The purpose of this paper is to discuss the flow of forced convection over a flat plate. The governing partial
differential equations are transformed into ordinary differential equations using suitable transformations.
The resulting equations were solved using a recent semi-numerical scheme known as the successive
linearization method (SLM). A comparison between the obtained results with homotopy perturbation method and numerical method (NM) has been included to test the accuracy and convergence of the method.
A ( )-Stable Order Ten Second Derivative Block Multistep Method for Stiff I...inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online
A ( )-Stable Order Ten Second Derivative Block Multistep Method for Stiff I...inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online
Numerical Methods was a core subject for Electrical & Electronics Engineering, Based On Anna University Syllabus. The Whole Subject was there in this document.
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Interpolating rational bézier spline curves with local shape controlijcga
The paper presents a technique for construction
of
C
n
interpolating
rational Bézier
spline curves by means
of blending
rational
quadric Bézier curves. A class of polynomials which satisfy special boundary
conditions is used for blending. Properties of the polynomials
are considered
.
The constructed spl
ine
curves have local shape control that make them useful in such geometric applications a
s
real
-
time
trajectory generation and fast curve sketching
.
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A six-step Continuous Block method of order (5, 5, 5, 5, 5, 5) T is proposed for direct solution of the second (2nd) order initial value problems. The main method and additional ones are obtained from the same continuous interpolant derived through interpolation and collocation procedures. The methods are derived by interpolating the continuous interpolant at 𝑥 = 𝑥𝑛+𝑗 , 𝑗 = 6 and collocating the first and second derivative of the
continuous interpolant at 𝑥𝑛+𝑗 , 𝑗 = 0 and 𝑗 = 2, 3, … 5 respectively. The stability properties of the methods are discussed and the stability region shown. The methods are then applied in block form as simultaneous numerical integrators. Two numerical experiments are given to illustrate the efficiency of the new methods.
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International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
NONSTATIONARY RELAXED MULTISPLITTING METHODS FOR SOLVING LINEAR COMPLEMENTARI...ijcsa
In this paper we consider some non stationary relaxed synchronous and asynchronous
multisplitting methods for solving the linear complementarity problems with their coefficient
matrices being H−matrices. The convergence theorems of the methods are given,and the efficiency
is shown by numerical tests.
Numerical Methods was a core subject for Electrical & Electronics Engineering, Based On Anna University Syllabus. The Whole Subject was there in this document.
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Interpolating rational bézier spline curves with local shape controlijcga
The paper presents a technique for construction
of
C
n
interpolating
rational Bézier
spline curves by means
of blending
rational
quadric Bézier curves. A class of polynomials which satisfy special boundary
conditions is used for blending. Properties of the polynomials
are considered
.
The constructed spl
ine
curves have local shape control that make them useful in such geometric applications a
s
real
-
time
trajectory generation and fast curve sketching
.
Derivation and Application of Six-Point Linear Multistep Numerical Method for...IOSR Journals
A six-step Continuous Block method of order (5, 5, 5, 5, 5, 5) T is proposed for direct solution of the second (2nd) order initial value problems. The main method and additional ones are obtained from the same continuous interpolant derived through interpolation and collocation procedures. The methods are derived by interpolating the continuous interpolant at 𝑥 = 𝑥𝑛+𝑗 , 𝑗 = 6 and collocating the first and second derivative of the
continuous interpolant at 𝑥𝑛+𝑗 , 𝑗 = 0 and 𝑗 = 2, 3, … 5 respectively. The stability properties of the methods are discussed and the stability region shown. The methods are then applied in block form as simultaneous numerical integrators. Two numerical experiments are given to illustrate the efficiency of the new methods.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
NONSTATIONARY RELAXED MULTISPLITTING METHODS FOR SOLVING LINEAR COMPLEMENTARI...ijcsa
In this paper we consider some non stationary relaxed synchronous and asynchronous
multisplitting methods for solving the linear complementarity problems with their coefficient
matrices being H−matrices. The convergence theorems of the methods are given,and the efficiency
is shown by numerical tests.
A SERIAL COMPUTING MODEL OF AGENT ENABLED MINING OF GLOBALLY STRONG ASSOCIATI...ijcsa
The intelligent agent based model is a popular approach in constructing Distributed Data Mining (DDM) systems to address scalable mining over large scale and ever increasing distributed data. In an agent based
distributed system, variety of agents coordinate and communicate with each other to perform the various
tasks of the Data Mining (DM) process. In this study a serial computing mode of a multi-agent system
(MAS) called Agent enabled Mining of Globally Strong Association Rules (AeMGSAR) is presented based
on the serial itinerary of the mobile agents. A Running environment is also designed for the implementation and performance study of AeMGSAR system.
Mafe De Baggis "L’Araba Fenice: rivitalizzare business maturi con i processi ...UNIONE INDUSTRIALE PRATESE
Mafe De Baggis, consulente di comunicazione freelance, è cofondatrice di Pleens, motore di ricerca per storie geolocalizzate. Lavora anche nella formazione, sempre in ambito comunicativo e cura la rubrica No Logo su Punto Informatico. Negli ultimi anni ha collaborato con Condé Nast (Style.it, Vogue.it e formazione ai giornalisti), Mondadori (Grazia e Donna Moderna), RCS (formazione), Barilla Center for Food & Nutrition, Danone, Lavazza, Bonduelle, Alfa Romeo, Indesit, Prénatal, ESA, Ferrari e eBookLabItalia.
On the cusp of ‘fourth revolution’, a new techno-dependent culture is emerging out of the ashes of a feudal-industrial society. However, the specter of change is fraught with ambiguities of hope. This presentation explores problems and possibilities that social work practice and education confront to adjust to and escape from realities transforming i) nature of ‘social’ and ‘work’’, ii) artificial intelligence and delivery of services; and iii) patterns of human-social development.
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In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type
with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson
extrapolation method is applied on the time direction and the fitted operator finite difference method on the
spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
examples, and the results are presented both in tables and graphs. Further, the present method gives a more
accurate solution than some existing methods reported in the literature.
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In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type
with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson
extrapolation method is applied on the time direction and the fitted operator finite difference method on the
spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
examples, and the results are presented both in tables and graphs. Further, the present method gives a more
accurate solution than some existing methods reported in the literature.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Adomian decomposition method for analytical solution of a continuous arithmet...TELKOMNIKA JOURNAL
One of the main issues of concern in financial mathematics has been a viable method for
obtaining analytical solutions of the Black-Scholes model associated with Arithmetic Asian Option (AAO).
In this paper, a proposed semi-analytical technique: Adomian Decomposition Method (ADM) is applied for
the first time, for analytical solution of a continuous arithmetic Asian option model. The ADM gives the
solution in explicit form with few iterations. The computational work involved is less. However, high level of
accuracy is not neglected. The obtained solution conforms with those of Rogers and Shi (J. of Applied
Probability 32: 1995, 1077-1088), and Elshegmani and Ahmad (ScienceAsia, 39S: 2013, 67–69). Thus, the
proposed method is highly recommended for analytical solution of other versions of Asian option pricing
models such as the geometric form for puts and calls, even in their time-fractional forms.
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variable method (FVM) is proposed to seek the exact traveling wave solutions of two higherdimensional
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(3+1)-dimensional space–time fractional Zakharov-Kuznetsov (ZK) equation and the (2+1)-
dimensional space–time fractional Generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony
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contain kink-shaped, singular kink, bell-shaped soliton, singular soliton and periodic wave
solutions, have many potential applications in mathematical physics and engineering. The
simplicity and reliability of the proposed method is verified.
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21st Mediterranean Conference on Control and Automation
The present paper is a survey on linear multivariable systems equivalences. We attempt a review of the most significant types of system equivalence having as a starting point matrix transformations preserving certain types of their spectral structure. From a system theoretic point of view, the need for a variety of forms of polynomial matrix equivalences, arises from the fact that different types of spectral invariants give rise to different types of dynamics of the underlying linear system. A historical perspective of the key results and their contributors is also given.
optimal solution method of integro-differential equaitions under laplace tran...INFOGAIN PUBLICATION
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Our efforts are mostly concentrated on improving the convergence rate of the numerical procedures both from the viewpoint of cost-efficiency and accuracy by handling the parametrization of the shape to be optimized. We employ nested parameterization supports of either shape, or shape deformation, and the classical process of degree elevation resulting in exact geometrical data transfer from coarse to fine representations. The algorithms mimick classical multigrid strategies and are found very effective in terms of convergence acceleration. In this paper, we analyse and demonstrate the efficiency of the two-level correction algorithm which is the basic block of a more general miltilevel strategy.
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When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
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Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
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Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
SUCCESSIVE LINEARIZATION SOLUTION OF A BOUNDARY LAYER CONVECTIVE HEAT TRANSFER OVER A FLAT PLATE
1. International Journal on Computational Sciences & Applications (IJCSA) Vol.5, No.3, June 2015
DOI:10.5121/ijcsa.2015.5308 105
SUCCESSIVE LINEARIZATION SOLUTION OF A
BOUNDARY LAYER CONVECTIVE HEAT TRANSFER
OVER A FLAT PLATE
Mohammed Abdalbagi, Mohammed Elsawi, and Ahmed Khidir
Department of Mathematics, Alneelain University, Khartoum, Sudan
ABSTRACT
The purpose of this paper is to discuss the flow of forced convection over a flat plate. The governing partial
differential equations are transformed into ordinary differential equations using suitable transformations.
The resulting equations were solved using a recent semi-numerical scheme known as the successive
linearization method (SLM). A comparison between the obtained results with homotopy perturbation
method and numerical method (NM) has been included to test the accuracy and convergence of the method.
KEYWORDS
Successive linearization method (SLM), Homotopy perturbation method, Forced convection.
1.INTRODUCTION
Many problems in fluid flow and heat transfer of boundary layers have attracted considerable
attention in the last decades. Most of these problems are inherently of nonlinearity and they do
not have analytical solution. Therefore, these nonlinear problems should be solved using other
numerical methods. The solution of some nonlinear equations can be found using numerical
techniques and some of them are solved using analytical methods such as Homotopy Perturbation
Method (HPM). This problem was proposed by Ji-Huan He [1] and it has been applied to find a
solution of nonlinear complicated engineering problems that cannot be solved by the known
analytical methods. Cai et al. [2], Cveticanin [3], and El-Shahed [4] have been applied this
method on integro-differential equations, Laplace transform, and fluid mechanics. Recently, there
are many different methods have introduced some ways to obtain analytical solution for these
nonlinear problems, such as the Homotopy Analysis Method (HAM) by Liao [5, 6], the Adomian
decomposition method (ADM) [7, 8, 9], the variational iteration method (VIM) by He [10], the
Differential Transformation Method by Zhou [11], Spectral Homotopy Analysis Method (SHAM)
by Motsa et al. [12] and recently a novel successive linearization method (SLM) which has been
used in a limited number of studies (see [13, 14, 15, 16, 17]) and it is used to solve the governing
coupled non-linear system of equations. Recently [18, 19, 20] have reported that the SLM is more
accurate and converges rapidly to the exact solution compared to other analytical techniques such
as the Adomian decomposition method, homotopy perturbation method and variation iteration
methods. Some of these methods, we should exert the small parameter in the equation. Therefore,
finding the small parameters and exerting it in the equation are deficiencies of these techniques.
The SLM method can be used in instead of traditional numerical methods such as Runge-Kutta,
shooting methods, finite differences and finite elements in solving high non-linear differential
equations. In this paper, we apply the Successive linearization method (SLM) to solve the
problem of boundary layer convective heat transfer over a horizontal flat plate. The obtained
results are compared with previous studies [21, 22, 23, 24, 25].
2. International Journal on Computational Sciences & Applications (IJCSA) Vol.5, No.3, June 2015
106
2.GOVERNING EQUATIONS
Let us consider the unsteady two-dimensional laminar flow of a viscous incompressible fluid.
Under the boundary layer assumptions, the continuity and Navier-Stokes equations are [26]:
0,
u v
x y
∂ ∂
+ =
∂ ∂
(1)
( )
2
2
1
,
u v dp u
u v g T T
x y dx y
ν β
ρ
∞
∂ ∂ ∂
+ = − + + −
∂ ∂ ∂
(2)
2
2
= .
T T T
u v
x y y
α
∂ ∂ ∂
+
∂ ∂ ∂
(3)
In the above equations, u and v are the components of fluid velocity in the x and y directions
respectively, ρ is the density of fluid, T is the fluid temperature, β is the coefficient of thermal
expansion, g is the magnitude of acceleration due to gravity, ν is the kinematic viscosity and
α is the specific heat. The initial and boundary conditions for this problem are
0, wu v T T= = = at 0y = ; ,u U T T∞ ∞= = at 0x =
,u U T T∞ ∞→ → as y → ∞ ;
Introducing:
0.5
Re ,x
y
x
η = (4)
( ) ,
w
T T
T T
θ η ∞
∞
−
=
−
(5)
where θ is a non-dimensional form of the temperature and the Reynolds number Re is defined
as:
Re .
u x
v
∞
= (6)
Using equations (1)-(5), the partial differential equations can be reduced to the following ordinary
differential equations
1
= 0,
2
f ff′′′ ′′+ (7)
1 1
= 0,
Pr 2
fθ θ′′ ′+ (8)
where f is related to the u velocity by
.
u
f
u∞
′ = (9)
The transformed boundary conditions for the momentum and energy equations are [27]:
( ) ( ) ( ) ( ) ( )0 0, 0 0, 0 1, 1, 0.f f fθ θ′ ′= = = ∞ = ∞ = (10)
3. International Journal on Computational Sciences & Applications (IJCSA) Vol.5, No.3, June 2015
107
3.METHOD OF SOLUTION
The system of equations (7) and (8) together with the boundary conditions (10) were solved using
a successive linearization method (SLM) (see [28, 29, 30]). The procedure of SLM is assumed
that the unknown functions ( )f η and ( )θ η can be written as
1 1
=0 =0
( ) = ( ) ( ), ( ) = ( ) ( ),
i i
i m i m
m m
f f Fη η η θ η θ η η
− −
+ + Θ∑ ∑ (11)
where mF and ( 1)m m ≥Θ are approximations which are obtained by solving the linear terms of
the system of equations that obtained from substituting (11) in the ordinary differential equations
(7) and (8). The main assumption of the SLM is that if and iθ are very small when i becomes
large, then nonlinear terms in if and iθ and their derivatives are considered to be very small and
therefore neglected. The initial guesses ( )0F η and ( )0 ηΘ which are chosen to satisfy the
boundary conditions
( ) ( ) ( ) ( ) ( )0 0 0 0 00 0, 0 0, 0 1, 1, 0,F F F′ ′= = Θ = ∞ = Θ ∞ = (12)
which are taken to be
0 0( ) = 1, ( ) = .F e eη η
η η η− −
+ − Θ (13)
We start from the initial guesses 0 ( )F η and 0 ( )ηΘ , the iterative solutions iF and iΘ are
obtained by solving the resulting of linearized equations. The linearized system to be solved is
1, 1 2, 1 1, 1,i i i i i iF a F a F r− − −
′′′ ′′+ + = (14)
1, 1 2, 1 3, 1 2, 1,i i i i i i ib F b b r− − − −
′′ ′+ Θ + Θ = (15)
together with the boundary conditions
( ) ( ) ( ) ( ) ( )0 0 0, 0 1,i i i i iF F F′ ′= = Θ ∞ = ∞ = Θ = (16)
where
1 1 1 1
1, 1 2, 1 1, 1 2, 1 3, 1
0 0 0 0
1 1 1 1 1
, , , , ,
2 2 2 Pr 2
i i i i
i m i m i m i i m
m m m m
a F a F b b b F
− − − −
− − − − −
= = = =
′′ ′= = = Θ = =∑ ∑ ∑ ∑
1 1 1 1 1 1
1, 1 2, 1
0 0 0 0 0 0
1 1 1
, .
2 Pr 2
i i i i i i
i m m m i m m m
m m m m m m
r F F F r F
− − − − − −
− −
= = = = = =
′′′ ′′ ′′ ′= − − = − Θ − Θ∑ ∑ ∑ ∑ ∑ ∑
The solutions of iF and iΘ , 1i ≥ can be found iteratively by solving equations (7) and (8).
Finally, the solutions for ( )f η and ( )θ η can be written as
( ) ( ) ( ) ( )
0 0
, ,
M M
m m
m m
f Fη η θ η η
= =
≈ ≈ Θ∑ ∑ (17)
4. International Journal on Computational Sciences & Applications (IJCSA) Vol.5, No.3, June 2015
108
where M is termed the order of approximation. Equations (7) and (8) are solved using
the Chebyshev spectral method which is based on the Chebyshev polynomials defined on
the region [ ]1,1− . We have to transform the domain of solution [ )0,∞ into the region
[ ]1,1− where the problem is solved in the interval [ ]0,L where L is a scale parameter used to
invoke the boundary conditions at infinity. Thus, by using the mapping
1
, 1 1.
2L
η ξ
ξ
+
= − ≤ ≤ (18)
The Gauss-Lobatto collocation points jξ is given by
cos , 0,1,2, , ,j
j
j N
N
π
ξ = = K (19)
The functions iF and iΘ are approximated at the collocation points as
( ) ( ) ( ) ( ) ( ) ( )
0 0
, , 0,1, , ,
N N
i i k k j i i k k j
k k
F F T T j Nξ ξ ξ ξ ξ ξ
= =
≈ Θ ≈ Θ =∑ ∑ K (20)
where kT is the th
k Chebyshev polynomial defined by
( ) ( )1
cos cos .kT kξ ξ−
= (21)
and
( ) ( )
0 0
, , 0, 1, , ,
r rN N
r ri i
kj i k kj i kr r
k k
d F d
F j N
d d
ξ ξ
η η= =
Θ
= = Θ =∑ ∑D D K (22)
where r is the order of differentiation and
2
D
L
=D with D being the Chebyshev spectral
differentiation matrix ( [31, 32, 33]), whose elements are defined as
( )
( )
2
00
2
2
2 1
,
6
1
, ; , 0,1, , ,
, 1,2, , 1,
2 1
2 1
.
6
j k
j
jk
k j k
k
kk
k
NN
N
D
c
D j k j k N
c
D k N
N
D
ξ ξ
ξ
ξ
+
+
=
−
= ≠ =
−
= − = −
−
+
= −
K
K
(23)
Substitute (18)-(22) into equations (14) and (15) gives the matrix equation
1 1.i i i− −=A X R (24)
where 1i −A is a ( ) ( )2 2 2 2N N+ × + square matrix and iX and 1i −R are ( )2 2 1N + ×
column vectors given by
5. International Journal on Computational Sciences & Applications (IJCSA) Vol.5, No.3, June 2015
109
1, 111 12
1 1
2, 121 22
, , ,
ii
i i i
ii
A A F
A A
−
− −
−
= = = Θ
r
A X R
r
(25)
where
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
0 1 1
0 1 1
, , , , ,
, , , , ,
T
i i i i N i N
T
i i i i N i N
F f f f fξ ξ ξ ξ
θ ξ θ ξ θ ξ θ ξ
−
−
=
Θ =
K
K
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
1, 1 1, 1 0 1, 1 1 1, 1 1 1, 1
2, 1 2, 1 0 2, 1 1 2, 1 1 2, 1
, , , , ,
, , , , ,
T
i i i i N i N
T
i i i i N i N
r r r r
r r r r
ξ ξ ξ ξ
ξ ξ ξ ξ
− − − − − −
− − − − − −
=
=
r
r
K
K
3 2
11 1, 1 2, 1
12
21 1, 1
2
22 2, 1 3, 1
,
,
,
.
i i
i
i i
A
A
A
A
− −
−
− −
= + +
=
=
= +
D a D a I
O
b I
b D b D
where T stands for transpose, ( ), 1 1,2 ,k i k− =a ( ), 1 1,2,3 ,k i k− =b and ( ), 1 1,2k i k− =r are
diagonal matrices, I is the identity matrix, and O is the zero. Finally, the solution is given by
1
1 1.i i i
−
− −=X A R (26)
4. RESULTS AND DISCUSSION
The non-linear differential equations (7) and (8) together with the conditions (10) have been
solved by using the SLM. We have taken 15, 60L Nη∞ = = = for the implementation of SLM
which gave sufficient accuracy. In order to validate our method, we have compared in Table 1
between the present results of ( )f η′ and ( )θ η corresponding to different values of η with
those obtained by Adomian Decomposition Method (ADM) [25], Homotopy Perturbation Method
(HPM) [24], and numerical method (NM) [21]. The results obtained by SLM are in excellent
agreement with a few order SLM series giving accuracy of up to six decimal places. In Figures 1
to 3 comparison is made between our results, HPM [23,24] and NM [21] methods. It is clear
from Figure 4 that, the temperature decreases with the increase in Prandtl number.
Table 1. The results of HPM, SLM, and NM methods for ( )f η′ and ( )θ η .
η ( )f η′ ( )θ η
HPM SLM NM HPM SLM NM
0 0 0 0 1 1 1
0.2 0.069907 0.066408 0.066408 0.930093 0.933592 0.933592
0.4 0.139764 0.132764 0.132764 0.860236 0.867236 0.867236
0.6 0.209441 0.198937 0.198937 0.790559 0.801063 0.801063
0.8 0.278723 0.264709 0.264709 0.721277 0.735291 0.735291
7. International Journal on Computational Sciences & Applications (IJCSA) Vol.5, No.3, June 2015
111
Figure 2. The comparison of the answers resulted by HPM [23], SLM, and NM for ( )f η′
Figure 3. The comparison of the answers resulted by HPM [23], SLM, and NM for ( )θ η
Figure 4. Effect of the Prandtl number Pr on ( )θ η
8. International Journal on Computational Sciences & Applications (IJCSA) Vol.5, No.3, June 2015
112
5.CONCLUSION
In this article, the SLM has been successfully applied to solve the problem of convective heat
transfer. The partial differential equations are reduced into ordinary differential equations using
similarity transformations. The present results indicate that this new method gives excellent
approximations to the solution of the nonlinear equations and high accuracy compared to the
other methods in solving non-linear differential equations. From the obtained results in the study,
it was found that the temperature profile generally decreases with an increase in the values of the
Prandtl number.
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