Numerical Heat Transfer, Part A, 53: 295–307, 2008
Copyright # Taylor & Francis Group, LLC
ISSN: 1040-7782 print=1521-0634 online
DOI: 10.1080/10407780701557931
APPLICATION OF THE HYBRID DIFFERENTIAL
TRANSFORM-FINITE DIFFERENCE METHOD
TO NONLINEAR TRANSIENT HEAT
CONDUCTION PROBLEMS
Hsin-Ping Chu1, and Cheng-Ying Lo2
1
Department of Mechanical and Automation Engineering, Kao Yuan University,
Luchu, Taiwan, Republic of China
2
Department of Aeronautical Engineering, National Formosa University,
Hu Wei, Taiwan, Republic of China
This article presents a hybrid differential transformation-finite difference method to analyze
nonlinear transient heat conduction problems. The differential transformation technique is
used to transform the governing equations from the time domain into the spectrum domain,
followed by use of the finite difference method to formulate discretized iteration equations
appropriate for rapid computation. Numerical examples provide reasonable results that well
explain the heat conduction phenomena of embedding a heat source in a square plate with
either constant or time-dependent thermal conductivity.
1. INTRODUCTION
Many transform methods often require complicated techniques and elaborate
work to overcome mathematic difficulties for nonlinear transient problems. The differential transformation technique, which is a numerical method based on the Taylor
series expansion, provides an alternative approach to such problems. Since it was
developed by Puhove [1], the method has been applied to various fields. For
example, Köksal and Herdem [2] introduced the method for the analysis of nonlinear
electrical circuits. Chen and Liu applied this method to solve two-boundary-value
problems [3]. Jang et al. used the two-dimensional differential transform method
to solve partial differential equations [4]. Yu and Chen applied the method to the
optimization of rectangular fins with variable thermal parameters [5, 6]. Chen and
Ho applied this method to the analysis of the free vibration modes of nonuniform
Timoshenko beams [7, 8].
The differential transform is also very suitable to combine with other numerical
techniques, as Yu and Chen [9] applied the hybrid method to solve the transient
thermal stresses distribution in a perfectly elastic isotropic annular fin. This article
Received 14 February 2007; accepted 4 June 2007.
Address correspondence to Cheng-Ying Lo, Department of Aeronautical Engineering, National
Formosa University, 64 Wen Hua Road, Hu Wei, Yunlin 63201, Taiwan, Republic of China. E-mail:
[email protected]
295
296
H.-P. CHU AND C.-Y. LO
NOMENCLATURE
c
~
D
f
F
g
G
h
H
K
Ko
N
q
Q
r
R
t
T
u
U
x
y
b
d
4
q
specific heat, J=kg K
differential operator
a real function
differential transform of function f
a real function
differential transform of function g
convective heat transfer coefficient,
W=m2 K
time span
thermal conductivity, W=m K
constant thermal conductivity, W=m K
number of meshes
power per unit volume, W=m3
differential transform of power q
a real function
differential transform of function r
time, s
differential transform operator
temperature function, C
differential transform of temperature u
dimensional coordinate, m
dimensional coordinate, m
constant coefficient
Dirac delta function
mesh step size, m
density, kg=m3
Subscripts
i
index
j
index
k
index
l
index
n
index
introduces a comprehensive procedure to integrate the differential transformation
technique with the finite difference method in the numerical investigation of
nonlinear transient heat conduction problems. This differential transformation
technique is used to transform the governing equations from the time domain into
the spectrum domain, followed by use of the finite difference method to generate
discretized iteration equations appropriate for rapid computation. Unlike the
traditional high-order Taylor series method, which requires a lot of symbolic computations, the present method involves iterative procedures in the spectrum domain.
The simulation results of the solution are obtained in the partial sum in the inverse
process.
In this article the differential transformation technique is outlined first and the
procedures for transforming and discretizing the governing equations as well as the
boundary conditions are given in two numerical examples. Numerical results are
achieved and compared for both linear and nonlinear cases. The ability of the differential transform method to solve nonlinear equations is also discussed.
2. DIFFERENTIAL TRANSFORM METHOD
The differential transform method used in the present study is outlined in this
section. Let f ðtÞ be an analytic function in the time domain. The Taylor series expansion of f ðtÞ with the center at t0 is of the form
f ðtÞ ¼
1
X
ðt t0 Þk d k f ðtÞ
k¼0
k!
dtk
ð1Þ
t¼t0
The resulting series is called the Maclaurin series of f ðtÞ if it expands about the
origin, i.e., t0 ¼ 0. Based on the above series expansion, the differential transformation
NONLINEAR TRANSIENT HEAT CONDUCTION
297
of the function f ðtÞ at t ¼ 0 is defined by the following operation:
H k d k f ðtÞ
F ðkÞ ¼ T½f ðtÞ ¼
dtk t¼0
k!
k ¼ 0; 1; 2; . . .
ð2Þ
The function F ðkÞ, also denoted by T½f ðtÞ, is called the differential transform of the
original function f ðtÞ about the origin. Conventionally, original functions are
denoted by lowercase letters and their differential transforms by the same letters
in uppercase, H is the time interval, or the time span, of the differential transformation. The differential transform F ðkÞ is also called the spectrum of f ðtÞ in the spectrum domain. By substituting the derivative parts in Eq. (2) back into the Maclaurin
series, the original function f ðtÞ can be retrieved through the operation of inverse
transformation given by
f ðtÞ ¼
1 k
X
t
k¼0
H
F ðkÞ
ð3Þ
in which f ðtÞ is also called the inverse of F ðkÞ. In practical applications, the value of
f ðtÞ is seldom determined by the sum of the infinite series of Eq. (3) but is approximated by its nth partial sum, or nth-order series, as
f ðtÞ ¼
n k
X
t
k¼0
H
F ðkÞ
ð4Þ
The differential transformation method provides a very effective way of solving
differential equations. The crucial idea is that the differential transformation replaces
operations of calculus by operation of algebraic iterations. For the purpose of establishing iteration equations, it is necessary to derive the relationship between the differential transforms of the original function f ðtÞ and its derivatives. This is given
directly from the definition of differential transformation and the Taylor series
expansion as
df ðtÞ
kþ1
~
DF ðkÞ ¼ T
F ðk þ 1Þ
k 2 0; 1; 2; . . .
ð5Þ
¼
dt
H
~ F ðkÞ. Another
where the differential transform of the derivative f 0 ðxÞ is denoted as D
important property needed in the solving process is the differential transformation of
products of functions. Let rðtÞ be the product of two k-time differentiable functions
f ðtÞ and gðtÞ, of which the differential transforms are F ðkÞ and GðkÞ, respectively.
Following directly from Leibnitz’s rule, the differential transform of rðtÞ is given by
RðkÞ ¼ T½f ðtÞgðtÞ ¼ F ðkÞ GðkÞ ¼
k
X
F ðlÞGðk lÞ
ð6Þ
l¼0
The symbol represents the sum of the product terms on the right side of Eq. (6).
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H.-P. CHU AND C.-Y. LO
The process of applying the differential transform method to differential equations consists of three major steps. First, the governing equations are transformed
into algebraic iteration equations in the spectrum domain. Second, the transformed
functions are determined from the resulting iteration equations. Third, the final solutions are achieved through the inverse transformation. Usually, to improve the convergent rate and accuracy of calculation, the entire problem domain is spilt into
several subintervals. Then the differential transform method is implemented in each
individual subinterval. The results of the previous subinterval are adopted as
the initial values in the next subinterval. The same procedures are repeated in all
subintervals until the solution for the whole domain is achieved.
3. MATHEMATICAL MODEL AND NUMERICAL PROCEDURE
This section discusses the mathematical model and the numerical procedures of
the proposed hybrid method for solving two-dimensional nonlinear heat conduction
problems. The general governing equation of heat conduction problems is described
by Fourier’s law as
qc
qu
¼ r ðKruÞ þ q
qt
ð7Þ
where u ¼ uðx; y; tÞ is the instantaneous temperature of a point ðx; yÞ at time t, K is
the thermal conductivity, q is the density, c is the specific heat, and q represents the
rate of energy generated per unit volume.
Assuming that the problems considered have constant density q, constant specific heat c, and the thermal conductivity K varies linearly with the temperature u
over a certain range, the straight-line relationship between the thermal conductivity
K and the temperature u may be expressed as
K ¼ K0 ð1 þ buÞ
ð8Þ
where K0 and b are constants for a specific material. Generally speaking, for practical materials with this kind of K u relationship, b is negative for good conductors
and positive for good insulators. Equations (7) and (8) lead to the nonlinear governing equations for such problems as
qc
qu
¼ K0 r ½ð1 þ buÞru þ q
qt
ð9Þ
The solving process begins with taking the differential transforms of both sides
of the governing equations from the time domain into the spectrum domain, i.e., taking the differential transformation with respect to the time variable t only. Setting the
time interval H ¼ 1 in the transformation process, Eq. (9) is transformed into the
NONLINEAR TRANSIENT HEAT CONDUCTION
299
following iteration formula based on the spectrum k:
qcðk þ 1ÞUðx; y; k þ 1Þ
(
"
k
X
q2 Uðx; y; kÞ q2 Uðx; y; kÞ
qUðx; y; k lÞ qUðx; y; lÞ
þ
þ
b
¼ K0
2
2
qx
qy
qx
qx
l¼0
"
#
k
k
X
X
qUðx; y; k lÞ qUðx; y; lÞ
q2 Uðx; y; lÞ
þ
Uðx; y; k lÞ
þb
qy
qy
qx2
l¼0
l¼0
#)
k
X
q2 Uðx; y; lÞ
þ
Uðx; y; k lÞ
þ Qðx; y; kÞ
qy2
l¼0
ð10Þ
where Uðx; y; kÞ and Qðx; y; kÞ are the differential transforms of uðx; y; tÞ and
qðx; y; tÞ, respectively. The finite difference method is then applied to Eq. (10), which
contains only derivatives with respect to the space coordinates x and y. Without loss
of generality, the following finite difference scheme is derived based on a square
domain. The whole domain is divided into N equal intervals along both the x axis
and the y axis. The x coordinates of the grid points are given by xi ¼ iD,
i ¼ 0; 1; 2; . . . ; N and the y coordinates by yj ¼ jD, j ¼ 0; 1; 2; . . . ; N, where D is
the mesh size. Using the central difference formula on the first and second derivatives in Eq. (10), the corresponding difference equation is
qcðk þ 1ÞUi;kþ1
j ¼ K0
þ
b
4D2
k
X
l¼0
X
k
kl
l
Uiþ1;
j Uiþ1; j þ
l¼0
l
Ui;kl
j1 Ui; j1
k
X
kl
l
Ui1;
j Ui1; j þ
l¼0
2
k
X
l¼0
kl
l
Uiþ1;
j Ui1; j
k
X
l
Ui;kl
jþ1 Ui; jþ1
l¼0
k
X
l
Ui;kl
jþ1 Ui; j1
l¼0
1
k
k
k
k
k
þ 2 ðUi1;
j þ Uiþ1; j þ Ui; j1 þ Ui; jþ1 4Ui; j Þ
D
k
k
k
X
X
b X
kl
l
kl
l
l
Ui1;
Uiþ1;
Ui;kl
þ 2
j Ui; j þ
j Ui; j þ
j1 Ui; j
D
l¼0
l¼0
l¼0
k
k
X
X
kl
l
kl l
þ
Ui; jþ1 Ui; j 4
Ui; j Ui; j
þ Qki; j
l¼0
ð11Þ
l¼0
where Ui;k j and Qki; j denote the differential transforms Uðx; y; kÞ and Qðx; y; kÞ at the
grid point ðxi ; yi Þ, respectively. Equation (11) is derived from the nonlinear governing equations by integrating the procedure of the differential transform method and
the finite difference method. When the thermal conductivity is constant, i.e., b ¼ 0,
Eq. (11) reduces to a linear problem. The higher-order terms of the differential transform spectrum can be achieved from the initial and the boundary conditions through
the iteration process. Ui;j0 are determined from the initial conditions as well as the
300
H.-P. CHU AND C.-Y. LO
boundary conditions. For k ¼ 0, Ui;1 j can be calculated using the iteration formula of
Eq. (11) together with the initial and boundary conditions, and Ui;kþ1
j for k 1 can
be achieved sequentially following the same iteration process. After sufficient iterative processes, the final numerical solutions of u(x, y, t) can be approximated by the
nth partial sum of Eq. (4).
4. NUMERICAL EXAMPLES
Two numerical examples of heat conduction phenomena in a square plate with
embedded heat sources are used to illustrate the effectiveness of the proposed hybrid
method for analyzing nonlinear heat transient conduction problems. In both examples, the plate has dimensions of 0.1 m on both sides and 10 mm in thickness and is
made of brass (30% Zn and 70% Cu) [10]. The in-plane square problem domain is
defined within 0 x 0:1; 0 y 0:1. The density and the specific heat are
q ¼ 8; 255 kg=m3 and c ¼ 385 J=kgK, respectively.
The first example deals with a fixed temperature boundary and a concentrated
heat source at the centroid of the plate, and the second example deals with a convective boundary and a distributed heat source. The boundary conditions as well as the
associated numerical procedures and their numerical results of these two examples
are given as the following two cases.
Case 1: A Fixed Temperature Boundary with a Concentrated
Heat Source
In the first case, all four sides of the plate are kept at a fixed temperature 20C
and the whole plate is assumed initially at the uniform temperature 20C. A concentrated heat source at the centroid ðx ¼ 0:05; y ¼ 0:05Þ) generates heat into the plate
at the constant rate of 200 W. Thus, the initial conditions, boundary conditions, and
heat loading are given by the following equations.
Boundary conditions:
uðx; 0; tÞ ¼ 20
uðx; 0:1; tÞ ¼ 20
uð0; y; tÞ ¼ 20
ð12Þ
uð0:1; y; tÞ ¼ 20
Initial conditions:
uðx; y; 0Þ ¼ 20
ð13Þ
q ¼ 200dðx 0:05; y 0:05Þ
ð14Þ
Heat loading:
where d is the spatial Dirac delta function. The differential transformation technique
is applied to Eqs. (12)–(14) and the resulting transformed equations are expressed as
NONLINEAR TRANSIENT HEAT CONDUCTION
301
follows, boundary conditions after transformation:
k
Ui¼0;
j
k
Ui¼N;
j
Ui;k j¼0
Ui;k j¼N
¼
20
0
20
¼
0
20
¼
0
20
¼
0
k¼0
j ¼ 0; 1; 2; . . . ; N
k ¼ 1; 2; 3; . . .
k¼0
j ¼ 0; 1; 2; . . . ; N
k ¼ 1; 2; 3; . . .
k¼0
j ¼ 0; 1; 2; . . . ; N
k ¼ 1; 2; 3; . . .
k¼0
j ¼ 0; 1; 2; . . . ; N
k ¼ 1; 2; 3; . . .
ð15Þ
Initial conditions after transformation:
Ui;0 j ¼ 20
i ¼ 0; 1; 2; . . . ; N
Heat loading after transformation:
Qki; j ¼ 200
0
j ¼ 0; 1; 2; . . . ; N
k ¼ 0; i ¼ j ¼ N=2 þ 1
otherwise
ð16Þ
ð17Þ
Case 2: A Convective Boundary with a Distributed Heat Source
The second case also assumes the uniform temperature distribution over the
entire plate at 20C. Thus, the initial conditions are the same as those in the first case.
The plate presumes the convection boundary conditions with a constant convection
heat transfer coefficient h ¼ 590 W=m2 K along all four sides. The temperature of
the convection flow is uo ¼ 20C. In the center of the plate is embedded a distributed
square heat source, 1=30 m in area and 10 mm in depth, which emits heat energy
uniformly through its volume. The boundary conditions and the heat loading are
given as follows.
Boundary conditions:
qu K0 ð1 þ buÞ
¼ hðu u0 Þ
qx x¼0
qu ¼ hðu u0 Þ
K0 ð1 þ buÞ
qx x¼0:1
qu K0 ð1 þ buÞ ¼ hðu u0 Þ
qy y¼0
qu ¼ hðu u0 Þ
K0 ð1 þ buÞ qy
ð18Þ
y¼0:1
Heat loading:
(
q¼
36 106
0
1
2
1
2
x
and
y
30
30
30
30
otherwise
ð19Þ
302
H.-P. CHU AND C.-Y. LO
The total power of the heat source given by Eq. (19) equals 400 W. Equations
(18)–(19) are transformed by the differential transform method and then discretized
by the finite difference method. The resulting discretized equations are as follows.
Boundary conditions after transformation and discretization:
k
U1;
j
k
k
X
X
1
K0 Dh k Dh
kl
l
l
¼
U
þ
u
b
U
U
þb
U0;kl
0
0; j
1; j 0; j
j U0; j
K0
K0
1þbU0;0 j
l¼1
l¼0
!
!
k
k
X
X
1
K
Dh
Dh
0
k
k
kl
l
kl l
UNþ1;
UN;
u0 b
UNþ1;
UN;
j¼
j
j UN; j þb
j UN; j
0
K0
K0
1þbUN;
j
l¼1
l¼0
!
ð20Þ
k
k
X
X
1
K0 Dh k Dh
k
kl l
kl l
Ui;1 ¼
Ui;0 þ u0 b
Ui;1 Ui;0 þb
Ui;0 Ui;0
0
K0
K0
1þbUi;0
l¼1
l¼0
!
k
k
X
X
1
K
þDh
Dh
0
k
k
kl
l
kl l
Ui;Nþ1
¼
Ui;N
u0 b
Ui;Nþ1
Ui;N
þb
Ui;N
Ui;N
0
K0
K0
1þbUi;N
l¼1
l¼0
Heat loading after transformation and discretization:
(
1
2
6 2
k
xi ; yj k ¼ 0 and
Qi;j ¼ 36 10 D
30
30
0
otherwise
ð21Þ
k
k
k
k
where U1;j
; UNþ1;j
; Ui;1
; and Ui;Nþ1
for i ¼ 1; 2; . . . ; N 1; j ¼ 1; 2; . . . ; N 1 are
virtual variables located outside the domain. The above equations can substitute
for the virtual variables in the previous governing difference equation, Eq. (11), to
determine the temperature along the edges and start the complete iteration procedure.
Two different kinds of thermal conductivities are used in each case, including a
temperature-dependent thermal conductivity given by a straight-line function
K ¼ 106 ð1 þ 0:0025uÞ W=m K, i.e., setting K0 ¼ 106; b ¼ 0:0025 in Eq. (8), and
a constant thermal conductivity K ¼ 109 W=m K, for which K0 ¼ 109; b ¼ 0. Both
examples are generically nonlinear problems if the thermal conductivity varies with
the temperature, though they can reduce to a linear problem if the thermal conductivity is independent of the temperature.
5. RESULTS AND DISCUSSION
When employing the hybrid differential transform-finite difference method,
three factors are important to ensure a quality solution. The first is the mesh size
used in the finite difference method. The others are the order of differential transform n and the time step Dt in the differential transformation technique. Fine meshes
and small time steps can improve simulation results. However, too small time steps
and too small meshes may unnecessarily require more computation time and may
cause divergence problems in numerical results. Such divergence can be suppressed
by increasing the differential transform order n. Increasing the order of the differential transform method can achieve more precise approximations. Higher-order terms
are necessary for fast-response problems.
NONLINEAR TRANSIENT HEAT CONDUCTION
303
In the present study, the rectangular problem domains, 0 x 0:1 and
0 y 0:1, are equally divided into 30 30 square meshes, i.e., D ¼ 1=300 m on
each side. The time step is chosen to be Dt ¼ 0.01 s. The order of differential transform is n ¼ 15. The above computation parameters are the same for all cases. The
numerical results for the first case are presented in Figures 1–3 and for the second
cases in Figures 4 and 5. Both cases include the results for linear and nonlinear heat
conduction behaviors.
The plate considered in the first case includes a concentrated heat source and
fixed-temperature boundaries. The sharp temperature rises are found at the center of
the plate, at which the concentrated heat source located. The temperature decreases
rapidly to 20C along the four sides of the plate. Figure 1 shows the temperature distribution over the plate at different times (t ¼ 1; 2; 4; 8 s) for the case with Ko ¼ 109,
b ¼ 0. The temperature distributions increase with the time. Figure 2 shows the temperature distribution over the plate for the case with Ko ¼ 106, b ¼ 0.0025. Figures 1
and 2 both show that the temperature in the region close to the heat source
(x ¼ 0.05, y ¼ 0.05) rises rapidly when the concentrative heat flow is applied. The
temperature rises more slowly as one moves farther away from the heat source.
The latter temperature increases at a slower rate than that of the former. Figure 3
shows that the temperature rise for the case with Ko ¼ 109, b ¼ 0 is faster than that
for the case with Ko ¼ 106, b ¼ 0.0025 at the center (x ¼ 0.05, y ¼ 0.05). After a
while, the temperature gradually becomes stable. The stable temperature at the center
for the case with Ko ¼ 109, b ¼ 0 is about 15C higher than that for the case with
Ko ¼ 106, b ¼ 0.0025.
Very similar numerical results are obtained for the second case, which has a
distributed heat source at the center and heat convection boundaries along all four
Figure 1. Temperature distribution for case 1 with Ko ¼ 109, b ¼ 0 at t ¼ 1, 2, 4, 8 s.
304
H.-P. CHU AND C.-Y. LO
Figure 2. Temperature distribution for case 1 with Ko ¼ 106, b ¼ 0.0025 at t ¼ 1, 2, 4, 8 s.
sides of the plate. The convection heat transfer coefficient h is set at h ¼ 590 W=m2 K.
The temperature distributions at t ¼ 1, 2, 4, and 8 s for the case with Ko ¼ 109, b ¼ 0
is shown in Figure 4 and for the case with Ko ¼ 106, b ¼ 0.0025 in Figure 5. Instead of
a sharp temperature rise, in the central region is a high-temperature hill. The
temperature distribution for the case with Ko ¼ 109, b ¼ 0 is still higher than that
Figure 3. Temperature response for case 1 at the center of the plate (x ¼ 0.05, y ¼ 0.05) with two different
thermal conductivities.
NONLINEAR TRANSIENT HEAT CONDUCTION
305
Figure 4. Temperature distribution for case 2 with Ko ¼ 109, b ¼ 0 at t ¼ 4, 8, 12, 16 s.
for the case with Ko ¼ 106, b ¼ 0.0025. Since the temperature is not fixed, a temperature rise along the boundary is observed as the heat flow across the boundary
of the plate. Figure 6 shows the temperature responses at the center for both
Figure 5. Temperature distribution for case 2 with Ko ¼ 106, b ¼ 0.0025 at t ¼ 4, 8, 12, 16 s.
306
H.-P. CHU AND C.-Y. LO
Figure 6. Temperature response for case 2 at the center of the plate (x ¼ 0.05, y ¼ 0.05) with two different
thermal conductivities.
thermal conductivities. It is shown that, at the center of the plate, the temperature
for the case with Ko ¼ 109, b ¼ 0 rises faster than for the case with Ko ¼ 106,
b ¼ 0.0025, due to the higher thermal conductivity. Compared with the first case,
the second case takes much longer to become stable and the stable temperature is
also higher than for the first case.
When solving the above nonlinear cases, the iteration procedure takes more
than 1 h to evaluate the solution on a 2-GHz computer. Compared with other methods, this method will not consume too much computer time when applied to nonlinear or parameter-varying systems. Nonlinear problems consume more computer
time than the linear cases because of the extra iteration procedures.
6. CONCLUSIONS
The present study has demonstrated the application of a hybrid differential
transform-finite difference method to two numerical examples of heat conduction
problems. The first case is a square plate with a concentrated heat source under
fixed-temperature boundaries. The other is a square plate with a distributed heat
source under convection boundaries. Nonlinear heat conduction phenomena have
been investigated by replacing the constant thermal conductivity (K ¼ 109) with a
temperature-dependent conductivity [K ¼ 106(1 þ 0.025u)] in both examples. Temperature distributions over the entire plate along the time frame have been obtained.
The former condition yields higher temperature distributions and faster temperature
rises, due to the larger thermal conductivity of the plates. The present numerical
technique can obtain reliable results over the whole domain without complex transformation procedures. This study has shown that integration of the differential
NONLINEAR TRANSIENT HEAT CONDUCTION
307
transform method and the finite difference method provides a very good tool to
solve linear or nonlinear heat conduction problems.
REFERENCES
1. G. E. Puhov, Differential Transformation and Mathematical Modeling of Physical Process,
Naukova Dumka, Kiev, 1986 (in Russian).
2. M. Köksal and S. Herdem, Analysis of Nonlinear Circuits by Using Differential Taylor
Transform, Comput. Electr. Eng., vol. 28, pp. 513–525, 2002.
3. C. L. Chen and Y. C. Liu, Solution of Two-Boundary-Value Problems Using the Differential Transformation Method, J. Optim. Theory Appl., vol. 99, pp. 23–35, 1998.
4. M. J. Jang, C. L. Chen, and Y. C. Liu, Two-Dimensional Differential Transform for
Partial Differential Equations, Appl. Math. and Comput., vol. 121, pp. 261–270, 2001.
5. L. T. Yu and C. K. Chen, The Solution of the Blasius Equation by the Differential Transformation Method, Math. Comput. Model., vol. 28, pp. 101–111, 1998.
6. L. T. Yu and C. K. Chen, Application of Taylor Transformation to Optimize Rectangular
Fins with Variable Thermal Parameters, Appl. Math. Model., vol. 22, pp. 11–21, 1998.
7. C. K. Chen and S. H. Ho, Free Vibration Analysis of Non-uniform Timoshenko Beams
Using Differential Transform, Appl. Math. Model., vol. 22 , pp. 219–234, 1998.
8. S. H. Ho and C. K. Chen, Free Transverse Vibration of An Axially Loaded Non-uniform
Spinning Twisted Timoshenko Beam using Differential Transform, Int. J. Mech. Sci.,
vol. 48, pp. 1323–1331, 2006.
9. L. T. Yu and C. K. Chen, Application of the Hybrid Method to the Transient
Thermal Stresses Response on Isotropic Annual Fins, ASME J. Appl. Mech., vol. 66,
pp. 340–346, 1999.
10. J. H. Lienhard, A Heat Transfer Textbook, p. 591, Prentice-Hall, Englewood Cliffs, NJ,
1987.