
Application of the Kirchhoff’s variable allows us to linearize the nonlinear heat
conductivity equation (5) and the second boundary condition (7) completely, whereas the
convective-radiation heat exchange condition is linearized in a part. Due to the nonlinear
expression
()
a
QT
, it is impossible to apply any classical method to solve the
boundary problem (10)–(13). Therefore, it is necessary to linearize the boundary condition
(11). In (Nedoseka, 1988; Podstrihach & Kolyano, 1972), the convective heat exchange
condition has been considered. Therefore, the nonlinear expression
()T
is simply
in
012
0Fo Fo Fo Fo)
n
, which divides the region of time variation into
1n intervals. Let us construct the spline
(0)
(Fo)
a
S with order 0, whose values coincide
with the values of expression
(Fo) ( )
aa
xa
QQT
at Fo Fo
i
and
1
() () ()
(0)
1
1
1
T (1,)in are the values of temperature
(,Fo)Tx , which are to be found on the surface xa
at the moments of time Fo
i
(the
unknown parameters of spline approximation),
()S
denotes the asymmetric unit
Heaviside function (H. Korn & T. Korn, 1977).
Having presented the nonlinear expression
()
a
xa
QT
by spline (14), the boundary
condition (11) becomes linear
(0)
(Fo) 0
a
xa
S
x
1
() () ()
(1)
1
1
1
(Fo) (Fo) (Fo) (Fo) (Fo Fo )
n
aaa
ai
ii
i
SP PPS
. (17)
Here the coefficients
() ()
,
aa
ii
kb of polynom
()
()a
i
Q
is expressed through
()a
i
T
by means of formula (15).
If
()
a
xa
QT
is expressed as the first-order spline (17), then boundary condition (11)
becomes linear
(1)
(Fo) 0
a
xa
S
x
(for specific dependence ( )
t
T
), the formula
for determination of the temperature
()
()
1
1
( , Fo,Fo , ,Fo , , ,
a
a
nn
Tfx T T (21)
can be obtained at arbitrary point
x and arbitrary moment of time Fo . For determination of
unknown values
()a
i
T in the expressions for temperature (21), the collocation method is
used. Assuming
Fo Fo
i
(1,)in
in (21), the system of equation for determination
(22)
is obtained. The structure of system (22) makes it possible to determine all unknown values
()a
i
T , starting from
()
1
a
T . Substitution of values, determined from (22), into the formula (21)
completes the solution procedure.
The temperature at given point
x and moment of time can be calculated in accordance to
the following scheme:
Heat Conduction – Basic Research
136
a. to divide the time axis by Fo
i
and then to determine the approximation parameters
()a
i
T
TT
, where
is the accuracy, then
the calculation is over. Otherwise, we shall return to the stage b.
The temperature can be computed with any given accuracy
for arbitrary segmentation of
the time axis. However, the increasing of number of time-segments decreases the
convergence of the proposed scheme. An appropriate choice of the initial moment of time
can be done by means of the estimated ‘a priory’ time-dependence of the temperature on the
surface
xa . We can also use the solution of corresponding boundary value problem for
the body of the same shape with constant characteristics. Then the initial choice for values
Fo
i
can be used as the appropriate one for the thermosensitive body.
The method of step-by-step linearization is applicable for determination of the temperature
fields in thermosensitive plates, half-space, solid and hollow cylinders or spheres, space
with cylindrical or spherical cavities, on the surfaces of which, the conditions of convective,
radiation or convective-radiation heat exchange may be given. This method has been
efficiently used for solving the two-dimensional steady problem in thermosensitive body.
3. Method of linearizing parameters
The method of step-by-step linearization makes it possible to determine the solutions to the
two-dimensional heat conductivity problems in thermosensitive bodies with simple
nonlinearity, when the nonlinear term in the condition of complex heat exchange for the
t
s
() ( ) 0
c
t
ttt
n
(24)
and initial
0
p
tt
(25)
Heat Conduction Problems of Thermosensitive Solids under Complex Heat Exchange
137
conditions, where
, (27)
Fo 0
0
(28)
is obtained, where
000
,,XxlYylZzl
are dimensionless coordinates;
0
,(,,,Fo)nnlqXYZ is the dimensionless function of heat sources. As a result, the initial
problem is partially linearized, meanwhile the condition (27) remains nonlinear. The latter
conditions have been obtained from the conditions of convective heat exchange due to
nonlinear expression
()T
on the surface S . For solving the problem (26)–(28) by using an
analytical method, it is necessary to linearize this condition. Let us prove the possibility of
such linearization.
Tk k T
. (31)
From the physical standpoint, the square root is chosen to be positive. After substitution of
the equation (31) into the boundary condition (27), the last one takes the form
12 1
Bi 0
pc
s
k
TT
nk
. (32)
cp
s
TT
n
(34)
instead of the nonlinear condition (32), where
is an unknown constant (linearized
parameter). Note that the boundary condition (34) coincides at 0
with the condition
(33). Since the problem (26), (28), (34) is linear, the appropriate classical analytical method
can be used for its solution. In addition to the original parameters of the problem
(
Po,Bi, , ,
c
p
TT
dimensions of the body, coordinates and time), the solution involves the
unknown linearized parameter
After some transformations, this equation can be given as
2
2
(1 )
s
k
. (36)
This equation holds for every moment of time Fo . After the paramenter
is found, we
substitute it into (35). In such manner, the expression for Kirchhoff’s variable is obtained.
The temperature in the body is then calculated by means of the relation (31).
Note that the boundary condition (34) can be represented as
Bi ( ) 0
c
s
T
n
variable for the case when the surface temperature of the thermosensitive body is equal to
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