The Islamic University of Gaza Faculty of Engineering Civil Engineering Department Numerical Analysis ECIV 3306

Chapter 1

Mathematical Modeling
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Part one : Approximation and Errors
Specific Study Objectives
• Recognize the difference between analytical and numerical solutions. • Recognize the distinction between truncation and round-off errors. • Understand the concepts of significant figures, accuracy, and precision. • Recognize the difference between true relative error t, approximate relative error a, and acceptable s error 10:51:09 PM

Chapter 1: Mathematical Modeling
Mathematical Model
• A formulation or equation that expresses the essential features of a physical system or process in mathematical terms. • Generally, it can be represented as a functional relationship of the form

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Mathematical Modeling

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Simple Mathematical Model
Example: Newton’s Second Law (The time rate of change of momentum of a body is equal to the resultant force acting on it)

a = acceleration (m/s2) ….the dependent variable m = mass of the object (kg) ….the parameter representing a property of the system. f = force acting on the body (N)

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Complex Mathematical Model
Example: Newton’s Second Law

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Where: c = drag coefficient (kg/s), v = falling velocity (m/s)

Complex Mathematical Model

At rest: (v = 0 at t = 0), Calculus can be used to solve the equation

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Analytical solution to Newton's Second Law
.

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Analytical solution to Newton's Second Law

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Analytical solution to Newton's Second Law

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Numerical Solution to Newton's Second Law

Numerical solution: approximates the exact solution by arithmetic operations. Suppose

...“Iterative Methods”
“Gauss and Gauss-Seidel”
Profesor | | Nieves Fonseca Ricardo |
Mentado Camacho Félix
Navarro Escamilla Erandy
Péloquin Blancas María José
Rubio Miranda Ana Luisa
Abstract
Many real life problems give us several simultaneous linear equations to solve. And we have to find a common solution for each of them. There are several techniques to use.
Instead of using methods that provide a solution to a set of linear equations after a finite number of steps, we can use a series of algorithms with fewer steps, but its accuracy depends on the number of times it is applied (also known as iterative methods). For large systems they may be a lot faster than direct methods.
We will expand on two important methods to find numerical solutions to linear systems of equations. There will be an introduction to each method, besides detailed explanations on each of them.
Normally each process is long, so they are ideal for programming.
Keywords
Iterative, algorithm, linear equation, convergence.
Objective
Understand the concepts of iterative methods, and convergence, besides the difference and usefulness between direct and iterative methods. To give a clear and understandable idea of Gauss
and Gauss-Seidel methods to solve systems of linear equations, and show how to apply them.
Investigation
Iterative method
An iterative method is one that computes approximations in a progressive way...

...DEPARTMENT OF COMPUTER SCIENCE
FACULTY OF INFORMATION TECHNOLOGY AND APPLIED SCIENCES
LEAD CITY UNIVERSITY, IBADAN
FIRST SEMESTER, 2013/2014 ACADEMIC SESSION
LECTURER-IN-CHARGE: PROF. B. A. OLUWADE
CSC 403: NUMERICAL COMPUTATION II (with MATLAB)
TUTORIAL QUESTIONS
1
(a)
What do you understand by the Euler method ?
(b)
Let
y/ = 3y + 1
y(0) = 2
be an initial value problem. Using Euler method, present an approximation of y(5)
using a step size of 1.
2
(a)
State Simpson’s rule.
(b)
Write MATLAB code for finding the numerical approximation of a
definite integral using (a) above.
3
(a)
Write the meaning of the following MATLAB commands and illustrate
with at least one example each:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
ceil (x)
round (x)
floor (x)
fix (x)
eye
zeros
ones
rem (x, y)
fprintf
xlabel ( )
(b)
With the aid of examples, explain the meaning and order of evaluation of
arithmetic operators and expressions in MATLAB.
1
(a)
Write a MATLAB code for evaluating sin(x)2 + cos(x)2
(b)
4
Write a MATLAB code for drawing the following :
y(x) = sin(3x)
3x
for -5 ≤ x ≤ 15
5
(a)
Write a MATLAB code for finding the approximate solution of an initial
value problem using the Euler method.
(b)
The midpoint rule, otherwise referred to as the rectangle method, is an
algorithm for computing an approximation to a definite integral by finding the area of...

...an introduction to the ﬁnite element solution of problems posed as partial diﬀerential equations. It is self-contained in that it requires no previous knowledge of the subject. Familiarity with the mathematics normally covered by the end of the second year of undergraduate courses in mathematics, physical science or engineering is all that is assumed. In particular, matrix algebra and vector calculus are used extensively throughout; the necessary theorems from vector calculus are collected together in Appendix B. The reader familiar with the ﬁrst edition will notice some signiﬁcant changes. I now present the method as a numerical technique for the solution of partial diﬀerential equations, comparable with the ﬁnite diﬀerence method. This is in contrast to the ﬁrst edition, in which the technique was developed as an extension of the ideas of structural analysis. The only thing that remains of this approach is the terminology, for example ‘stiﬀness matrix’, since this is still in common parlance. The reader familiar with the ﬁrst edition will notice a change in notation which reﬂects the move away from the structural background. There is also a change of order of chapters: the introduction to ﬁnite elements is now via weighted residual methods, variational methods being delayed until later. I have taken the opportunity to introduce a completely new chapter on boundary element methods. At the time of the ﬁrst edition, such methods were in their...

...Preparation Guide NumericalAnalysis
This preparation guide helps you to prepare for numerical aptitude assessments. It provides guidance on how best to approach the assessment, allowing you to give your best possible performance.
Why are Aptitude Assessments used?
Employers often use aptitude assessments as part of their assessment procedures for the selection and development of staff. Research has shown that they are powerful predictors of performance at work.
Assessments help you to:
- demonstrate your strengths - be assessed fairly on job relevant criteria - find out more about your strengths and development needs - make future career decisions based on your abilities
Assessments help employers to:
- select people best suited to the demands of the job - identify areas where individuals might benefit from further development - obtain objective information about people’s abilities
Instructions
On the following pages are some practice questions which are similar to those you will be asked in the assessment. Completing these will help you understand the types of questions used and gain experience in taking ability tests. These questions are designed to assess your ability to understand numerical information. You will be presented with a series of tables and graphs, each followed by several questions. Your task is to choose the best answer to each question from the options given. Have a pen and paper...

...~~~~~~~~~~~~~~~~~~~~ www.MathWorks.ir ~~~~~~~~~~~~~~~~~~~~
An Introduction to Programming
and Numerical Methods in MATLAB
~~~~~~~~~~~~~~~~~~~~ www.MathWorks.ir ~~~~~~~~~~~~~~~~~~~~
S.R. Otto and J.P. Denier
An Introduction to
Programming and
Numerical Methods
in MATLAB
With 111 Figures
~~~~~~~~~~~~~~~~~~~~ www.MathWorks.ir ~~~~~~~~~~~~~~~~~~~~
S.R. Otto, BSc, PhD
The R & A
St Andrews
Fife
KY16 9JD
Scotland
J.P. Denier, BSc (Hons), PhD
School of Mathematical Sciences
The University of Adelaide
South Australia 5005
Australia
British Library Cataloguing in Publication Data
Otto, S. R. (Stephen Robert)
An introduction to programming and numerical methods in
MATLAB
1. MATLAB (Computer file) 2. Numericalanalysis — Data
processing
I. Title II. Denier, J. P.
518′.02855
ISBN 1852339195
Library of Congress Control Number: 2005923332
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as
permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of
the publishers, or in the case of reprographic reproduction in accordance with the terms of licences
issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms
should be sent to the publishers.
ISBN-10:...

...Cao
Numerical SPDES
November 20, 2009
1 / 35
Outline
1
Introduction
Brownian sheet and elliptic SPDE
Yanzhao Cao
Numerical SPDES
November 20, 2009
2 / 35
Outline
1
Introduction
Brownian sheet and elliptic SPDE
2
SPDE with discretized white noise
˙
Discretization of white noise Wh
Error estimate
Yanzhao Cao
Numerical SPDES
November 20, 2009
2 / 35
Outline
1
Introduction
Brownian sheet and elliptic SPDE
2
SPDE with discretized white noise
˙
Discretization of white noise Wh
Error estimate
3
Finite element approximation of elliptic SPDE
Finite element solution Uh
Yanzhao Cao
Numerical SPDES
November 20, 2009
2 / 35
Outline
1
Introduction
Brownian sheet and elliptic SPDE
2
SPDE with discretized white noise
˙
Discretization of white noise Wh
Error estimate
3
Finite element approximation of elliptic SPDE
Finite element solution Uh
4
Finite element approximations of stochastic Stokes equations
Stochastic Stokes Equations
Yanzhao Cao
Numerical SPDES
November 20, 2009
2 / 35
Outline
1
Introduction
Brownian sheet and elliptic SPDE
2
SPDE with discretized white noise
˙
Discretization of white noise Wh
Error estimate
3
Finite element approximation of elliptic SPDE
Finite element solution Uh
4
Finite element approximations of stochastic Stokes...

...CHAPTER 3
LOAD FLOW ANALYSIS
[CONTENTS: Review of solution of equations, direct and iterative methods,
classification of buses, importance of slack bus and YBUS based analysis,
constraints involved, load flow equations, GS method: algorithms for finding the
unknowns, concept of acceleration of convergence, NR method- algorithms for
finding the unknowns, tap changing transformers, Fast decoupled load flow,
illustrative examples]
REVIEW OFNUMERICAL SOLUTION OF EQUATIONS
The numericalanalysis involving the solution of algebraic simultaneous equations
forms the basis for solution of the performance equations in computer aided electrical
power system analyses, such as during linear graph analysis, load flow analysis
(nonlinear equations), transient stability studies (differential equations), etc. Hence, it
is necessary to review the general forms of the various solution methods with respect
to all forms of equations, as under:
1. Solution Linear equations:
* Direct methods:
- Cramer’s (Determinant) Method,
- Gauss Elimination Method (only for smaller systems),
- LU Factorization (more preferred method), etc.
* Iterative methods:
- Gauss Method
- Gauss-Siedel Method (for diagonally dominant systems)
2. Solution of Nonlinear equations:
Iterative methods only:
- Gauss-Siedel Method (for smaller systems)
- Newton-Raphson Method (if corrections for...