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Development of Efficient Numerical Methods for Solving Differential Equations using Hes Variational Iteration Technique
A Thesis
Presented to the
Graduate Faculty of the
University of Louisiana at Lafayette
In Partial Fulfillment of the
Requirements for the Degree
Master of Science
Manoj Chand
Fall 2013
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Manoj Chand
2013
All Rights Reserved
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Development of Efficient Numerical Methods for Solving Differential Equations using Hes Variational Iteration Technique
Manoj Chand
APPROVED:
________________________________ ________________________________
Yucheng Liu, Chair Jim Lee Assistant Professor of Mechanical Professor of Mechanical Engineering Engineering
________________________________ ________________________________
William E. Simon Mary Farmer-Kaiser Professor of Mechanical Engineering Interim Dean of the Graduate School
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ACKNOWLEDGMENTS
I would like to express my sincere appreciation to all of the people who have
contributed to formulating this thesis.
First of all, my deepest gratitude goes to, Dr. Yucheng Liu, my advisor, for his
incessant support and motivation throughout my graduate study at University of Louisiana at
Lafayette. He helped and guided me in every aspect of my study throughout this journey.
Without his support, encouragement and excellent guidance, this manuscript might not have
been completed. I would like to thank my defense committee members, Dr. Jim Lee and
Dr.William E. Simon for providing positive criticism in order to improve my thesis.
The project work was supported by Louisiana Space Consortium (LaSPACE). I
would like express my thanks to Dr. John Wefel and other personnel in that organization for
their help. I would like to recognize my friends at University of Louisiana at Lafayette,
Purrostam Sigdel and Ashok Khadka, who always helped me to step forward for my goal and
objective. I would like to thank my hardworking parents for supporting me in every aspects
of my life. They always devoted their love and generosity to me. I would like to remember
my late mom for giving me birth who passed on just before I came to USA. I would not have
made it this far without her love and encouragement. I wished she could have lived to see my
graduation and success. I will always miss her.
Last but not the least, I would like to recognize my sister, Sarita Chand, who always
helped me and encouraged me during my hard time of life and during my study at University
Louisiana at Lafayette. She has been my best friend all my life. She was always there making
me happy and always stood by me through the good times and bad, I love her dearly.
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TABLE OF CONTENTS
ACKNOWLEDGMENTS ..................................................................................................... iv
LIST OF TABLES ................................................................................................................ vii
LIST OF FIGURES ............................................................................................................. viii
CHAPTER 1: INTRODUCTION .......................................................................................... 1 1.1 Background ................................................................................................................ 1 1.2 Motivation and Objectives ......................................................................................... 3 1.3 Organization ............................................................................................................... 4
CHAPTER 2: DEVELOPMENT OF AN EFFICIENT NUMERICAL METHOD FOR SOLVING HEAT EQUATIONS APPLYING HES VARIATIONAL ITERATION TECHNIQUE .................................................................................................. 5
2.1 Introduction ................................................................................................................ 5 2.2 Heat Equation............................................................................................................. 7 2.3 Hes Variational Iteration Method (HVIM) ............................................................... 8 2.4 Technical Approach and Validation .......................................................................... 9
2.4.1 Example 1 ................................................................................................. 10 2.4.2 Example 2 ................................................................................................. 14
2.5 Discussion ................................................................................................................ 18 2.6 Conclusion ............................................................................................................... 19 References ....................................................................................................................... 20
CHAPTER 3: DEVELOPMENT OF AN EFFICIENT NUMERICAL METHOD FOR SOLVING ONE-DIMENSIONAL PARABOLIC EQUATIONS APPLYING HES VARIATIONAL ITERATION TECHNIQUE ........................................................ 22
3.1 Introduction .............................................................................................................. 22 3.2 Hes Variational Iteration Method (HVIM) ............................................................. 25 3.3 Technical Approach ................................................................................................. 25 3.4 Illustrative Examples ............................................................................................... 28
3.4.1 Example 1 ................................................................................................. 28 3.4.2 Example 2 ................................................................................................. 29 3.4.3 Example 3 ................................................................................................. 30
3.5 Comparison with Adomian Decomposition Method (ADM) .................................. 32 3.6 Discussion and Conclusion ...................................................................................... 33 References ....................................................................................................................... 34
CHAPTER 4: CONCLUSION............................................................................................. 36
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vi
ABSTRACT ........................................................................................................................... 38
BIOGRAPHICAL SKETCH ............................................................................................... 40
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LIST OF TABLES
Table 2.1. Comparison between the developed algorithm solutions and the exact solutions for Eqn. (2.2) for c=500 at different times ................................................12
Table 2.2. Comparison between the presented algorithm solutions and the exact solution, Eqn. (2.32) for c=500 at different times ....................................................16
Table 3.1. Comparison of number of iterations between presented method and ADM for test example 1. .....................................................................................................32
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LIST OF FIGURES
Figure 2.1. Graph generated from exact solution, Eqn. (2.20), u(x, t) .....................................13
Figure 2.2. Graph generated from presented method, approximate solution, u1(x, t) ..............13
Figure 2.3. Graph generated from exact solution, Eqn. (2.32), u(x, t) .....................................17
Figure 2.4. Graph generated from presented method, approximate solution, u1(x, t) ..............17
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CHAPTER 1: INTRODUCTION
1.1 Background
Differential equations play a prominent role in different engineering branches from
modeling engineering structures, describing important phenomena, to simulating the
numerical behavior of engineering dynamic systems. They can be used as powerful
numerical tools to model a variety of engineering systems and scenarios. Various scientific
laws, engineering principles and systems such as biological processes, celestial motion,
hydraulic flow, heat transfer, vibration isolation, electric circuits, etc. can be described,
modeled and predicted by differential equation systems. Currently, engineering-related
research and study have been dramatically developed and a variety of differential equations
are required to model the processes encountered in research and studies. It is well known that
a large number of engineering structures, systems, and processes are modeled with
differential equations. For example, continuum beams are applied in many flexible space
structures, such as aircraft wings, satellite antenna, solar panels of a high aspect ratio, vehicle
chassis and other engineering architectures.
Differential equations are used to model these devices and estimate their natural and
forced frequencies. In a Heating Ventilation and Air Conditioning (HVAC) system, heat
transfer and cooling processes are needed to control climate, and to keep occupants
comfortable by regulating temperature and air flow. Such processes are described using
thermal system equations, which are mathematically differential equations. Meanwhile, the
heat equations are even used for thermal-based damage detection of porous materials.
Moreover, feedback control systems play a critical role in automotive, aerospace, and ocean
vehicles as well as other engineering architectures. Numerical models of those systems, as
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well as many other physical systems, are also modeled based on the differential equation
systems. Similarly, differential equations are used to model population growth, chemical
reactions, chemical mixtures, etc.
Because of the importance of the differential equations in many engineering
disciplines, a number of analytical and numerical methods have been proposed to accurately
solve for various linear and nonlinear differential equations. For example, Adomian
decomposition method (ADM), one of the most popular methods, has been extensively
applied for solving a variety of differential equations such as free vibration of continuum
beam equations, fractional differential equations, heat equations and cooling problems,
Blasius equations, and other higher-order nonlinear differential equations. Other popular
solution methods include homotopy perturbation method, differential transform method,
differential quadrature method, collocation method, etc.
Among the different numerical methods, a recently proposed analytical method, Hes
variational iteration method (HVIM), is considered as the most powerful, effective, and
convenient way for solving simple to complex higher order linear and nonlinear differential
equations. This method has been proved to be versatile, which can effectively, easily, and
accurately solve a large class of nonlinear problems with rapid convergence after a less
number of iterations. Therefore, the variational iteration technique can be applied to
efficiently solve the differential equations, which is extremely important in engineering
research. In this study, efficient numerical methods were developed for solving two types of
differential equation problems, a heat transfer equation and a parabolic equation by using the
HVIM. The developed methods can be widely used in different engineering branches for
engineering systems design and modeling.
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1.2 Motivation and Objectives
A variety of engineering systems can be modeled and described as differential
equation systems. Therefore, in order to model, evaluate, and predict the behavior of those
engineering systems correctly and efficiently, innovative numerical methods for solving the
differential equation systems are in high demand. For instance, in the space and aerospace
industry, aerospace systems are becoming more complex. It takes more simulation time to
complete a maneuver on space vehicles, and more logical decisions are required in the
various flights of high performance missiles. These complicated engineering systems require
more computational work to simulate and predict their behavior, and it is almost impossible
to achieve a real-time solution with traditional numerical algorithms.
Therefore, efficient algorithms are highly demanded which can solve the differential
equation systems swiftly and correctly by using fewer iterations. Unfortunately, different
types of deficiencies are found in existing numerical methods, such as limited convergence,
divergent results, unrealistic assumptions, huge computational work, and non-compatibility
with the versatility of physical problems. In order to fill this gap, the present research
presents an innovative method for solving the differential equations by using the HVIM.
As verified from previous research of the authors as well as other investigators, the
HVIM is a very powerful method in solving differential equation systems and is not
encumbered with the aforementioned deficiencies encountered in other numerical methods.
Previous results showed that the HVIM lead to rapid convergence within a considerably
smaller number of iterations.
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4
Thus, the overall goal of this research is to develop efficient methods for solving a
heat equation and a parabolic equation employing the HVIM. Specific tasks of this study are
as follows:
1. To develop a computationally efficient algorithm for solving a one-
dimensional heat transfer equation and validate its accuracy and efficiency.
2. To develop a computationally efficient algorithm for solving a one-
dimensional parabolic equation and validate its accuracy and efficiency.
1.3 Organization
This thesis was written in a compilations style format, which is a collection of the
authors two published journal papers. This thesis was organized according to the Guidelines
for the Preparation of Theses and Dissertations. The remaining chapters of this thesis are
organized as follows: Chapter 2 (the first paper) presents and validates a method of using
HVIM to solve for one-dimensional heat equation. Chapter 3 (the second paper) presents a
method to solve a parabolic equation by using the HVIM and validates its efficiency. Chapter
4 concludes this study and suggests future work to further extend and improve the present
research.
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1This chapter was prepared based on a published paper: Y.-C. Liu and M. Chand, Development of an efficient numerical method for solving heat equations applying Hes variational iteration method, Applied Mathematical Sciences, 7(2), 2013, 93-102.
CHAPTER 2: DEVELOPMENT OF AN EFFICIENT NUMERICAL METHOD FOR
SOLVING HEAT EQUATIONS APPLYING HES VARIATIONAL ITERATION
TECHNIQUE1
Abstract: This paper presents a numerical method that solves heat equations using Hes
variational iteration method (HVIM). It shows that the solutions obtained from the developed
method converged rapidly to the exact solutions within three iterations. It is also found that
the HVIM results trivial solutions for nonlinear differential equations with zero initial
condition.
Keywords: Heat Equation, Hes variational iteration method, differential equation,
numerical algorithm, initial condition.
2.1 Introduction
The heat equation is an important partial differential equation which describes the
distribution of heat (or variation in temperature) in a given region over time. Many important
engineering processes, such as the heat transferring and cooling process, are described
through heat equations, today, heat equations have been applied in thermal-based damage
detection in porous materials. Due to its importance in engineering design and research, a
number of investigators have proposed analytical methods to find promising approximate
solutions for such equations. Dawson et al. [2.3] presented a finite difference domain
decomposition algorithm for a numerical solution of the heat equation. That method was later
developed and proposed by Dehghan [2.4] for solving the one-dimensional heat equation
subject to the specification of mass. Similarly, Khan et al. [2.10] used two-step Adomian
decomposition method (ADM) to solve for the heat equation and Alizadeh et al. [2.2] found
solutions for the cooling problem using ADM. Lu et al. [2.14], however, developed a novel
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analytical approach for heat equations in a multi-dimensional composite slab subject to time-
dependent boundary changes of the first kind. It was found that the presented method
involves no iterative computation such as numerically searching for eigenvalues and no
residue evaluation, therefore it is considered very efficient.
Besides the existing numerical methods, an analytical method, Hes variational
iteration method (HVIM), is considered as an effective and convenient method for solving
both weakly and strongly nonlinear equations. HVIM was originally developed by a Chinese
Mathematician, Ji-Huan He, for solving different differential equation systems [2.5-2.8]. This
method is modification of a general Lagrange multiplier method [2.5-2.8]. One author of this
paper has extensively applied HVIM for solving broad types of analytical problems, such as
nonlinear differential difference equations [2.11], Blasius equation [2.12], and the free
vibration of an Euler-Bernoulli beam [2.13].
Based on the current progress made using HVIM for solving nonlinear differential
equations and the authors experience, we plan to develop an efficient algorithm for solving
the heat transfer equations by employing HVIM. The developed algorithm can later be
implemented into computer programs so as to significantly improve the computers
performance in design and simulation of heat systems. The remaining sections of this paper
are organized as follows: Section 2.1 introduces the heat transfer equation that will be solved
in this paper; section 2.2 briefly reviews the HVIM method; section 2.3 explains detailed
approach of solving the heat equations using HVIM and validates the presented method and
the discussion that is presented in section 2.4. Finally, the paper is concluded and ended by
section 2.5.
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2.2 Heat Equation
The heat equation is an important partial differential equation which describes the
distribution of heat in a given region over time. In this paper, we only consider the one-
dimensional heat equation with variable properties, as shown in Eq. (2.1). The developed
analytical method then can be expanded to solve for a real heat equation with three spatial
variables (x, y, z).
0,)()( >
=
tx
uxk
xt
uxc
(2.1)
Where, c; specific heat capacity of the material, ; the mass density and k; the thermal
conductivity are taken as functions of x.Then ,the Eqn. (2.1) becomes a standard one
dimensional heat equation if and k are constants, which had been successfully solved by
variational iteration methods including HVIM [2.14-2.18]. In this paper, we will continue to
apply HVIM to develop numerical solutions for Eqn. (2.1), where and k are functions of x
and such equation has a wide application in material design. Eqn. (2.1) with zero initial
condition had been solved by Khan et al. [2.10] using modified ADM. Unfortunately, if
HVIM was used to solve such equation with zero initial condition, only trivial solutions
would be obtained. This is because that in using HVIM, the successive terms of iterations
fully depend upon the initial condition, which could not be zero if meaningful iterations are
wanted. A new technique for finding initial conditions has been proposed by Ali [2.1]. In this
paper, the heat equation Eqn. (2.1) with nonzero initial conditions will be solved by HVIM,
which will lead to significant results.
Specifically, If (x) = 1/x and k(x) = x, and assume x varies from 1 to e, then Eqn.
(2.1) takes the form
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0, >
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9
use of the integral operator; as a result it becomes very convenient and fast to find the
solution of differential equations while still maintaining the high level of accuracy. Hence,
VIM has a wide range of application as compared to other methods. In variational technique,
first of all the differential equation is written in terms of correction function by introducing
Lagrange Multiplier, which is to be found by variation theory. Consider the general nonlinear
differential equation given in the form
Lu (t) + Nu (t) = g (t) (2.4)
Where, L is a linear operator, N is a nonlinear operator, and g (t) is a known function.
By using the variational iteration method, a correction functional can be constructed as
++=+t
nnnn dguNLututu 01 ))()(~)(()()(
(2.5)
Where, is a general Lagrange multiplier, which can be determined optimally via
variational theory; the subscript n means the nth approximation; nu
~
is a restricted variation
and nu
~ = 0. u1(t), u2(t), un(t) can then be found to from Eqn. (2.5). The solution to
the Eqn. (2.5) then can be obtained as
u(t) = limnun( t) (2.6)
2.4 Technical Approach and Validation
The steps of solving the heat equations using HVIM can be summarized as: 1) find
the correctional function and simplify it (Eqn. (2.5)); 2) take variation on both sides of the
simplified correctional function with respect to un and from there to derive the stationary
conditions; 3) find an appropriate Lagrange multiplier from the stationary conditions; 4)
substitute back to the correctional function and the final solution then can be found using
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Eqn. (2.6). Following this procedure, analytical solutions will be found for Eqns. (2.2) and
(2.3) and will be compared with the exact solutions for validation.
2.4.1 Example 1
Comparing Eqn. (2.2) with (2.4), the correctional function can be written as
++=+t
nnnn dxgxuNxLutxutxu 01 )),(),(~),((),(),(
(2.7)
Substituting Eqn. (2.2) into (2.7) and we obtain
+=+
t
nn
nn dxux
xxc
xxutxutxu
01),(),((),(),(
(2.8)
where is the Lagrange multiplier which can be identified by imposing stationary
conditions.
Next, variation was taken on both sides of correctional function with respect to un to
derive the stationary conditions.
+=+
t
nn
nn dxux
xxc
xxutxutxu
01),(~),((),(),(
(2.9)
+=+
tn
nn dxu
txutxu01
0),((),(),(
(2.10)
Where u~ is considered as restricted variation, i.e. 0~ =nu
Integrating Eqn. (2.10) by part and we can have
+= =+t
ntnnndxuxutxutxu
01),('),(),(),(
(2.11)
Comparing the coefficients of and , the stationary conditions then can be found
based on Eqn. (2.11), which are
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11
t
t
=
=
=
=+
0)('0)(1
(2.12)
From Eqn. (2.12), the Lagrange multiplier was therefore determined as -1. So
substituting it back to Eqn. (2.8) and the variation iteration formula can be obtained as
=+
t
nn
nn dxux
xxc
xxutxutxu
01),(),((),(),(
(2.13)
Taking the initial condition: sin(lnx)),(0 =txu
The other ),( txu can be determined from the iteration formula Eqn. (2.5) as
=
=
tdxu
xx
xc
xxutxutxu
0 00
01 ),(),((),(),(
( ) ( ) ( ) ( ) ( ),lns lnslns)lns(lns0
xinc
txindxin
xx
xc
xxinxin
t
=
(2.14)
( ) ( ) ( ) ,!2lns
!1lnslns),(
2
2xin
c
txinc
txintxu
+
= (2.15)
( ) ( ) ( ) ( ) ,!3ln
!2ln
!1lnln),(
32
3xin
c
txinc
txinc
txintxu ssss
+
= (2.16)
( ) ( ) ( ) ( ) ( ) ,!4ln
!3ln
!2ln
!1lnln),(
432
4xin
c
txinc
txinc
txinc
txintxu sssss
+
+
= (2.17)
( ) ( ) ( ) ( ) ( )
( ),
!5lns
!4lns
!3lns
!2lns
!1lnslns),(
5
432
5
xinc
t
xinc
txinc
txinc
txinc
txintxu
+
+
+
=
(2.18)
In this manner, the rest of the terms can be calculated. Summing up, the series
solution is given by
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12
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )..............
!8ln
!7ln
!6ln
!5ln
!4ln
!3ln
!2ln
!1lnln),(
8765
432
+
+
+
+
+
+
+
=
xinc
txinc
txinc
txinc
t
xinc
txinc
txinc
txinc
txintxun
ssss
sssss
(2.19)
Then the closed form solution is given as
( ) ctexintxu /*ln),( = s (2.20)
This is also exact solution of the Eqn. (2.2). Based on Eqn. (2.6) ),( txun will
converge to the exact solution when n . In this case the exact solutions are found after
running three iterations. The numerical solutions ),( txun are compared with the exact
solutions ( ) ctexintxu /*ln),( = s and listed in Table 2.1 and plotted in Fig.2.1 and Fig.2.2 to
validate the present method.
Table 2.1. Comparison between the developed algorithm solutions and the exact solutions for Eqn. (2.2) for c=500 at different times
t x Exact solution u1(x,t) u2(x,t) u3(x,t) 0.25 0.3 -0.93300468439 -0.93300456773 -0.93300468441 -0.93300468439
0.6 -0.48865325131 -0.48865319020 -0.48865325132 -0.48865325131
0.9 -0.10511312245 -0.10511310930 -0.10511312245 -0.10511312245
0.50 0.3 -0.93253829865 -0.93253783207 -0.93253829881 -0.93253829865
0.6 -0.48840898575 -0.48840874139 -0.48840898583 -0.48840898575 0.9 -0.10506057902 -0.10506052646 -0.10506057904 -0.10506057902
0.75 0.3 -0.93207214605 -0.93207109642 -0.93207214658 -0.93207214605 0.6 -0.48816484230 -0.48816429257 -0.48816484258 -0.48816484230
0.9 -0.10500806186 -0.10500794361 -0.10500806192 -0.10500806186 1.00 0.3 -0.93160622647 -0.93160436077 -0.93160622771 -0.93160622647
0.6 -0.48792082089 -0.48791984375 -0.48792082154 -0.48792082089
0.9 -0.10495557096 -0.10495536077 -0.10495557110 -0.10495557096
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13
Figure 2.1. Graph generated from exact solution, Eqn. (2.20), u(x, t)
Figure 2.2. Graph generated from presented method, approximate solution, u1(x, t)
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14
2.4.2 Example 2
Comparing Eqn. (2.3) with (2.5), the correction function is obtained as Eqn. (2.7);
after substituting Eqn. (2.3) into Eqn. (2.7), we have
+=+
t
n
n
nn dxux
xxc
xxutxutxu
02
2
1 ),()1(}1{),((),(),(
(2.21)
Similarly, by taking variation on both sides of Eqn. (2.17) with respect to un, the
stationary conditions were derived as
+=+
t
n
n
nn dxux
xxc
xxutxutxu
02
2
1 ),(~11),((),(),(
(2.22)
+=+
tn
nn dxu
txutxu01
0),((),(),(
(2.23)
Just like Example 1, u~ is set as 0.
To determine the Lagrange Multiplier , Eqn. (2.23) was integrated by part to obtain
+= =+t
ntnnndxuxutxutxu
01),('),(),(),(
(2.24)
The stationary conditions are found as (which is same as Example1)
t
t
=
=
=
=+
0)('0)(1
The Lagrange multiplier was therefore determined as = -1.Substitting back to Eqn.
(2.20) and the variational iteration formula was obtained as
=+
t
nn
nn dxux
xxc
xxutxutxu
02
2
1 ),(11),((),(),(
(2.25)
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15
Taking the initial condition )1),( 20 xtxu = , the other u(x, t) can be determined
from the iteration formula Eqn. (2.5) as
=
tdxu
xx
xc
xxutxutxu
0 02
20
01 ),(11),((),(),(
=t
dxx
xxc
xxx
022
222 11111
( ) ,1 1 22 xc
tx
= (2.26)
,
!21
!111),(
2222
2x
c
tx
c
txtxu
+
= (2.27)
,
!31
!21
!111),(
232222
3x
c
tx
c
tx
c
txtxu
+
= (2.28)
,
!41
!31
!21
!111),(
24232222
4x
c
tx
c
tx
c
tx
c
txtxu
+
+
= (2.29)
,
!51
!41
!31
!21
!111),(
25
24232222
5
x
c
t
x
c
tx
c
tx
c
tx
c
txtxu
+
+
+
=(2.30)
In this the manner, the rest of the components of the series can be obtained. Summing
up, the series solution is given by
..........
!81
!71
!61
!51
!41
!31
!21
!111),(
28272625
24232222
5
+
+
+
+
+
+
+
=
x
c
tx
c
tx
c
tx
c
t
x
c
tx
c
tx
c
tx
c
txtxu
(2.31)
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16
Then the closed form of solution from the above equation can be written as
( ) ctextxu /2 *1),( = (2.32)
Based on the Eqn. (2.6) ),( txun will converge to exact solution, which is Eqn. (2.32).
Comparing the iteration results to the exact solution of the heat equation (Eqn. (2.3)) with
specified boundary and initial conditions (Eqn. (2.32)), it was found that the numerical
results converged very fast to the exact solutions only in three iterations, which is displayed
in Table 2.1 and plotted in Fig. 2.3 and Fig. 2.4.
Table 2.2. Comparison between the presented algorithm solutions and the exact solution, Eqn. (2.32) for c=500 at different times
t x Exact(x,t) VIM,u1(x,t) VIM,u2(x,t) VIM,u3(x,t) 0.25 0.3 0.95346235104 0.95346223182 0.95346235106 0.95346235104
0.6 0.79960009998 0.79960000000 0.79960010000 0.79960009998
0.9 0.43567200388 0.43567194941 0.43567200389 0.43567200388
0.5 0.3 0.95298573903 0.95298526222 0.95298573919 0.95298573903
0.6 0.79920039987 0.79920000000 0.79920040000 0.79920039987
0.9 0.43545422233 0.43545400446 0.43545422240 0.43545422233
0.75 0.3 0.95250936526 0.95250829261 0.95250936580 0.95250936526
0.6 0.79880089955 0.79880000000 0.79880090000 0.79880089955
0.9 0.43523654964 0.43523605951 0.43523654989 0.43523654964
1.00 0.3 0.95203322962 0.95203132301 0.95203323089 0.95203322962
0.6 0.79840159893 0.79840000000 0.79840160000 0.79840159893
0.9 0.43501898576 0.43501811457 0.43501898635 0.43501898576
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Figure 2.3. Graph generated from exact solution, Eqn. (2.32), u(x, t)
Figure 2.4. Graph generated from presented method, approximate solution, u1(x, t)
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18
2.5 Discussion
As can be seen from the above table and figure, for the heat equation with two
variables and nonzero initial condition, the presented methods results quickly converged to
the exact solutions within three iterations. The surfaces shown in Fig. 2.1 and Fig. 2.3 are
generated from the exact solutions and Fig. 2.2 and Fig. 2.4 are generated from the first
iteration equation of the corresponding heat equations 2.2 and 2.3 respectively. The graphs
are so close to each other. This verified that the approximate solution is very close to the
exact solution. The comparison verified that the developed method is accurate and fast. The
present study also verifies that even the original HVIM was developed for solving nonlinear
differential equations with only one variable, the HVIM approach can also be used to solve
differential equations with two variables (x, and t as seen from Eqns. (2.2) and (2.3)). For
convenience, c is assumed to be 500 for the illustrative examples, the value of c can vary
according to the material required and the tables only list results for certain ts and xs.
From the result it is clear that the solution is converged to the exact solution. The
obtained closed form solution is satisfied by the given boundary conditions and the initial
conditions so it can be concluded that the obtained solutions are exact solutions for the
corresponding heat equations. To validate its accuracy, the numerical results of the exact
solution are shown in graph and compared with the different stages of the iterations. It was
found that there is no necessary to proceed further in iterations after the third step, since the
solutions converged to the exact solution just in three iterations in case of both examples.
The presented equations with the particular boundary conditions and the initial
conditions in this paper have not been solved previously by any other method. Hence, the
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19
results are compared with the exact solutions rather than any other methods. Since the
solutions converged to the exact solution this proves that this method is the correct method.
2.6 Conclusion
In this paper, an efficient approach is presented to implement HVIM to accurately
solve for heat equations with two variables x and t, and whose material conductivity and
density are functions of x. On comparing the HVIM results to the exact solutions, it was
found that the presented approach leads to a rapid convergence to the exact solutions in less
than three iterations. Through the illustrative examples, it is therefore concluded that HVIM
is an efficient numerical tool which can be used for solving the heat equations to greatly
reduce the size of calculations while maintaining a high level of accuracy and efficiency.
Also, it could be found that for the heat equations ((2.2) and (2.3)), if the initial conditions
are zero, the application of HVIM will lead to a series of trivial solutions, which are of no
use. Therefore, it is revealed that as an efficient tool for solving nonlinear differential
equations, HVIM is not applicable for the differential equations with zero initial condition.
The approach presented in this study can be developed as an efficient algorithm to model and
analyze the heat and cooling systems, which are governed by the heat equations and
extensively used in different engineering architectures.
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20
References
[2.1] E.J. Ali, Modified treatment of initial boundary value problems for one dimensional heat-like and wave-like equations using variational iteration method, Applied Mathematical Sciences, 6(33), 2012, 1613-1626.
[2.2] E. Alizadeh, K. Sedighi, M. Farhadi and H.R. Ebrahimi-Kebria, Analytical approximate solution of the cooling problem by Adomian decomposition method, Communications in Nonlinear Science and Numerical Simulation, 14, 2009, 462-472.
[2.3] C.N. Dawson, Q. Du and T.F. Dupont, A finite difference domain decomposition algorithm for numerical solution of the heat equations, Mathematics of Computation, 57 (195), 1991, 63-71.
[2.4] M. Dehghan, The one-dimensional heat equation subject to a boundary integral specification, Chaos, Solitons & Fractals, 32(2), 2007, 661-675.
[2.5] T.H. Hao, Search for Variational Principles in Electrodynamics by Lagrange Method, Int.J.of Nonlinear Sciences and Numerical Simulation, 6(2), 2005, 209-210.
[2.6] J.-H. He, Variational iteration method a kind of non-linear analytical technique: some examples, International Journal of Non-Linear Mechanics, 34(4), 1999, 699-708.
[2.7] J.-H. He, Variational iteration method for autonomous ordinary differential systems, Applied Mathematics and Computation, 114(2/3), 2000, 115-123.
[2.8] J.-H. He, Variational iteration method for delay differential equations, Communications in Nonlinear Science and Numerical Simulation, 2(4), 1997, 235-236.
[2.9] J.-H. He, Variational iteration method some recent results and new interpretations, Journal of Computational and Applied Mathematics, 207(1), 2007, 3-17.
[2.10] D.N. Khan Marwat and S. Asghar, Solution of the heat equation with variable properties by two-step Adomian decomposition method, Mathematical and Computer Modelling, (48), 2008, 83-90.
[2.11] Y.-C. Liu and C.S. Gurram, Solving nonlinear differential difference equations using Hes variational iteration method, Applied Mathematical and Computational Sciences, 3(1), 2011, 33-46.
[2.12] Y.-C. Liu and S.N. Kurra, Solution of Blasius equation by variational iteration, Applied Mathematics, 1(1), 2011, 24-27.
-
21
[2.13] Y.-C. Liu and C.S. Gurram, The use of Hes variational iteration method for solving free vibration of Euler-Bernoulli beam, Mathematical and Computer Modelling, 50(11/12), 2009, 1545-1552.
[2.14] X. Lu, P. Tervola and M. Viljanen, A new analytical method to solve the heat equation for a multi-dimensional composite slab, Journal of Physics A: Mathematical and General, 38(13), 2005, 2873.
[2.15] Mo. Miansari, D.D. Ganji and Me. Miansari, Application of Hes variational iteration method to nonlinear heat transfer equations, Physics Letters A, 372(6), 2008, 779-785.
[2.16] M. Tatari and M. Dehghan, Hes variational iteration method for computing a control parameter in a semi-linear inverse parabolic equation, Chaos, Solitons & Fractals, 33(2), 2007, 671-677.
[2.17] S.-Q. Wang, J.-H. He, Variational iteration method for solving integro-differential equations, Physics Letters A, 367(3), 2007, 188-191.
[2.18] R.Yulita Molliq,M.S.M. Noorani,I.Hashim,Variational iteration method for fractional heat-and wave-like equations, Nonlinear Analysis, Real World Applications,10, 2009, 1854-1869.
-
2This chapter was prepared based on a published paper: Liu,Yucheng and Chand, M., The Use of Hes Variational Iteration Method for Solving the One-Dimensional Parabolic Equation with Non-Classical Boundary Conditions, Applied Mathematical Sciences, 2013 7(85) pp. 4213 4221.
CHAPTER 3: DEVELOPMENT OF AN EFFICIENT NUMERICAL METHOD FOR
SOLVING ONE-DIMENSIONAL PARABOLIC EQUATIONS APPLYING HES
VARIATIONAL ITERATION TECHNIQUE2
Abstract: Over the last 15 years, Hes variational iteration method (HVIM) has been applied
to obtain formal solutions to a wide class of differential equations. This method leads to
computable and efficient solutions to linear and nonlinear operator equations. The parabolic
partial differential equations with non-classical boundary conditions model various physical
problems. The aim of this paper is to investigate the application of HVIM for solving the
second-order linear parabolic partial differential equation with non-classical boundary
conditions. HVIM provides a reliable technique that requires less work when compared with
the traditional techniques such as the Adomian decomposition method (ADM). The present
approach can be used and extended for investigating more scientific applications.
Keywords: Hes variational iteration method, one-dimensional parabolic equation, non-
classical boundary conditions, numerical algorithm, closed form solutions
3.1 Introduction
One-dimensional parabolic equation is a type of second-order partial differential
equation (PDE) which describes a wide family of problems in science and engineering
including heat diffusion, ocean acoustic propagation, etc. Such equations with non-classical
boundary conditions has many important applications in chemical diffusion, thermo
elasticity, heat conduction processes, population dynamics, vibration problems, nuclear
reactor dynamics, inverse problems, control theory, medical science, biochemistry, and
certain biological processes [3.2-3.5, 3.9, 3.14, and 3.15].
In this study, such problem takes the form
-
23
Ttxx
u
t
u
-
24
(ADM) to solve such problem. It was found that in comparison to the traditional techniques,
the ADM is more powerful and efficient in finding exact solutions for the one-dimensional
parabolic equation with non-classical boundary specifications. Three examples were
demonstrated to test the ADM approach. Dehghans work presents a benchmark for this
study, which is aimed at developing and validating a new approach of solving such parabolic
equation employing Hes variational iteration method (HVIM).
Besides the aforementioned numerical methods, HVIM is an effective and convenient
method for solving both weakly and strongly nonlinear equations, which was originally
presented by He for solving differential equation systems [3.10-3.13]. The author has
extensively applied HVIM for solving broad types of differential equation systems, such as
heat equations, free vibration of Euler- Bernoulli beam, the nonlinear differential difference
equations, Blasius equations, and [3.16-3.19]. From those works, it is found that both HVIM
and ADM are efficient and powerful methods which can lead to correct solutions in closed
form.
Comparing to ADM, HVIM requires less calculations and gives a direct approximate
solution at each iteration step. Based on the previous results, this paper aims at using HVIM
to solve the one-dimensional parabolic partial differential equation with non-classical
boundary conditions in rapidly convergent series. The developed approach is validated by
comparing the analytical results with those obtained from ADM. In the future, the algorithms
of applications of HVIM can be implemented into computer programs in order to improve
the computers performance in analyzing and simulating the differential equation systems.
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25
3.2 Hes Variational Iteration Method (HVIM)
In this section, the concept of Hes variational iteration method is briefly introduced.
Consider the general nonlinear differential equation given in the form
Lu (t) + Nu (t) = g (t) (3.5)
where L is a linear operator, N is a nonlinear operator, and g(t) is a known function.
By using the variational iteration method, a correction functional can be constructed as
++=+t
nnnn dguNLututu 01 ))()(~)(()()(
(3.6)
where is a general Lagrange multiplier, which can be determined optimally via
variational theory; the subscript n means the nth approximation; un is a restricted variation
and un = 0. u1 (t), u2 (t) un(t) can then be found from Eqn. (3.6). The solution to the Eqn.
(3.6) then can be obtained as u(t) = limnun( t) (3.7)
3.3 Technical Approach
The steps of using HVIM for solving Eqns. (3.1) to (3.4) are described as follows.
Comparing Eqn. (3.1) with (3.5) to obtain the correctional function:
++=+t
nnnn dxgxuNxLutxutxu 01 )),(),(~),((),(),(
(3.8)
un+1(x, t) is expressed by substituting Eqn. (3.1) into (3.8) as:
+=+
tnn
nn dx
xuxutxutxu
0 2
2
1),(),((),(),(
(3.9)
In above equations, is the Lagrange multiplier, which can be identified by imposing
stationary conditions.
Next, the stationary conditions were derived by taking variation on both sides of the
correctional function (Eqn. (3.8)) with respect to un.
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26
+=+
tnn
nn dx
xuxutxutxu
0 2
2
1),(~),((),(),(
(3.10)
+=+
tn
nn dxu
txutxu01
0),((),(),(
(3.11)
Where nu~
is considered as restricted variation, i.e. 0~ =nu .
Afterwards, Eqn. (3.11) was integrated by part and we have:
+= =+t
ntnnndxuxutxutxu
01),('),(),(),(
(3.12)
The stationary conditions then can be found from Eqn. (3.12) as:
t
t
=
=
=
=+
0)('0)(1
(3.13)
According to Eqn. (3.13), the Lagrange multiplier was found as -1. Using that value
for Eqn. (3.9) and the variation iteration formula was obtained as:
=+
tnn
nn dx
xuxutxutxu
0 2
2
1),(),((),(),(
(3.14)
Applying the initial condition (Eqn. (3.2)) into the iteration formula Eqn. (3.9) to
determine u(x, t), and un(x, t) can be solved based on the recurrence relation (Eqn. (3.14)) as:
!1)()(),(
,
)()()(
),(),((),(),(
21
0 2
2
0 2
20
01
txfxftxu
ordx
xfxfxf
dx
xuxutxutxu
t
tn
+=
=
=
(3.15)
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27
,
!2)(
!1)()(),(
242
2t
xftxfxftxu ++= (3.16)
,
!3)(
!2)(
!1)()(),(
36
242
3t
xftxftxfxftxu +++= (3.17)
,
!4)(
!3)(
!2)(
!1)()(),(
48
36
242
4t
xftxftxftxfxftxu ++++= (3.18)
,
!5)(
!4)(
!3)(
!2)(
!1)()(),(
510
48
36
242
5t
xftxftxftxftxfxftxu +++++= (3.19)
In the similar manner, the rest of the components can be obtained as per the
requirement
For n number of iterations, the series solution can be written as
........................
!8)(
!7)(
!6)(
!5)(
!4)(
!3)(
!2)(
!1)()(),(
816
714
612
510
48
36
242
++++
+++++=
txftxftxf
txftxftxftxftxfxftxun
(3.20)
Finally the closed form of the solution can be written as follow
=
+=1
2
!)()(),(
n
nn
nn
txfxftxu
(3.21)
Here f(x) can be any kind mathematical functions. The superscript of function f
denotes the nth derivative of that function. It can be found from here that HVIM is performed
based on the initial condition only and no specific boundary condition is required. Instead,
given boundary conditions can be directly implemented into the original equation to simplify
the problem, if required. This suggests another advantage of HVIM in solving numerical
problems.
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28
3.4 Illustrative Examples
Three illustrative examples were demonstrated and solved applying the present
HVIM approach. The examples had been solved before by Dehghan [3.7] using ADM. The
results were compared to validate the accuracy and efficiency of the present approach.
3.4.1 Example 1
Considering equations (3.1) (3.4), Assuming function,
f(x) = e-x, 0 < x < 1; (3.22)
1=
e
e , (3.23)
11
=e
, (3.24)
which is easily seen to have exact solution
u(x, t) = et-x. (3.25)
Using Eqn. (3.21) by assuming u0(x, t) = f(x) = e-x, we can write:
,
!1),(
,)(
),(),((),(),(
1
0 2
2
0 2
20
01
teetxu
ordx
eexf
dx
xuxutxutxu
xx
t xx
tn
+=
=
=
(3.26)
,
!2!1),(
2
2t
et
eetxu xxx ++= (3.27)
,
!3!2!1),(
32
3t
et
et
eetxu xxxx +++= (3.28)
,
!4!3!2!1),(
432
4t
et
et
et
eetxu xxxxx ++++= (3.29)
-
29
until
+++++= .............
!4!3!2!11),(
432 ttttetxu x (3.30)
The closed form solution can be easily expressed as u(x, t) = e-xet = et-x, which
coincides with the exact solution as well as the solution obtained from ADM. From that
example it can be seen that the analytical solution obtained from presented approach rapidly
converges to the exact solution as in the case of ADM.
3.4.2 Example 2
Considering the equations from (3.1)-(3.4), Assuming the function, f as algebraic
function as below
f(x) = x2/2, (3.31)
166+
=t
t , (3.32)
3616
++
=t
t , (3.33)
This is easily seen to have the theoretical solution
u(x, t) = x2/2 + t. (3.34)
Using Eqn. (21) by assuming u0(x, t) = f(x) = x2/2, and other components can be
obtained using the functional formula as:
-
30
txtxu
ordx
xx
xf
dx
xuxutxutxu
t
tn
+=
=
=
21
0 2
222
0 2
20
01
21),(
,21
21
)(
),(),((),(),(
(3.35)
txtxutxu +=+= 212 210),(),( (3.36)
txtxutxu +=+= 223 210),(),( (3.37)
And so on, the closed form solution can be written as
txtxun +=2
21),( (3.38)
It can be seen that in some cases the series fast converge to the exact solution after the
first iteration. This method is suitable in context of the simple example for which the exact
solution is known and the exact solution can be used for comparison. This example proves
that this method is excellent approximation to the exact solutions.
3.4.3 Example 3
In this, example the trigonometric function has been tested as shown below
f(x) = cos(x), 0 < x < 1; (3.39)
= csc (1), (3.40)
= cotg (1), (3.41)
which is easily seen to have exact solution
u(x, t) = e-tcos(x). (3.42)
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31
Similarly, let u0(x, t) = f(x) = cos(x) and substituting it into Eqn. (3.14) to obtain
following equations:
!1)cos()cos(),(
)cos()cos()(
),(),((),(),(
1
0 2
2
0 2
20
01
txxtxu
dx
xxxf
dx
xuxutxutxu
t
tn
=
=
=
(3.43)
,
!2)cos(
!1)cos()cos(),(
2
2t
xt
xxtxu += (3.44)
,
!3)cos(
!2)cos(
!1)cos()cos(),(
32
3t
xt
xt
xxtxu += (3.45)
,
!4)cos(
!3)cos(
!2)cos(
!1)cos()cos(),(
432
4t
xt
xt
xt
xxtxu ++= (3.46)
and so on.
Summing up all the components, series form solution is given by
+++= .............
!5!4!3!2!11)cos(),(
5432 tttttxtxu (3.47)
It is obviously that as n , un(x, t) u(x, t) = e-tcos(x), which is the exact
solution. From the illustrative examples, it can be found that by using HVIM to solve the
differential equations, there is no boundary condition involved, and usually the boundary
conditions are implemented into the original equation to simplify the problem.
-
32
3.5 Comparison with Adomian Decomposition Method (ADM)
Taking, test example 1.
Table 3.1. Comparison of number of iterations between presented method and ADM for test example 1.
No of Iterations
Adomian Decomposition Algorithm(ADM)[3.7]
Presented Algorithm
1 !1
),(1t
etxu x= !1
),(1t
eetxu xx +=
2 !1
),(2t
eetxu xx += !2!1
),(2
2t
et
eetxu xxx ++=
3 !2!1
),(2
3t
et
eetxu xxx ++= !3!2!1),(
32
3t
et
et
eetxu xxxx +++=
4 !3!2!1),(
32
4t
et
et
eetxu xxxx +++=
Similarly, other examples can be compared with the resented method to compare the
number of iterations. From the above example, it is clear that developed method needs less
iterations than ADM. As noted above, three iterations in presented algorithm equals four
iterations in ADM. In other words, if a result is obtained in fourth iteration in case of ADM,
the same result is obtained in the third iteration in case of this developed method. In a
generalized way, it can be concluded that, n iterations in this method equals to n+1
iterations in Adomian decomposition method (ADM). Hence, we can conclude that this
-
33
approach requires less number of iterations, and in addition, there is no need of calculations
of polynomial Adomian Polynomials [3.7], which greatly reduce the computational work.
3.6 Discussion and Conclusion
This paper presents a method of using HVIM to solve for the one-dimensional
parabolic equation with non-classical boundary conditions and validates the accuracy and
efficiency of this method through three illustrative examples. Comparing the developed
method to ADM, it can be found that both methods yield results that fast converge to the
exact solutions. However, by using the present method, there is no need to calculate the
Adomian coefficients and polynomials, which further simplify the solution process by
reducing computing efforts. From the illustrative examples, it can also be found that the
proposed method results in convergence to the exact solutions faster than the ADM results,
because this method needs less iteration than the ADM.
Furthermore, by using the HVIM, the solution process is executed only based on the
preceding terms and initial conditions, not on the boundary conditions. Instead, the boundary
conditions can be implemented into the original equations to simplify them. Thus, comparing
to other numerical methods, this method using HVIM can handle differential equations with
complex boundary conditions, which is another advantage of the HVIM as an efficient and
powerful tool in numerical analysis. The present method can be used by more researchers to
investigate more scientific applications.
-
34
References
[3.1] W.T. Ang, A method of solution for the one-dimensional heat equation subject to a nonlocal condition, SEA Bulletin of Mathematics, 26(2), 2002, 197-203.
[3.2] J.R. Cannon and H.M. Yin, On a class of non-classical parabolic problems, Journal of Differential Equations, 79, 1989, 266-288.
[3.3] V. Capasso and K. Kuniseh, A reaction-diffusion system arising in modeling man-environment diseases, Quarterly of Applied Mathematics, 46, 1988, 431-449.
[3.4] W.A. Day, Existence of a property of solutions of the heat equation subject to linear thermo elasticity and other theories, Quarterly of Applied Mathematics, 40, 1982, 319-330.
[3.5] W.A. Day, A decreasing property of solutions of a parabolic equation with applications to thermo elasticity and other theories, Quarterly of Applied Mathematics, 41, 1983, 468-475.
[3.6] M. Dehghan, Numerical solution of a parabolic equation with non-local boundary specifications, Applied Mathematics and Computations, 145(1), 2003, 185-194.
[3.7] M. Dehghan, The use of Adomian decomposition method for solving the one-dimensional parabolic equation with non-local boundary specifications, International Journal of Computer Mathematics, 81(1), 2004, 25-34.
[3.8] G. Ekolin, Finite difference methods for a non-local boundary value problem for the heat equation, BIT Numerical Mathematics, 31(2), 1991, 245-261.
[3.9] A. Friedman, Monotonic decay of solutions of parabolic equation with nonlocal boundary conditions, Quarterly of Applied Mathematics, 44, 1986, 468-475.
[3.10] J.-H. He, Variational iteration method a kind of non-linear analytical technique: some examples, International Journal of Non-Linear Mechanics, 34(4), 1999, 699-708.
[3.11] J.-H. He, Variational iteration method for autonomous ordinary differential systems, Applied Mathematics and Computation, 114(2/3), 2000, 115-123.
[3.12] J.-H. He, Variational iteration method for delay differential equations, Communications in Nonlinear Science and Numerical Simulation, 2(4), 1997, 235-236.
[3.13] J.-H. He, Variational iteration method some recent results and new interpretations, Journal of Computational and Applied Mathematics, 207(1), 2007, 3-17.
[3.14] B. Kawohl, Remark on a paper by D. A. Day on a maximum principle under nonlocal boundary conditions, Quarterly of Applied Mathematics, 45, 1987, 751-752.
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35
[3.15] Y. Lin, S. Xu and H.M. Yin, Finite difference approximations for a class of nonlocal parabolic equations, International Journal of Mathematics and Mathematical Sciences, 20(1), 1997, 147-164.
[3.16] Y.-C. Liu and M. Chand, Development of an efficient numerical method for solving heat equations applying Hes variational iteration method, Applied Mathematical Sciences, 7(2), 2013, 93-102.
[3.17] Y.-C. Liu and C.S. Gurram, The use of Hes variational iteration method for solving free vibration of Euler-Bernoulli beam, Mathematical and Computer Modelling, 50(11/12), 2009, 1545-1552.
[3.18] Y.-C. Liu and C.S. Gurram, Solving nonlinear differential difference equations using Hes variational iteration method, Applied Mathematical and Computational Sciences, 3(1), 2011, 33-46.
[3.19] Y.-C. Liu and S.N. Kurra, Solution of Blasius equation by variational iteration, Applied Mathematics, 1(1), 2011, 24-27.
[3.20] S. Wang and Y. Lin, A finite difference solutions to an inverse problem determining a control function in parabolic partial differential equations, Inverse Problems, 5, 1989, 631-640.
[3.21] S.Wang and Y. Lin, A numerical method for the diffusion equation with nonlocal boundary specifications, International Journal of Engineering Science, 28, 1990, 543-546.
-
CHAPTER 4: CONCLUSION
In this study, two types of differential equations have been successfully solved by
using variational iteration technique. The second chapter presents a unique algorithm for
solving one-dimensional heat equations. In that chapter, two heat equations with variable
properties were successfully solved by using variational approach and based on which
algorithms for solving that type of equations were developed. The accuracy of the method
was found by comparing the numerical results with the exact solution.
In the third chapter, another algorithm was developed for solving one-dimensional
parabolic equation by using Hes variational technique. The efficiency of the method was
validated through three numerical examples, which had already been solved by Dehghan
[3.7] using Adomian Decomposition Method (ADM). The results were compared with those
obtained from ADM, and it was discovered that the presented method possesses two unique
merits against the decomposition method. At first, this developed method requires fewer
calculations compared to the ADM method. For instance, the result obtained in n+1
iterations in decomposition method, can be obtained in only n iterations by using the
developed algorithm. Secondly, the developed method eliminates the volume of calculations
by removing the necessity of calculating Adomain polynomials, so the iteration is more
direct and straight forward. In other words, each iteration in the developed algorithm gives a
direct approximation to the solution; however, by using ADM, each iteration only gives the
components of the approximate solution and since the number of iterations needed to achieve
a required accuracy is not known beforehand, those components have to be summed up
continually at the end of each iteration in order to obtain the full term of the approximate
solution. In addition, the use of ADM requires us to generate Adomian polynomials at each
-
37
iteration step. As the number of iterations increases, the amount of calculations needed to
obtain the new Adomian polynomial also increases; and, therefore, the process of obtaining
those polynomials becomes more tedious and time consuming. It is evident that all the
mentioned drawbacks can be eliminated by using the presented method, which does not
require the extra calculation for the Adomian polynomials. Thus, it is concluded that the
method developed in this study is more efficient than the existing decomposition method.
Because of the extensive applications of the differential equations for modeling
engineering systems, the developed computationally efficient algorithm can be further
implemented into engineering software packages to improve the efficiency and accuracy of
testing, assessment, and analysis of the engineering systems. Based on the outcomes of this
study, more powerful algorithms can be developed based on the presented method for solving
other differential equation systems and reaching broad applicability in a variety of industrial
sectors such as aerospace, automotive, shipbuilding, and architectural industry.
-
Chand, Manoj. Bachelor of Engineering, Tribhuvan University, Spring 2009; Master of Science, University of Louisiana at Lafayette, Fall 2013
Major: Engineering, Mechanical Engineering option Title of Thesis: Development of Efficient Numerical Methods for Solving Differential
Equations using Hes Variational Iteration Technique. Thesis Director: Dr. Yucheng Liu Pages in Thesis: 48; Words in Abstract: 236
ABSTRACT
Differential equations play a prominent role in engineering and research fields in
modeling engineering structures, describing important phenomena, and simulating
mathematical behavior of engineering dynamical systems. Because of the increasing
complexity of modern engineering systems, computationally efficient methods are demanded
for solving these differential equations. In order to meet this challenge, this thesis presents
two efficient algorithms for solving two types of differential equations: a one-dimensional
heat equation with variable properties, and a one-dimensional parabolic equation, both of
which are very popular and important in current engineering systems. In this study, the two
equations were successfully solved using Hes variational iteration technique, and efficient
algorithms have been developed. Detailed procedures for developing these algorithms are
presented.
At first, a unique algorithm for solving the one-dimensional heat equations was
developed by using the iteration variational approach. The accuracy of this algorithm was
found by comparing the obtained solutions with the exact ones.
And similarly, using variational iteration approach, another efficient algorithm for
solving the one-dimensional parabolic equation was developed. Three illustrative numerical
problems were solved and the obtained results were compared with those yielded from the
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39
Adomian decomposition method (ADM) to verify the efficiency and accuracy of the
developed algorithm.
With the encouraging results obtained from this study, it is expected that, in the
future, developed algorithms can be extended to solve other differential equation systems,
thus achieving a broader applicability in engineering and other research fields.
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BIOGRAPHICAL SKETCH
Manoj Chand was born in July 3, 1985 in Mahendranagar, Kanchanpur, Nepal to Padam
Bahadur Chand and the late Govindi Chand. He earned his Bachelor of Engineering in
Mechanical Engineering from Tribhuvan University in 2009. After that, he worked in
Hyundai Motors Company as a service engineer in Nepal for one and half years. After that,
he started his masters program in the Department of Mechanical Engineering at University
of Louisiana at Lafayette and worked as a Research Assistant under the supervision of Dr.
Yucheng Liu since Spring 2012. He is expected to graduate in December 2013.
