Key Implications Of The Solow Model

Key Implications Of The Solow Model
The Solow model is widely considered to be the standard neoclassical economic growth model which serves as the basis for understanding economic
growth. I will first introduce the two basic equations that the Solow model is built around, discussing the main assumptions made along the way. I
will then present the key equation of the Solow model and discuss its results and implications. I will then address why it is desirable to use
log–linearization, and how it can be used to study the dynamics near the steady–state. We will then briefly look into the empirical results of Mankiw et
al. (1992) and what they mean for the Solow model. We begin with the aggregate production function of the model. Let us denote output (Y), capital
(K), labour (L) and the index of productive efficiency (A). The production function can then be written as:
Y=F(K,AL) ... Show more content on Helpwriting.net ...
To the contrary, if a is high then the rate of convergence is low because diminishing returns set in slowly.
The differential equation for output convergence is quite similar. Assuming Cobb–Douglas technology we can show that y=k^a which then implies that
y М‡/y=a k М‡/k by taking logs and differentiating with respect to time. Here a is translating a given growth rate of capital into a given growth rate of
output. Substituting into the previous differential equation we get a new one for the growth rate of output:
y М‡/yв‰€
–О»[logy–logy^*]
The growth rate in each of these expressions is linear in the gap from the steady–state measured in logarithms.
Solving both differential equations and re–arranging we get: logy(t) =(e^(–О»t) )logy(0)+(1–e^(–О»t) )logy^* (5) k(t)– k^*=(e^(–О»t) )[k(0)–k^*]
... Get more on HelpWriting.net ...

Singularly Perturbed Problem Essay
section{Parameter Uniform} In the context of singularly perturbed problems posed on non–rectangular domains, the finite element method is a
natural choice for many researchers (e.g. cite{118}). However, in the context of singularly–perturbed convection–diffusion problems, it is difficult to
construct monotone methods on highly anisotropic meshes, which are desirable when thin layers are present. Moreover, energy norms or other norms
based on the $L^2–$norm, are the natural norms associated with variational methods. Error analysis in the pointwise norm is not straightforward in a
finite element framework, especially when one is dealing with a problem that is not self–adjoint. Given that their interest lies in parameter uniform
numerical... Show more content on Helpwriting.net ...
The presence of a convective term means that the data of the problem across the entire domain has an influence on the outflow boundary layer, in
contrast to reaction diffusion problems where it is only the data local to a boundary that has an influence on the boundary layer. Hegarty and O'Riordan
cite{115} constructed a parameter–uniform numerical methods for the singularly perturbed problem posed on a circular domain. Asymptotic
expansions for the solutions to such a problem have been established in cite{120,121,122,123}. Analytical expressions for the exact solution, in the
case of constant data, are given in cite{123} as a Fourier series with coefficients written in terms of modified Bessel functions. In cite{120,121,122}
sufficient compatibility conditions are identified so that the accuracy of the asymptotic expansion can be estimated in the $L^2–$norm or in a suitably
weighted energy norm. These expansions are derived without recourse to a maximum principle. The smooth case, where significant compatibility is
assumed, is examined in cite{120}; the non–compatible case with a polynomial source term is studied in cite{125} and for a more general source
term in cite{122}. Based on these asymptotic expansions, a numerical method is constructed in cite{124} for the problem, which uses a
quasi–uniform mesh (which is, hence, not layer–adapted). By enriching the finite element subspace, with certain exponential boundary–layer basis

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

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 '

Mesh Dependency And Simp Method
This approach is called Solid Isotropic Material with Penalty (SIMP), in which material is assumed to be dependent upon density linearly. In
accordance to the SIMP technique, the design variables are a sum of material densities denoted as (ПЃ) that relates the stiffness of an element straight
to the density of that element (James , et al., 2014). According to (Gomes , et al., 2013) the three main mathematical encounters with SIMP method are
mesh dependency, checkboard patterns and local minima. To compensate for mesh–dependency and checkboard patterns instabilities, a well–known
sensitivity filter brought forth by Bendose Sigmund is used (Gomes , et al., 2013). The local minima problem is dealt with by a continuation method
with various... Show more content on Helpwriting.net ...
In the construction of the topology optimisation problem, a material is assigned a density (ПЃ) which is associated with a stiffness (E, E(ПЃ,q) = ПЃq
E). This material density (ПЃ) can then be used by each finite element in the initial topology problem design space with E as the stiffness for the
isotropic material. The problems can be solved as void if ПЃ = 0 or associated material if ПЃ =1. Therefore, if ПЃ tends to be zero, the stiffness in that
given element is zero, which means the element can be deleted because it is no longer important for the structure (Rao, et al., n.d.). However, if the
density reaches one then that element is of dire importance to the structure and cannot be deleted (Rao, et al., n.d.). If this simple formulation is used,
the total elastic energy measure (U) can be used as the objective function of the optimisation problem. This formulation is written below: Minimise: U
n=1.....,N Subject to: в€‘_(N
–1)^Nв–’гЂ–гЂ–ПЃгЂ—_n V_n=V_0 гЂ— (2.1) 0 в‰¤ ПЃn в‰¤ 1 n = 1......,N This formulation in equation (1) and
can also be extended to suite multiple load case problems by minimising the weight of the total elastic energies. The following equation expresses
formulation for a multi–load case topology problem, using a weighted sum function for

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

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

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

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

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

Assignment 1: Finite Element Analysis Essay
[pic] ENGD3016 Solid Mechanics Assignment 1: Finite Element Analysis Name: Wei Zhang ID: P14021978 Date: Dec17th 2015 Abstract 1.0
Introduction 2.0 Objectives 3.0 Matlab 4.0 Solidworks 4.1 Model of truss 1 4.2 Model of truss 2 5.0 Comparison of the two trusses 6.0 Comparison
between MATLAB and SOLIDWORKS 6.1 Comparison of results 6.2 Advantages and disadvantages 7.0 Conclusion Appendix Abstract The purpose
of this report is to based on two different 2D pictures of trusses to finite element analysis from two ways. First carried... Show more content on
Helpwriting.net ...
To construct a part to be analysed on Algor software and to find relevant deflections stresses and strains using the software. To compare the two
methods of finite element analysis outlining strengths and drawbacks of each. To decide on an appropriate box size, and material for the truss taking
into account its intended use. 3.0 Matlab First, the truss of materials selection, use of the material information shown in the following tableAISI 4130
Steel, annealed at 865C |Elastic |Poisson's Ratio|Shear Modulus |Mass Density |Tensile |Yield Strength |Thermal |Specific | |Modulus | | | |Strength |
|conductivity |Heat | |2.05e+011N/m^|0.285N/A |8e+010N/m^2 |7850kg/m^3 |560000000N/m^2 |460000000N/m^2 |42.7W/(mВ·K) |477J/(kgВ·K) |
|2 | | | | | | | | E = 2.05E11 The thickness of the steel pipe 2mm, so that the outer diameter of 20mm, an inner diameter of 16mm. Architecture sectional
area A=20*20–16*16=144 mmВІ=0.000144

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 М…(t)=Hv М…(t)1.6
, where H is a constant matrix and v М…(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

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

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.
1.1.1Description 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

Notes On Relation Between Latex And Latex
documentclass[11pt]{article}
usepackage{graphicx} usepackage{a4wide} ewtheorem{theorem}{Theorem}[section]
ewtheorem{corollary}[theorem]{Corollary}
ewtheorem{lemma}[theorem]{Lemma}
ewtheorem{proposition}[theorem]{Proposition}
ewtheorem{definition}[theorem]{Definition}
ewtheorem{remark}[theorem]{Remark}
ewtheorem{assumption}[theorem]{Assumption}
ewtheorem{conjecture}[theorem]{Conjecture}
ewtheorem{example}{underline{Example}} setlength{parindent}{0mm} %=============================================
%
% Comments in latex are marked with a % and do not print
%
%=============================================
%
% To include figures save then either as eps or as jpg or pdf.
%
%=============================================... Show more content on Helpwriting.net ...

vspace{0.5cm}
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.
vspace{0.5cm}
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

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 differentialheat equation: u_t=в€
ќu_xx (1)
Here, u signifies the temperature of the entire body and в€
ќ 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_в€
ћ) 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

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

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

Comparing Calculus 3 And Differential Equations
I have been very blessed in my life by excelling in school. Although I do very well in my classes, there are a couple classes that I have had to work
twice as hard in. To be specific, I have had to work exceptionally harder in math and science. Most of my friends would be surprised to hear that I
have to work so much harder in these classes. Many of them know that I am in Advanced Placement classes, so they automatically assume I am
naturally gifted in all subject areas and that the course matter comes easily to me. I am actually in the highest math class offered at my school, a
combination of Calculus 3 and Differential Equations. This course is two grade levels higher than the math class average senior takes. Although I am
very ... Show more content on Helpwriting.net ...
I go to my teachers and ask them questions about different lessons. I also have had to study more for these specific classes. For example, one thing
I do that is quite different from many students is that I make my own quizzes and tests to study for Calculus 3 and Differential Equations. Many of
my friends say I am crazy for putting this amount effort into it, but I believe that it has truly helped me do well on quizzes and tests. Additionally, I
look up numerous educational videos on YouTube and Khan Academy to get a different way of obtaining more knowledge on certain subjects. I do
this exclusively for AP Biology. My AP Biology teacher gives us review books for the AP test, so I often take the practice tests and look at the
answers to see if I know the information. All of these tactics have helped me immensely. Although I eventually understand the content, I do have trouble
applying the information. I can know how to do a Chi–square test with fruit flies as well as how to do triple integrals, but knowing when exactly to do
them is quite difficult for me. For both of these classes, I also form study groups with my friends to review material. It is very beneficial because we
help each other with different concepts. Math and science will always be challenging for me, but I am working hard to do

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

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

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).
в€‡(cв€™в€‡u)+bв€™в€‡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

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 ...
ApothГ©loz'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. ApothГ©loz'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

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

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, PoincarГЁ 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 П‰ 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

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,

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 orderordinary differential equations.
Key words ; Power series, differential equations, Frobenius Method, Lengendre polynomials
1.0 INTRODUCTION
1.1BACKGROUND 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).
1.2STATEMENT OF THE PROBLEM
The research work seeks to find solutions of second–order ordinary differential equations using the power series method.
1.3AIM 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

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.

Taking a Look at ANSYS
1.Introduction
1.1ANSYS
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].
1.1.1Brief 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

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

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

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= (Оё Оё М‡ )^T
To be able to apply the LQR technique on the system, linearization is important. Therefore the nonlinear model of the system turns into: Оё М€=
–D^(–1) CОё М‡–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
М‡=(в– (0&I@0&–D^(–1) C))x+(в– (0@гЂ––DгЂ—^(–1) C))+(в– (0@D^(–1) H))u (15)
Where I and 0 are identity and zero matrices respectively.
The system equation can be rewritten as: x М‡=f(x)+g(x)u (16)
Where
f(x)=(в– (0&I@0&–D^(–1) C))x+(в– (0@гЂ––DгЂ—^(–1) C)) (17) g(x)=(в– (0@D^(–1) H)) (18)
5.2 Linear

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) = п°„ n n,П‰j n H n,П‰j i,j ОІi,j П†i (x,t), where ОІi,j's are defined in each computational time interval and
П†i (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
пїј1 0.9
0.8П‰E K 0.7
0.6
0.5
0.4
0.3
KK 1П‰2 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
) =
п°†
О© fn+1v в€’ п°†
п°† un+1+un+1в€’un

+ fixed fixed v v + fixed fixed
О© n+1
в€†t
+ ufixed) В· в€‡v.
(4)
+ Оєв€‡(u+ n+1
О©
6
For any fixed basis П†n,П‰j 's this equation is zero as the fixed solution is constructed by setting the k n,П‰j equation zero for each fixed basis.
Using П†k,+ 's в€
Ђk, 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 О±П‰k = в€
ҐRnП‰k в€
Ґ/в€
ҐRnв€
Ґ,where Rn is the global residual vector and RnП‰k
is the local residual vector in П‰k (as mentioned earlier) and L1 norm is used .
Let NП‰ be the average number of subregions where additional basis will added. Furthermore, О±П‰j О±п°‡ j = п°„j О±П‰j NП‰, (5) П‰ With
probability proportional to О±п°‡ j в€§ 1, the region П‰j , is selected and Jj = 1 if the region is П‰ п
їјselected and zero otherwise. Given subregion j
is selected the k th extra basis is

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

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 oilв€’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, О», for a mixture of heavy oil and light hydrocarbons where Оіo and Оіs are
the specific gravities of the crude oil and liquid solvent, respectively.
The diffusion coefficient is usually correlated with viscosity asвЂЁ
DпЂЁuпЂ©пЂЅauпЂb (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 Dasв€’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 пЃ„пЃІпЂ пЂЅпЂ пЃІm пЂпЂ пЃІsвЂЁHeavy oilв€’solvent mixture is commonly treated as
ideal solution and its density, ПЃm, is calculated by
пЃІm пЂЅcsпЃІs пЂ«coпЃІo (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

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

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

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

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}.

Key Implications Of The Solow Model

Recommended

Recommended

More Related Content

Similar to Key Implications Of The Solow Model

Similar to Key Implications Of The Solow Model (20)

More from Laura Brown

More from Laura Brown (10)

Recently uploaded

Recently uploaded (20)

Key Implications Of The Solow Model