There Are Two Basic Principles That A System Can Be...

There are two basic principles that a system can be...
There are two basic principles that a system can be approached by; the continuous matter or modular
approach and the discrete matter or lumped mass approach (Holroyd, 2007). Generally, when a mass
can be defined as a rigid body or, in other words, when a system have a finite number of degrees of
freedom, it is more efficient to be modeled as a discrete (lumped parameter system). On the other
hand, when a mass is non–uniform or, in other words, when a system have an infinite number of
degrees of freedom (e.g. because it includes continuous elastic members), it is best to be modeled as
a continuous (distributed parameter system). Furthermore, there are hybrid models which combine
lumped and distributed parameters and provide more realistic ... Show more content on
Helpwriting.net ...
However, the behaviour and interaction of individual components of an electromechanical system is
not possible to be examined with lumped parameter models. Finally, lumped parameter models
require modifications in their whole lumped model when changes in any system component occur.
As already mentioned, distributed (modularized) models are solved by a set of partial differential
equations due to all the dependent variables consist of more than one independent variable.
However, these equations can be homogeneous or non–homogenous (inhomogeneous) equations. In
practice, the solution of a homogeneous equation with the appropriate boundary conditions
illustrates the behavior of the system after it has been properly set in motion and then subject to no
further force. In addition to this, the solution depicts the trend of the system to vibrate at a number
of natural frequencies. On the contrary, the solution of a non–homogeneous equation depicts the
behavior of the system to specific forces (Holroyd, 2007). The forced–damped method can be used
for solving the non–homogeneous equation of motion. According to this method, the steady–state
response to exciting forces is calculated by transfer matrices. Moreover, this method uses fewer
elements than the lumped mass approach in order to create a realistic model. This method contains
terms which are dependent on frequency, thus it requires the
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All the Mathematics You Missed but Need to Know for...
All the Mathematics You Missed
Beginning graduate students in mathematics and other quantitative subjects are expected to have a
daunting breadth of mathematical knowledge, but few have such a background. This book will help
students see the broad outline of mathematics and to fill in the gaps in their knowledge. The author
explains the basic points and a few key results of the most important undergraduate topics in
mathematics, emphasizing the intuitions behind the subject. The topics include linear algebra, vector
calculus, differential geometry, real analysis, point–set topology, differential equations, probability
theory, complex analysis, abstract algebra, and more. An annotated bibliography offers a guide to
further reading and ... Show more content on Helpwriting.net ...
. . . .
2 and J Real Analysis 2.1 Limits . . . . . 2.2 Continuity... 2.3 Differentiation 2.4 Integration .. 2.5 The
Fundamental Theorem of Calculus. 2.6 Pointwise Convergence of Functions 2.7 Uniform
Convergence . 2.8 The Weierstrass M–Test 2.9 Weierstrass ' Example. 2.10 Books .. 2.11 Exercises .
E
20 21 21
23 23 25 26 28 31 35 36 38
40
43
44 47
47
3
Calculus for Vector–Valued Functions 3.1 Vector–Valued Functions . . . 3.2 Limits and Continuity . .
. . . 3.3 Differentiation and Jacobians . 3.4 The Inverse Function Theorem 3.5 Implicit Function
Theorem 3.6 Books .. 3.7 Exercises . . . . Point Set Topology 4.1 Basic Definitions . 4.2 The
Standard Topology on R n 4.3 Metric Spaces . . . . . . . . . . 4.4 Bases for Topologies . . . . . . 4.5
Zariski Topology of Commutative Rings 4.6 Books .. 4.7 Exercises . Classical Stokes ' Theorems 5.1
Preliminaries about Vector Calculus 5.1.1 Vector Fields . 5.1.2 Manifolds and Boundaries. 5.1.3 Path
Integrals .. 5.1.4 Surface Integrals 5.1.5 The Gradient .. 5.1.6 The Divergence.
49

50 53 56
60 60
63 63 66 72 73
75 77 78 81 82 82
4
5
84
87
91 93 93
CONTENTS
5.1.7 The Curl . 5.1.8 Orientability . 5.2 The Divergence Theorem and Stokes ' Theorem 5.3
Physical Interpretation of Divergence Thm. . 5.4 A Physical Interpretation of Stokes ' Theorem 5.5
Proof of the Divergence Theorem . . . 5.6 Sketch of a Proof for Stokes ' Theorem 5.7 Books .. 5.8
Exercises .
6 Differential Forms and Stokes '

The Evolution Of The Topic
Subject Topic Create a written narrative of the evolution of the topic. Include significant
contributions from cultures and individuals. Describe important current applications of the topic that
would be of particular interest for students.
Number Systems Complex Numbers The earliest reference to complex numbers is from Hero of
Alexandria's work Stereometrica in the 1st century AD, where he contemplates the volume of a
frustum of a pyramid.
The proper study first came about in the 16th century when algebraic answers for roots of cubics
and quartics were revealed by Italian mathematicians Tartaglia and Cardano. For example,
Tartaglia's formula for a cubic equation x^3=x gives the solution as 1/√3 ((√(–1))^(1/3)+1/(√(–
1))^(1/3) ). ... Show more content on Helpwriting.net ...
For example, the treatment of resistors, capacitors, and inductors are unified by combining them in a
single complex number called the impedance, which is the measure of the opposition that a circuit
presents to a current when a certain voltage is applied.
Algebra The Quadratic Formula Early methods for solving quadratic equations were purely
geometric.
Babylonian tablets contained problems which could be reduced to solving quadratic equations. The
Egyptian Berlin Papyrus (2050–1650 BC) contains the solution to a two–term quadratic equation.
Euclid (300 BC) used geometric methods to solve quadratic equations in his book Elements.
In Arithmetica, Diophantus (250 BC) solved quadratic equations with methods which more closely
resembled algebra. However, his solution only gave one root, even when both roots are positive.
Brahmagupta (597–668 AD) explicitly described the quadratic formula in words instead of symbols
in Brahmasphutasiddhanta in 628 AD. His solution of ax^2+bx=c equated to the formula: x=
(√(4ac+b^2 )–b)/2a
In the 9th century, Persian mathematician al–Khwarizmi solved quadratic equations algebraically.
The quadratic formula which covered all cases was first described by Simon Stevin in 1594.
The quadratic formula that we know today was published by Rene Descartes in La Geometrie in
1637.

The first appearance of the general solution in modern mathematical literature was in an 1896 paper
by Henry

Function Of A Value Of X Essay
Up to now if I gave you an equation, and asked you to solve it for x you would be, in general,
looking for a value of x which solved the equation. Given:
x^2+3x+2=0
You can solve this equation to find two values of x.
I could also give you an equation which linked x and y explicitly, and you could find a relationship
between the two which, given a value of x would give you a value of y. You've been doing this now
for many years. Now we're going to add a hugely powerful tool to our mathematical arsenal. We're
going to allow our equations to include information about gradients of the function...let's see what
this means...
We're going to take everything that you learnt about integration and turn it into a way to model and
understand the world around us. This is a very powerful statement and indeed differential equations
are without a doubt the most powerful mathematical tool we have to understand the behaviour of
everything from fundamental particles to populations, economies, weather, flow of wealth, heat,
fluids, the motion of planets, the life of stars, the flight of an aircraft, the trajectory of a meteor, the
way a pendulum swings, the way a ponytail swings (see paper on this here), the way fish move, the
way algae grow, the way a neuron fires, the way a fire spreads...and so much more.
So what is a differential equation? It is an equation which contains one or more derivatives of a
function.
Let's look at a very simple example, of population growth. We might want to

Differential Equations Of A Nonlinear System
5.1 Linearization
It can be seen clearly by the system's equation that the model belongs to a nonlinear system. Normal
differential equations can be created by the conversion of the system into state space model format.
When a control law is designed, Lagrange equations of motion (9) are reformatted. To be able to
carry this out, a state vector is introduced which is as follows. x= (θ θ ̇ )^T
To be able to apply the LQR technique on the system, linearization is important. Therefore the
nonlinear model of the system turns into: θ ̈= –D^(–1) Cθ ̇–D^(–1) G+D^(–1) Hu (14)
After putting the variables of the system matrices in the above generalisation and their derivatives,
the system equation is as follows: x ̇=(■(0&I@0&–D^(–1) C))x+(■(0@〖–D〗^(–1) C))+
(■(0@D^(–1) H))u (15)
Where I and 0 are identity and zero matrices respectively.
The system equation can be rewritten as: x ̇=f(x)+g(x)u (16)
Where
f(x)=(■(0&I@0&–D^(–1) C))x+(■(0@〖–D〗^(–1) C)) (17) g(x)=(■(0@D^(–1) H)) (18)
5.2 Linear

Statement Of Structural Engineering
The determintaion for seeking my future in Structural Engineering comes from my interest in
Structures after I joined undergraduate studies in Civil Engineering at Kathmandu University. I'd
found my interest growing already in Civil Engineering when I was in high school. I was following
works of my brother while he was studying Civil engineering. I could understand and draw Plan and
sections before I was got enrolled in undergraduate studies. During my undergraduate studies, I
particularly enjoyed the subjects related to structural Engineering and material science such as
structural analysis, RCC structure, and steel structures etc. Therefore, Joining the MS in Structural
Engineering program at Lehigh University will help me to advance my career in Structural
Engineering. ... Show more content on Helpwriting.net ...
I also enjoyed programming in C, C++, and Matlab. The Final year Project in my Undergraduate
studies, which was a group project, was on the analysis of support system in Hydropower Tunnels in
H igher Himalayas of Nepal. It studies the deformation of the tunnel, support interaction with rock
mass, pressure on the support system, and stiffness of the support required etc. This project has
proven to very beneficial to me as I came to know about the power of Final Element Analysis
because of use of a powerful 2D finitef element software called RS2 (Phase2) for numerical
analysis. During my Undergraduate studies, as a part of the course, I also interned at Upper
Tamakoshi Hydro–Electric Project. This internship actually boosted my desire to choose my career
path to Structural Engineering when I studied the plans and sections and work on the construction of
a dam that was being

Essay On Fourier Analysis
In numerical analysis, explicit and implicit approaches are used to obtain numerical approximations
of time dependent ordinary and partial differential equations. Fractional order differential equations
are used widely for finance market analysis. Implicit solution methods require more computational
efforts and are complex to program. In order to overcome these difficulties, explicit method for
fractional order differential equation has been introduced which is one of the most recently
developed areas in the world of finance. The main aim of this paper is to investigate stability of
Fractional Explicit method for qth order time fractional Black–Schols equation by the well known
Fourier analysis method and a numerical experiment is presented for comparison of European call
option prices for different values of 'q'.
Keywords– Fractional calculus; Fractional Explicit Method; stability; European call options; time
fractional Black–Schols equation; Fourier analysis.
MSC 2010 No.: 26A33, 65M06, 65TXX.
Introduction
In Numerical analysis, the use of Fractional calculus is increasing day by day. The field of fractional
calculus is not new for mathematicians. It is as old as in the year 1695 , when L'Hopital sent a letter
to Leibniz asking him an important question about the order of the derivative, " What would be the
result if order of derivative is ... Show more content on Helpwriting.net ...
After introduction 1, the next section 2, will review the working of Fractional Explicit method.
Section 3 is based on the stability analysis of the method. In section 4, there is a numerical
experiment analyzing the performance of Fractional Explicit method for different values of 'q'. Data
for this experiment is taken from historical data section of NSE website of jet airways of the period
from 1st November 2016 to 30th November 2016. Graphical representation is given for the more
precise comparison. Finally in section 5 there is concluding remarks for the

The Inner Surface Temperature Against Time And Tile Thickness
From Fig.2, it shows that the inner surface temperature against time and tile thickness. On the right
hand side, it is shown that the Forward and the Dufort–Frankel method are very unstable. Both
methods had an infinite inner surface temperature at start, which is not ideal. For Backward and
Crank Nicolson methods, both of them had a smooth curve and did not have much fluctuation.
On the left hand side, it shows the temperature at the inner surface against time. Forward and
Dufort–Frankel deviated very soon after the tile get heated. This time, the Backward method moved
away from the starting temperature which is a sign of unstable. Therefore, Crank Nicolson was
selected to be the most appropriate method to solve this problem.
In theory, forward differencing and Dufort–Frankel methods were explicit method, and backward
differencing and Crank Nicolson were implicit methods. It was suggested that the implicit method
was more stable than the explicit as it solved the equation involving both the current state and the
next step rather than just using the current state.
The dx, dt were found using Fig.2. dt was found where the Crank Nicolson line started to fluctuate
heavily at around 14s (Fig.3), and dx was found when the line started to bend on the right hand side.
Using the maximum temperature and tile thickness, the parameters, nt and nx, were calculated and
used in the 'shuttle' function.
The left hand side of Fig.4 shows that the inner surface temperature across a range

Method Of Separation Of Variable ( Developed By J. Fourier
Chapter 3
METHODOLOGY
3.1 Method of Separation of Variable (developed by J. Fourier)
The method of separation is based on the expansion of an arbitrary function in terms of the Fourier
series. This method is applied by assuming that the dependent variable is a product of a number of
functions and each function being a function of a single independent variable. This reduces the
partial differential equation to a system of ordinary differential equations, each being a function of a
single independent variable. For the transient conduction in a plain wall, the dependent variable is
the solution function θ(X, F0), which is expressed in terms of θ(X, F0) = F(X)G(t), and the
application of this method results in to the two ordinary differential equations, one in terms of X and
the other one in F0.
Now we demonstrate the use of the method of separation of variables by applying it to the one–
dimensional transient heat conduction problem given in Eqs. (1). First,
Dimensionless differential equation is given by: ( ∂²θ)/∂X²=∂θ/(∂F_0 ) eq. (1a)
Boundary conditions: (∂T(0,t))/∂x=0
And –k (∂T(L,t))/∂x=h[T_((L,t))–T_a] eq. (1b)
Dimensionless initial condition is θ(X, 0) = 1

Applying An Analytical Model Of A Plane Wall
Abstract
To conduct a proper analysis of the 1–D transient conduction in a plane wall we must take the
necessary mathematical procedures to obtain an analytical model that accurately represents the heat
transfer that occurs. The equation must accurately model a plane wall that has a thickness L, is well–
insulated on one side, but is still vulnerable to convection on the other side. In order to complete the
model, one must scale the problem in terms of both a length scale and a time scale to transform the
variables to a dimensionless form that allows for a set of solutions that can be narrowed down to the
simple parameter, Bi=hL/k.
Introduction & Mathematical Model
This analysis looks into the phenomena of 1–D transient conduction in a plane wall of thickness L
that is insulated on one side and subject to convection on the other. The conduction is governed by
the differential heat equation: u_t=∝u_xx (1)
Here, u signifies the temperature of the entire body and ∝ signifies the thermal diffusivity.
Furthermore, the differential heat equation above must respect the following boundary conditions:
u_x |_(x=0)=0
–ku_x |_(x=L)=h(u|–T_∞) u|_(t=0)=T_i In the above boundary conditions, k represents a material
property commonly referred to as thermal conductivity, whereas T_i represents the initial
temperature throughout the wall. In this instance the flow conditions are such that they sustain
constant

Global Finite Element Matrix Construction Based on a...
Introduction
Many physical phenomena in stationary condition such as electrical and magnetic potential, heat
conduction, fluid flow and elastic problems in static condition can be described by elliptic partial
differential equations (EPDE). The EPDE does not involve a time variable, and so describes the
steady state of problems. A linear EPDE has the general form as presented in Eq. (1), where a,b,c are
coefficient functions, the term f is a source (excitation) function and u is the unknown variable. All
of this function can vary spatially (x,y,z).
∇(c∙∇u)+b∙∇u+au=f (1)
EPDE can be solved exactly by mathematical procedures like Fourier series [1]. However, the
classical solution frequently no exists and for those problems where is possible the use of these
analytical methods, many simplifications are done [2]. Consequently, several numerical methods
have been developed to efficiently solve EPDE such as the finite element method (FEM), finite
difference and others.
The FEM have several advantages over other methods. The main advantage is that it is particularly
effective for problems with complicated geometry using unstructured meshes [2]. One way to get a
suitable framework for solving EPDEs by using FEM is formulate them as variational problems also
called weak solution.
The variacional formulation of an EPDE is a mathematical treating for converting the strong
formulation to a weak formulation, which permits the approximation in elements or subdomains,
and the EPDE

Non Linear Behaviour And Chaos
ABSTRACT
In this report non–linear behaviour and chaos have been explored through Duffing Equation
computationally. Key features of the chaos theory such as attractors, Poincarè sections and phase–
space diagrams have been analysed and discussed. The programing language of choice for this
experiment was Fortran 90, which has been written explicitly for the purposes of acquiring a chaotic
system and solving the Duffing equation.
Introduction
The Duffing Oscillator named by the German electrical engineer Georg Duffing is a non–linear,
second–order differential equation, periodically forced and includes a damping term proportional to
particle's velocity. The equation can show different types of oscillations such as a limit cycles and
chaos. Given its characteristics Duffing oscillators are often used to produce similar behaviours in
nature. The equation in this experiment has been studied through the dynamics of a particle under a
potential field, driven by an applied periodical force. The equation of motion for the particle in this
system is
(1)
Where constant A gives the strength of a non–linear term, B gives the strength of the linear term, C
controls the size of damping, D controls the periodic driving force, and ω is the driving frequency.
The changes in the environment caused by the periodic force are sufficient to lead a chaotic
behaviour, as is the case with Duffing Oscillators.
Most of the systems in universe have a non–linear nature. Although a majority of

Essay On Multiscale Basis
2.2 Modeling the solution using multiscale basis
Selecting the dominant scale corresponding to the small eigen values gives rise to a fixed basis sets
and using fixed basis to solve the weak form produces the fixed solution un,fixed(x,t) = ‫ﰄ‬ n n,ωj n
H n,ωj i,j βi,j φi (x,t), where βi,j's are defined in each computational time interval and φi (x,t) are
fixed basis functions. Fixed solution at n + 1 th time point is computed by solving equation
(3) by setting un as the fixed solution at n th time point and writing un+1 in the space of HH
5
1 0.9
0.8ωE K 0.7
0.6
0.5
0.4
0.3
KK 1ω2 i 0.2 KK
0.1
0
0 0.2 0.4 0.6 0.8 1
Figure 3: Illustration of fine grid, coarse grid, coarse neighborhood and oversampled domain. fixed
basis at n + 1 th time ... Show more content on Helpwriting.net ...
The true solution is assumed to be normal around the fixed solution with small variance. Finally, this
structure enables us to compute the posterior or conditional distribution of the basis selection
probability and conditional solution of the system given the observation and the pde model.
Residual and selection probability on the subregion and basis
From equation (3), the residual is defined as
Rn(un+1,un+1 ,un
) =
‫ﰆ‬
Ω fn+1v − ‫ﰆ‬
‫ﰆ‬ un+1+un+1−un
+ fixed fixed v v + fixed fixed
Ω n+1
∆t
+ ufixed) · ∇v.
(4)
+ κ∇(u+ n+1

Ω
6
For any fixed basis φn,ωj 's this equation is zero as the fixed solution is constructed by setting the k
n,ωj equation zero for each fixed basis. Using φk,+ 's ∀k, j in the residual function one can compute
the residual for additional basis and writing down the residual as a long vector over subregions and
basis the following quantity is defined. Let αωk = ∥Rnωk ∥/∥Rn∥, where Rn is the global residual
vector and Rnωk is the local residual vector in ωk (as mentioned earlier) and L1 norm is used .
Let Nω be the average number of subregions where additional basis will added. Furthermore, αωj
α‫ﰇ‬ j = ‫ﰄ‬j αωj Nω, (5) ω With probability proportional to α‫ﰇ‬ j ∧ 1, the region ωj , is selected and Jj
= 1 if the region is ω selected and zero otherwise. Given subregion j is selected the k th extra
basis is

Numerical Modeling And Tropical Meteorology
My general areas of interest are numerical modeling and tropical meteorology. For example,
numerical simulation models of tropical cyclones is the problem that interest me most in
atmospheric science. Based on one or two problems in tropical meteorology or related fields, I hope
to work based on the existing models, and make unique contributions. My ultimate goal is to
develop skills to become a sophisticated researcher and teacher in the field, while pursuing a
doctorate degree in atmospheric sciences.
My interest on PDEs dates back to high school. Since typhoons frequently struck my hometown, I
developed interest in the forecast of tropical cyclones. Guided by a net–pal with a master's degree in
meteorology, I started to systematically study the basics of weather prediction. However, when
reading Principles of Meteorological Analysis, I encountered PDEs describing dynamics of
atmosphere, which was incomprehensible for me then. Realizing that a solid math foundation would
be critical to further study in meteorology, I chose to study mathematics in University of California,
Los Angeles (UCLA).
After finishing basic math courses, I challenged myself with a year–long series of honors algebra
course, in which group theory, ring theory and Galois theory were covered. The homework sets and
take–home exams in this course horned my skills of solving hard math problems. With limited clues
for each problem, I had to review related definitions and theorems carefully, and

The Final Infinite Interval And Exponential Gegenbauer...
In this paper, we introduce two new functions on the semi–infinite interval namely Rational
Gegenbauer and Exponential Gegenbauer functions and we apply them as basis functions in Tau
method to solve the boundary layer flow of a magneto–micropolar fluid on a continuous moving
plate with suction and injection. These functions are a general case of rational Chebyshev and
Legendre functions and this is the first time that they are used in Tau method. The operational
matrices of derivative and product of rational and exponential Gegenbauer functions are also
presented. These new matrices together with the Tau method are then utilized to reduce the solution
of the governing equations to the solution of a system of nonlinear algebraic ... Show more content
on Helpwriting.net ...
Many problems of physics and engineering lead naturally to the resolution of differential equations
in unbounded domains and semi–infinite domains. In the context of spectral methods, a number of
approaches have been proposed and investigated for treating these problems. The most common
method is the use of polynomials that are orthogonal over unbounded domains, such as the Hermite
and Laguerre spectral methods [10, 11, 12, 13].
Guo [14, 15] proposed a method that proceeds by mapping the original problem in an unbounded
domain to a problem in a bounded domain, and then using suitable Jacobi polynomials to
approximate the resulting problems. Another approach is using the domain truncation method by
choosing sufficiently large for replacing the infinite domain with and the semi–infinite interval with
[16].
There is another effective direct approach for solving such problems which is based on the rational
approximations. Christov [17] and Boyd [18, 19] developed some spectral methods on unbounded
intervals using mutually orthogonal systems of rational functions. Boyd [18] defined a new spectral
basis, named rational Chebyshev functions on the semi–infinite interval, by mapping to the
Chebyshev polynomials. Guo et al. [20] introduced a new set of rational Legendre functions which
are mutually orthogonal in . They applied a spectral scheme using the rational Legendre functions
for solving the Korteweg–de Vries equation on the half–line.

Equation: A Comparative Analysis: Definition Of...
CHAPTER 1
INTRODUCTION
Definition of Differential Equation
A differential equation is an equation which consists of derivatives or differentials of one or more
dependent variables with respect to one or more independent variables (Abell & Braselton, 1996).
Differential equation generally can be classified into two, which are ordinary differential equation
and partial differential equation. If a differential equation consists of ordinary derivation of one
dependent variable with respect to only one independent variable, it is known as ordinary
differential equation. Meanwhile, if a differential equation consists of partial derivative of one or
more dependent variables with respect to more than one independent variable, it is known as partial
differential equation.
Ordinary Differential Equation
In general, an ordinary differential ... Show more content on Helpwriting.net ...
It is simply expressed as the matrix product of two factors, a variable vector with a constant matrix,
x ̅(t)=Hv ̅(t) 1.6
, where H is a constant matrix and v ̅(t) is a time–variable vector. This approach, using the matrix
theory, gives a numerical solution to the systems of homogeneous or non–homogeneous of linear
differential equation with constant coefficient. Any systems of linear differential equation which can
be solved by Laplace Transformation can also be solved by this approach.
Problem Statement
The basic approach recalculates the partial fraction expansion coefficients of a rational function
from the very beginning whenever the initial conditions are changed, which is very time–
consuming. This study finds an alternative approach to compute the solution with a minimal and
straightforward effort of re–computation upon the initial conditions changed.
Objective of

Alternative Methods Of The NIPG DGFEM Superconvergence
subsection{NIPG DGFEM Superconvergence 2011}
Now that the continuous standard Galerkin solution over a quasi–uniform mesh may oscillate as
$epsilon to 0$. An alternative tool will be a discontinuous Galerkin (DG) method where the
oscillation can be avoided provided that an appropriate mesh refinement is applied, to capture the
boundary layer behavior. The origins of the DG methods can be traced back to the seventies where
they had been proposed as variational methods for numerically solving initial–value problems and
transport problems. It is well known that the DG methods, in particular the local DG (LDG) method
cite{210}, are highly stable and effective for convection diffusion problems cite{211}. Whereas,
the main feature of the ... Show more content on Helpwriting.net ...
cite{212,213} and Zhang et al. cite{214} adopted the local discontinuous Galerkin (LDG) method
to solve convection diffusion equations and analyzed the corresponding superconvergence
properties. On the other hand, non–symmetric discontinuous Galerkin method with interior penalty
(the NIPG method), originally designed for elliptic equations, is analyzed by Zarin and Roos
cite{87} for convection–diffusion problems with parabolic layers.
A disadvantage of DG method is that the method produces more degrees of freedom than the
continuous finite element method (CFEM). With this motivation, this work derives and analyzes a
coupled approach of LDG and CFEM on a layer adapted Shishkin mesh for singularly perturbed
convection–diffusion problems. By splitting the domain into the coarse and the fine part, we adopt
the CFEM with linear elements in the fine part where the mesh size is comparable with $epsilon$,
and use LDG method in the coarse part for its stabilization.
The idea of combining DG and CFEM to obtain the advantages of both methods is not new. A
coupled LDG–CFEM approach has also been studied by Perugia and Sch{"o}tzau cite{215} for
the modeling of elliptic problems arising in electromagnetics. Roos and Zarin cite{79}, Zarin
cite{82} analyzed the NIPG–CFEM coupled method on Shishkin mesh for two–dimensional
convection diffusion problems with exponentially layers or characteristic layers. In cite{209}, Zhu
et

Characterization Of One Dimensional Vapex
Characterization of one–dimensional VAPEX
Figure 1 shows a model of Vapex Process modeled in a vertical thin sandpack (cylindrical), which is
saturated with heavy oil. Solvent Injector and oil producer are placed at the bottom of the sandpack.
When solvent is injected, it moves upwards due to buoyancy and comes in contact with the heavy
oil and heavy oil is extracted and diluted and drained downward by gravity force.
We make these following assumptions for our 1D VAPEX Process:
1. We have 2 areas: 2 phase area and 1 phase area, and they are divided by the interface between
solvent chamber and transition zone;
2. The solvent chamber is filled with diluted oil (saturated oil) which is in liquid phase and the
gaseous solvent vapor;
3. ... Show more content on Helpwriting.net ...
The correlation between diffusion coefficient and the concentration of solvent (light hydrocarbon) in
crude oil is usually expressed via the viscosity of the heavy oil−solvent mixture. The dependence of
the viscosity on the solvent concentration was proposed by Lederer [3] (3) where Shu [4] formulated
the following correlation to determine the weighting factor, λ, for a mixture of heavy oil and light
hydrocarbons where γo and γs are the specific gravities of the crude oil and liquid solvent,
respectively.
The diffusion coefficient is usually correlated with viscosity as
Duaub (4) where a and b are both constants depending on the properties of oil and gas
sample as well as the operation condition (pressure and temperature). Hayduk and and Das−Butler
proposed different correlations for normal paraffin solute/solvent system and propane/heavy oil
system, respectively.
Symbol v in Eq. (2) denotes convection velocity between solvent vapour and diluted oil in the
transition zone. Darcy's law [5] is commonly used to depict the fluid flow rate in porous medium,
(5) where m s Heavy oil−solvent mixture is commonly treated as ideal solution
and its density, ρm, is calculated by
m css coo (6)
Moving boundary of transition zone
The transition zone is assumed bounded by two interfaces [6]: one is next to the solvent chamber
and the other is neighboring the untouched heavy oil zone. The former interface is defined as the
plane where

Essay on Chaos Theory Explained
Chaos Theory Explained "Traditionally, scientists have looked for the simplest view of the world
around us. Now, mathematics and computer powers have produced a theory that helps researchers to
understand the complexities of nature. The theory of chaos touches all disciplines." –Ian Percival,
The Essence of Chaos Part I: The Basics of Chaos. Watch a leaf flow down stream; watch its
behavior within the water... Perhaps it will sit upon the surface, gently twirling along with the
current, dancing around eddies, slightly spinning, then all of a sudden, it slaps into a rock or gets
sucked beneath the water by a small whirlpool. After doing this enough times one will realize it is
nearly impossible to accurately ... Show more content on Helpwriting.net ...
In the last years of the 19th century French mathematician, physicist and philosopher Henri
Poincare' stumbled headlong into chaos with a realization that the reductionism method may be
illusory in nature. He was studying his chosen field at the time; a field he called 'the mathematics of
closed systems' the epitome of Newtonian physics. A Closed system is one made up of just a few
interacting bodies sealed off from outside contamination. According to classical physics, such
systems are perfectly orderly and predictable. A simple pendulum in a vacuum, free of friction and
air resistance will conserve its energy. The pendulum will swing back and forth for all eternity. It
will not be subject to the dissipation of entropy, which eats its way into systems by causing them to
give up their energy to the surrounding environment. Classical scientists were convinced that any
randomness and chaos disturbing a system such as a pendulum in a vacuum or the revolving planets
could only come from outside chance contingencies. Barring those, pendulum and planets must
continue forever, unvarying in their courses.2 It was this comfortable picture of nature that Poincare'
blew apart when he attempted to determine The stability of our solar system... For a system
containing only two bodies, such as the sun and

Reliability And Availability Evaluation Of A System Switched
RELIABILITY AND AVAILABILITY EVALUATION OF A SYSTEM SWITCHED TO
ANOTHER SIMILAR, SUBSTITUTE OR DUPLICATE SYSTEM ON TOTAL FAILURE
ABSTRACT
A two–unit standby system is considered with two types of repair facilities. One facility repairs one
unit at a time and other facility repairs both the units simultaneously. When both the units fail, if unit
can be repair in short time then repair will be continued, otherwise in order to improve availability
another substitute system taken from outside is used, which is guaranteed for failure free operation.
Assuming failure and repair times as exponentially distributed, Expressions for the mean time to
system failure (MTSF), the steady state availability and busy period for system are derived using
linear first order differential equations. A particular case for the proposed system is discussed in
which substitute system was not considered. Also comparison is performed graphically to observe
the effect of the proposed system on Availability.
Keywords: Availability, Linear first order differential equation, Mean Time to System Failure,
Reliability, Steady State Availability.
1) INTRODUCTION
Competition exists in every field, to keep ahead a major challenge is availability improvement of a
system, as less availability has negative impact. People often use "availability" and "reliability"
interchangeably. In fact, however, the two terms are related but have distinct meanings. Reliability
(as measure of the mean time between system

Critically Looking At The Research
Critically Looking at the Research
While this research project was taking place, a method(s) of research had to be chosen. Multiple
options were considered such as online research, questioners, and interviews. It was found that the
questionnaires would not help the research because the information that was needed could not be
extrapolated out of a questionnaire. Secondly, interview(s) were ruled out of the question because
there was no one that could be found and contacted that was creditable. There was, however, an
exception to do with the interview later on in the research. This was to do with the required help on
some of the complicated math faced in the project. This was accomplished with the knowledge of
one of the schools mathematics ... Show more content on Helpwriting.net ...
When conducting the research, difficulties and challenges were experienced. One of the big
problems faced was finding a way to convert the differential equation into a function. Research
online helped narrow down the problem, and then specifically having a talk/interview with math's
personal at the school allowed for a solution to be found.
Throughout the research the most useful source was, Population Dynamics of Western Atlantic
Bluefin Tuna: Modeling the Impacts of Fishing using Differential Equations. The source was the
most useful because of the connections that was shown between population models and
mathematics. The reliability was judged to be high because of the authors Esther Bowen, Marie
Hoerner, and Cassie Kontur. The information stated was backed up by other sources and judged to
be accurate. Lastly the information displayed was up to date and on topic.
If this research task was to be redone, there are a few improvements that could be made. Firstly the
topic was a little broad if the topic was narrowed down more, a more informative research project
could have been created. Secondly, more time with a specialist in mathematics, and more help
specifically with differential equation and calculus would have been externally productive. The
mathematical aspect of the research project would have been a lot stronger with more/longer access
to math experts. For example an improvement of converting the differential equation to a function
that has the

Availability Improvement For Single Unit System With Two...
AVAILABILITY IMPROVEMENT IN SINGLE UNIT SYSTEM WITH TWO TYPES OF REPAIR
FACILITIES
Gurvindar Kaur and pooja vinodiya
School of Studies in Statistics
Vikram University Ujjain (M.P.)
Email:gkbhatti2289@gmail.com
ABSTRACT
This study deals with the reliability, availability, and busy period characteristics of single unit
system. On failure of the system, if system can be repaired in short time then repair will be
continued, otherwise in order to improve availability another substitute system taken from outside is
used, which is guaranteed for failure free operation and an expert repairman is called for fast repair
of the unit. Assuming failure and repair times as exponentially distributed, expressions for the mean
time to system failure (MTSF), the steady state availability and busy period for system are derived
using linear first order differential equations. A particular case for the proposed system is discussed
in which substitute system was not considered. Also comparison is performed graphically to observe
the effect of the proposed system on Availability.
Keywords: Availability, Linear first order differential equation, Mean Time to System Failure,
Reliability, Steady State Availability.
1) INTRODUCTION
Competition exists in every field, to keep ahead a major challenge is availability improvement of a
system, as less availability has negative impact. People often use "availability" and "reliability"
interchangeably. In fact, however, the two terms are related but have

Taking a Look at ANSYS
1. Introduction
1.1 ANSYS
ANSYS is a software package that allows various simulations in a range of different fields and
industries to be modelled and analysed. The main fields within the program include, computational
fluid dynamics, structural mechanics, and electromagnetics to mention a few. The use of this
software allows an individual or business to test various cases of product use, eliminating the outlay
cost of building and testing many prototypes. This saves on time and costs and is a lot faster and
more accurate then computing a hand analysis. In certain circumstances it allows for test cases that
would otherwise not be possible to set up in a lab [1] [2].
1.1.1 Brief History of ANSYS
Modern finite element method may be traced back as early as the 1900's with the first models being
represented and calculated by means of discrete equivalent elastic bars. The individual that has been
credited with developing the finite element method is R. Courant. He made use of piecewise
polynomial interpolation over triangular sub regions to analyse problems involving torsion [3].
Boeing was the first major company to make use of triangular stress elements to model their
airplane wings during the years from 1950–1962. It was during the 1960's that the finite element
method use was broadened to other industries including heat transfer. ANSYS was officially
released in 1971 [3]. It was not until 1995 when Microsoft released Windows 95, which the
acceptance of computers as a

Summary Of The Movie Hidden Figures
The movie Hidden Figures is based on the remarkable true story of African American women
working for NASA in Hampton, Virginia 1961. With all their hard work and determination, they did
the calculations and equations for the shuttle launches of Friendship 7, Apollo 11, and other Space
missions. One out of these women was a brilliant mathematician named Katherine Gobel– Johnson.
As a young child, Gobel she was recognized for her high intellect and was recommended to an
alternative school so they can see what she can really do. At first, Katherine's parents were hesitant
because of the cost to send an African–American child during that time but the teachers made a
collection and full scholarship for Katherine to go. During the movie, young Katherine was
presented to solve the equation Katherine solved the equation and the answer was x=1, –7, 3, and –
1/2. Katherine graduated high school at age 18, then went to West Virginia State College now called
West Virginia State University. She graduated summa cum laude with degree in Mathematics and
French in 1937. Next, she became the first female African–American to attend West Virginia
University Graduate School. With her superb mathematical intellect, Katherine became a
"computer" at NASA. On the article called Human Computers found on NASA, "The term
'computer' referred to people, not machines. It was a job title designating someone who performed
mathematical equations and calculations by hand. Over the next thirty years, hundreds

Power Series Method For Solving Linear Differential Equations
ABSTRACT In this work, we studied that Power Series Method is the standard basic method for
solving linear differential equations with variable coefficients. The solutions usually take the form of
power series; this explains the name Power series method. We review some special second order
ordinary differential equations. Power series Method is described at ordinary points as well as at
singular points (which can be removed called Frobenius Method) of differential equations. We
present a few examples on this method by solving special second order ordinary differential
equations.
Key words ; Power series, differential equations, Frobenius Method, Lengendre polynomials
1.0 INTRODUCTION
1.1 BACKGROUND OF THE STUDY
The attempt to solve physical problems led gradually to Mathematical models involving an equation
in which a function and its derivatives play important role. However, the theoretical development of
this new branch of Mathematics –Differential Equations– has its origin rooted in a small number of
Mathematical problems. These problems and their solutions led to an independent discipline with
the solution of such equations an end in itself (Sasser, 2005).
1.2 STATEMENT OF THE PROBLEM
The research work seeks to find solutions of second–order ordinary differential equations using the
power series method.
1.3 AIM AND OBJECTIVES
The aim and objectives of the study are to: Describe the power series method. Use it to solve linear
ordinary differential equations with

A flat plate solar collector has a dynamic behaviour in...
A flat plate solar collector has a dynamic behaviour in response to variations in the intensity of solar
radiation at different times of the day and also variations in weather conditions. The characteristics
governing the input–output behaviour of a flat plate collector can be described by a mathematical
model which serves as a prerequisite for simulation and control. The steady state and transient
characteristics of flat plate solar collectors have been studied in
cite{Hilmer1999Solar,Dhariwal2005Solar,deRon1980,RodriguezHidalgo2011,Refaie1980}.
Depending on the complexity of the flat plate solar collector under observation, deriving a
mathematical model may lead to a high order model which requires high computational effort and
longer ... Show more content on Helpwriting.net ...
Improvements to the original frequency weighted balanced truncation by Enns have been described
in cite{Ghafoor2007PFE,SreeramSahlan2012,SreeramSahlanMudaPFE2013}. Schelfhout and
Moor pointed out that many specifications and robustness requirements yield natural frequency
domain weighting functions but in other cases time domain weighting functions are more
appropriate which led to the introduction of time weighted controllability and observability
gramians for the balanced truncation algorithm cite{SchelfhoutMoor1995}. Sreeram had defined
the frequency response error bounds for time weighted balanced truncation
cite{TimeWeightedSreeram2002}. More recently Shaker and Tahavori had introduced time
weighted balanced stochastic truncation cite{TahavoriShaker2012,TahavoriShaker2011}. Cross
gramians matrices contain information regarding both controllability and observability of a system
in a single matrics cite{FernandoNicholson1983,FernandoNicholson1984}. Instead of computing
two separate gramians for controllability and observability, states which are the least controllable
and observable can be identified from a single cross gramian matrics and these states can be
truncated. Model reduction using cross gramians does not involve balancing which is an advantage
since balancing may be ill conditioned for systems with almost uncontrollable and unobservable
states cite{Aldhaheri2006}.

The Challenges: Advantages And Disadvantages Of Flexible...
1.1. Introduction
Flexible robots consist of manipulators that are made of flexible and lightweight materials. These
manipulators are operated by using some actuator that may be a dc motor, some robots use electric
motors and solenoids as actuators, while some have a hydraulic system, and some others may use a
pneumatic system. Lightweight flexible robots are widely used in space applications due to their
increased payload carrying capacity, lesser energy consumption, cheaper construction, faster
movements, and longer reach compared to conventional industrial rigid robots. However due to light
weight they undergo vibrations and hence the control mechanism of the flexible robot becomes
more challenging.
1.1.1 Description of flexible robots
Flexible ... Show more content on Helpwriting.net ...
On contrary flexible robot position is not constant and hence partial differential equation is used to
represent the distributed nature of position. Further due to sudden change in payload there may be a
large variation in manipulator parameters. Thus control with constant gain controllers is difficult and
adaptive methods must be used.
2. OBJECTIVE OF THE WORK
The objectives of the thesis are as follows.
1. To study the dynamics of a flexible beam and have a knowledge of Assumed mode method
(AMM), for the modelling of a flexible robot manipulator system.
2. To derive a mathematical model of a physical TLFM set–up and to validate the obtained model .
3. To study fuzzy identification and obtain a fuzzy model of the system.
4. To design and implement control strategies like PID, Linear Quadratic Regulator and Model
Predictive Control for controlling the tip trajectory

Analysis Of Restricted Boltzmann Machines
Analysis of RNNs revealed that the hidden–to–output function, hidden–to–hidden transition, and
input–to–hidden function must be made deeper [3]. Based on the following input sequence: x = (x1;
: : : ; xT), a standard RNN is responsible for computing the vector sequence: h = (h1; : : : ; hT) as
well as the output vector sequence: y = (y1; : : : ; yT) using two equations (depicted below) from t =
1 to T [3]. (1) ht = H(Wxhxt+Whhht–1+bh) (2) yt = Whyht+by H. Restricted Boltzmann Machines
An RBM is a specialized Boltzmann Machine comprised of two respective layers, a layer of visible
and hidden units, without hidden–hidden and visible–visible connections. Each hidden and visible
unit within the network has a bias and either a binary or ... Show more content on Helpwriting.net ...
Apothéloz's proposal is of importance in understanding the artificial intelligence models of
argumentation since it coincides with the properties of a square of opposition [6]. Fig. 5. The square
of opposition. Apothéloz's square of opposition proclaims makes the following claims [6]: (1) A and
O as well as E and I both serve as negations of each other (2) A and E entails, I and O, respectively
(3) Although A and E cannot be true together, A and E can be false together I and O cannot be false
together yet can be true together. J. Sentimental Analysis Sentimental analysis tries to figure out
how the presenter feels about the subject material being presented. This analysis helps the NLP
formulate a more accurate and appropriate response. Many sentimental analyses work by looking at
each sentiments of the sentence by giving positive or negative points to each word. Points are then
summed up for each sentence and based on that score it is deemed either positive, negative, or
neutral. But sentiments are often very subtle and cannot be detected using simple point analysis [1].
To better grasp sentiments in NSL computer scientist once again turned to the deep learning process
and developed a tree–structured long short–term memory analysis (LSTM). LSTM combines deep
learning with the points system. After the deep learning process has assigned meaning to a word it is
given a weight based upon positive or negative feel. It is then placed

Evaluation of Various Numerical Methods for Option Pricing...
In finance, a derivative is a financial instrument whose value is derived from one or more
underlying assets. An option is a contract which gives the owner the right, but not the obligation, to
buy or sell the asset at a specified strike price at the specified date. The derivative itself is just a
contract between two or more parties. Its value is determined by fluctuations in the value of the
underlying asset. This price is chosen so that the value of the contract to both sides is zero at the
outset, which means that the price is fair, so neither party is taking advantage of the other. Hence,
numerical methods are needed for pricing options in cases where analytic solutions are either
unavailable or not easily computable. The subject of ... Show more content on Helpwriting.net ...
This method is widely used as it is able to handle a variety of conditions. Finite difference methods
were first applied to option pricing by Eduardo Schwartz in1977. In general, finite difference
methods are used to price options by approximating the differential equation that describes how the
option price moves over time by a set of difference equations. This method arises since the option
value can be modeled by partial differential equations, such as the Black–Scholes PDE. This
approach has the same level of complexity tree methods. The application of Monte Carlo method to
option pricing was by Phelim Boyle in 1977. In terms of theory, Monte Carlo valuation relies on risk
neutral valuation. The technique is to generate several thousand possible random price paths for the
underlying asset and via simulation, and to calculate the average payoff of each path. This approach
is particularly useful in the valuation of options with complicated features, which would be difficult
to value through straightforward Black–Scholes style or tree model. ( reference [3] Valuation of
Options)
Each of these methods has its own advantages and disadvantages. The comparison of accuracy and
consistence are presented and suitable method for each situation is discussed.
Then the report briefly goes through some exotic options and implements the numerical solutions
with binomial tree method. These options, includes American option which can be exercised any
time before the

The Lotka Volterra Predator Prey Model
Introduction The Lotka–Volterra equations are used in biology, chemistry, and many other sciences
that deal with two populations whether that be in our case animal, or chemical, where two species
both rely on a single source to stay alive, called a Competitive model (Appendix A), or where one
species relies on another species to stay alive, called a Predator/Prey Model (Appendix B). Initially,
the Lotka–Volterra predator–prey model was stated by Alfred K. Lotka. This was a similar equation
to the logistic equation which was proposed by Pierre–François Verhulst in 1838. By 1926, Vito
Volterra, who was a physicist and a mathematician, also had published the same set of equations
identical to that of Volterra. Later on the model was ... Show more content on Helpwriting.net ...
In this case we have a saddle point where if both species use the resources in the same proportions,
they can coexist but any small change to the system, which in reality is perfectly normal, will push
the point to either side and allow one species to overtake the other, and this saddle point is also
represented by the point at: We also have two stable steady states at (K1,0) and (0,K2), and an
unstable steady state at (0,0): (See Appendix E for a graph with various initial conditions applied
under this system) Beta1 Beta2 K1 K2 5 1 250 100 Now in our second scenario, we have two
species also on an island, the Rabbits and the Foxes. The fox relies on the rabbit for food, and the
rabbit relies on grasses (a constant unchanging source) to survive. The growth rate of the rabbits and
foxes can be modelled by

Differential Equation : Mathematical Function
Balanchard Differential Equation
An ODE is an equation that contains ordinary derivatives of a mathematical function. Solutions to
ODEs involve determining a function or functions that satisfy the given equation. This can entail
performing an anti–derivative i.e. integrating the equation to find the function that best satisfies the
differential equation. There are several techniques developed to solve ODEs so as to find the most
satisfactory function. This discussion seeks to explore some of these techniques by providing
worked out examples. Bernoulli Equation
Bernoulli equation is named after Daniel Bernoulli who was a Swiss Mathematician.
Bernoulli equation takes the form, , where p(x) and q(x) are continous real functions and n is not a
complex number. These functions are defined within a given interval (Greenberg 35). It is worth
noting that if n=1 or n=0 the equations becomes linear. Therefore, when solving Bernoulli equations,
the main aim is to find solutions for numerical values of n except 0 and 1.
The solution to this equation entails dividing the equation by y^n to get
................................................................1
Next, we perform a substitution to change it into a differential function in terms of an arbitrary term
v=y^(1–n). This kind of substitution gives a differential equation that is possible to solve. However,
care is given when dealing with derivatives of the form, y^t. In this case, it is practical to determine
what y^t is with

A Summary On The Intellectual Merits And. Broader Impact...
Before explaining the obtained results in details we start a summary on the intellectual merits and
broader impact of the project.
The PI initiates a new approach (in items 2,5, 6), using the precise large time
asymptotic behavior of solutions of a parabolic equation to study the geometric property of K
manifolds, and to solve the Poincar Lelong equation. The method is effective in proving
sharp and optimal result. The method reminisces the celebrated ergodic theorem of Birkhoff
which connects the space average of a continuous function on the phase
space of a Hamiltonian system with its time average taking along the trajectory (see the second part
for
detailed descriptions). This connection is also in some way related to other ... Show more content on
Helpwriting.net ...
In item
11 comparison result for viscosity solutions of some first and second order PDEs are proved. This
immediately yields the celebrated Levy–Gromov isoperimetric inequality and its generalization
as consequences. In item 14, a classification result on four dimensional gradient shrinking
solitons with nonnegative isotropic curvature was proved. This result generalizes the earlier result of
Naber, which proves a classification under the stronger assumption of bounded nonnegative
curvature operator.
The research conducted in items 2, 3, 4, 5, 6 are related to Birkhoff ergodic

theorem (which was applied by H. Weyl to understand the retreats and advances of glaciers. Further
understanding of this connection shall be sensational to the subject of partial differential equation
and
dynamic system. The research in items 9, 12, 13 are related to the concept of entropy in
thermodynamics, which have impacts to other sciences beyond mathematics. The work in item 1
contributes an advancement in the high energy physics.
In promoting teaching, training, and learning, the PI
advised (including some current students) nine Ph.D students, including two female graduate
students, and served/serves as the faculty mentor for several postdoc visitors, including one SEW
assistant professor at UC San Diego. At UC San Diego the PI teaches the courses

Essay On Homotopy Analysis
Assignment of Research Methodology Student Name: Nisha Shukla Enrollment No.: 14408002
Department: Mathematics Topic: Review of thesis Thesis details Authors Name: Erik Sweet Topic
of Thesis: Analytical and Numerical solutions of differential equations arising in fluid flow and Heat
transfer University: University of Central Florida Orlando, Florida Year: 2009 1. Brief Summary of
Thesis: In this thesis, Homotopy analysis method (HAM) has been applied to obtain the solutions of
nonlinear differential equations arising in fluid flow and Heat transfer. This method ... Show more
content on Helpwriting.net ...
There are many other analytical methods exist for solving nonlinear differential equations, for
example: Adomin's decomposition method, Homotopy perturbation method (HPM), Liapunov's
artificial method etc. But HAM is more general in comparison of all of these methods. In this thesis,
author presents a difference between HPM and HAM, but he has not compared the results of any
problem obtained by HAM and HPM. He presents only a theoretical description of comparison of
methods. In HAM, we have to choose an initial guess, a linear operator and a convergence control
parameter h. According to Liao, we can choose these parameters freely. This thesis provides a way
to select an appropriate linear operator and value of convergence parameter. Non–uniqueness of
linear operator has also mentioned, which shows the independency of choosing a linear operator.
Author has applied HAM to solve a system of nonlinear partial and ordinary differential equations,
which shows a significant application of HAM. But he has not solved any system of more than two
equations; this is a drawback of this thesis. In chapter 3, some theorems are given to show the
existence of solution. In fluid flow problems, many differential equations have multiple solutions,
but in this thesis this case has not

Horse Jockeys: Why Do Horse Racing Study
On May 1, 2004, a horse named Smarty Jones won the Kentucky Derby. That same afternoon, I
graduated from Pepperdine University with a degree in economics. A day later, I stepped onto the
Hollywood Park racetrack aboard a horse named Dubai Dolly to ride my first race as a professional
jockey. At first glance, there appears to be little overlap between economists and thoroughbred horse
jockeys. Practically speaking, this is probably true, as most economists study things other than the
horse racing, and most jockeys study nothing but the Daily Racing Form. I have always been
passionate about both, however, and after riding more than 4,000 horse races, I am drawn back to
the study of economics. While most jockeys give little thought to economics, the economics of
health care do directly impact them, as virtually all jockeys suffer racing related injuries, many of
them quite severe. After riding races professionally for over eleven years, I have likewise found
myself well acquainted with hospitals and the health care system. In addition to a myriad of minor
injuries, in 2009, I broke my pelvis in six places, my ... Show more content on Helpwriting.net ...
The program was headed by a family friend whom I respected tremendously, and designing ships
sounded unique and appealing. Unfortunately, the program turned out to be a poor fit for me, and I
realized that my true academic interests had always been economics and mathematics. While I was
coming to terms with the fact that enrolling in the program had been a mistake, my injuries were
resolving better than had been expected. Once I was physically able to ride, I chose to withdraw
from my classes except for Differential Equations and head back to the racetrack. I spent the rest of
the semester commuting from Ann Arbor, MI to Thistledown racetrack in Cleveland,

Stochastic Model For Energy Spot Price Analysis
In this chapter, we construct stochastic model for energy spot price by using e of Ordinary Least
Square Regression Model. At this point, it is imperative to discuss seasonality, which is a commonly
observed characteristic in energy markets. In order to assess whether there is actually an underlying
pattern prevailing in the return an autocorrelation test can be easily carried out for verification. As
explained in [10], the evidence of high autocorrelation manifests an underlying seasonality. On the
contract, if the returns were independents, as assumed by the Black – Scholes model, the correlation
coefficient would be very close to zero indicating insufficiently evidence for an underlying
seasonality. Here, would follow an approach where ... Show more content on Helpwriting.net ...
As mentioned earlier, to address this issue, the most commonly used methods include the OLSR.
The discretized equation has been tailor made for an Autoregrassion. However, the idea behind it
into an OLSR model is to subtract the term from both sides of the equation which then gives, (39)
As we observe, we can now analyze this equation as an algebra equation given by, (40) Where, (41)
In order to look at the above equation as a system of Linear equation, take (42) Where, , is an (n–1)
dimensional vector containing the difference of the log prices where 'n' is the number of
observations. matrix with 1's in the first column and the log prices in the second. matrix with the
first coefficient as the intercept and the second coefficient as the slope of the regression line. =
Noise or Residual term. In particular, (43) Firstly, we observe that the slope of the regression line
must equal the coefficient of the log prices. In particular, as indicated above, (44) Taking natural
logarithm of both sides gives, (45) (46) Secondly, the deterministic part of equation (40) must equal
the intercept of the regression line, which gives, (47) (48) Finally, we need to formulate an equation
for

I Can Add Numbers Using A Number Line
Good Moring class, today we are going to be learning about how to add with a number line. Raise
you hand if you have ever used a number line to add? If not that perfectly acceptable because after
today, you will know how! Your I can statement for to is "I can add numbers using a number line."
We will start out with a demonstration of a number line and vocabulary, then we will use our bodies
to understand number lines, we will practice and then you all will complete a worksheet
independently. Let get our thinking caps on and get ready to work hard! Teacher will play a video of
the jumping jelly bean which demonstrates how to use the number line to add. After the video is
complete the teacher will draw a number line and an equation. The teacher will tell students that a
number line had arrows at each end, and dashes along it to show where each number goes. The
number line will go to 20. The teacher will explain that 4+8= is an addition equation and that each
number is called an addend, the + and = are signs and the answer is the sum. The teacher will then
demonstrate how to use the number line to add two addends. Once this is complete students will line
up to go outside.
Middle
Direct instruction ("I do it" – Teacher modeling, direct teaching of content)
Activity 1: Once students are outside the teacher will draw a number line up to 20 with chalk. The
teacher will give an equation to model to the students so they will understand how to do this activity.
The teacher will say

Solving The Time Fractional Coupled Burger 's Equations
HPM for Solving the Time–fractional Coupled Burger's Equations
Khadijah M. Abualnaja
Department of Mathematics and Statistics, College of Science, Taif University, Taif, KSA
dujam@windowslive.com
ABSTRACT
This paper is devoted to derive the explicit approximate solutions for the time–fractional (Caputo
sense) coupled Burger's equations with implementation of the homotopy perturbation method. The
numerical results are compared with the exact solution at special cases of the fractional derivatives.
The results reveal that the proposed method is very effective and simple.
Keywords: Fractional differential equations; Caputo's derivative; Homotopy perturbation method.
MSC 2010: 65N20; 41A30.
INTRODUCTION
Ordinary and partial fractional differential equations (FDEs) have been the focus of many studies
due to their frequent appearance in various applications in damping laws, motion in Newtonian
fluid, dynamical systems, viscoelasticity, biology, physics and engineering ([3], [6]). Consequently,
considerable attention has been given to the solutions of fractional differential equations of physical
interest. Most fractional differential equations do not have exact solutions, so numerical techniques
([12], [18]) must be used. Also, some untraditional approximate methods have recently been
developed by scientists and engineers ([1], [14]).
In this paper we will implement one of these methods, namely, homotopy perturbation method
(HPM) which was firstly presented by He

The Problem Of Differential Equations
Predicting the future is a big topic that many people have attempted and failed. Many people try to
predict things such as the end of the world, the next stock market crash, and the weather. Many
people are also scared of the future and wonder what it will hold, such as the prophet Jonah. When
the Lord told Jonah to go to Nineveh, he was terrified of the future and fled from the Lord. God later
showed Jonah that he is the only one in control and Jonah couldn't run from God. Christians believe
that there is only one person that is in control of the future and can predict it, and that would be God.
While they know this to be certain, mathematicians believe that by using math, specifically
differential equations, they can predict how things such as population, the stock market, and the
weather can be somewhat accurately predicted. In order to decide whether differential equations can
predict future events, it is important to know exactly what a differential equation is. A differential
equation is an equation involving derivatives of a function or functions.. The functions usually
represent some quantities, and the derivatives represent their rates of change. The differential
equation that results from the two relates the derivative and the function to be used as a productive
equation. The rate of change according to time can be a pivotal part in trying to predict some aspects
of the future. When mathematicians think of using differential equations to predict

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The flowmap denoted $phi_{t,t_{0}}in Diff(R^{N})$ where $Diff(R^{N})$ denotes the group of
diffeomorphisms of $R^{N}$ is defined as a map that takes the initial solution, that is the solution at
$t_{0}$ to the solution at any time $t$, this can be expressed mathematically by
egin{center}

$phi_{t,t_{0}}: Y_{t_{0}}longmapsto Y_{t}$ end{center} That is to say, given any initial data
$Y_{0}in R^{N}$, the solution $ Y_{t}$ at any later time can be easily specified. This can be done
by applying the action of the flowmap to the initial data $Y_{0}$ in order to $Y_{t}=phi_{t,t_{0}}
circ Y_{t_{0}}$.
Consider a function $fin Diff(R^{N})$, by using the chain rule, we obtain
$ frac{d}{dt}(f(Y))=V(Y).partial_{Y} f(Y)$
This means that action of the vector fields on $Diff(R^{N})$ is as first order partial differential
operators since $V(Y).partial_{Y} f(Y)$ is considered as first order partial differential operators.
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The evolution of $fcircphi_{t}$ is given by
egin{center}
$ frac{d}{dt}(fcircphi_{t})=Vcirc fcircphi_{t}$ end{center} The above equation is an autonomous
linear functional differential equation for $fcircphi_{t}$. Such equation has a solution
$fcircphi_{t}=exp(tV)circ fcircphi_{t}$ as $phi_{0}=id$
Setting $f=id$ provides the representation of the flowmap as follows
egin{center}
$phi_{t}=exp(tV)$ end{center} Hence, in this considered case, the flowmap is the exponential of
the vector field. By compositing the above equation

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