lec 01 geng 300 numerical methods
DESCRIPTION
numerical lecturesTRANSCRIPT
GENG 300 NUMERICAL METHODS
Dr. Mohammad Aman Ullah
BOOKS 6/27/2015
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Course Textbook: o Applied Numerical Methods with MATLAB
for Engineers and scientists, Chapra S.C., 3rd edition, McGraw- Hill, 2012
Reference(s): o Chapra S.C. and R.P. Canale (2006)
Numerical Methods for Engineers, 5th edition, McGraw- Hill.
CONTENTS 6/27/2015
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1. Modeling, Computers, and Error analysis 2. Roots of equations 3. Linear algebraic equations 4. Curve fitting and interpolation 5. Numerical differentiation & integration 6. Ordinary differential equations
GRADES 6/27/2015
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HWs and Quizzes: 20% Midterm: 20 % Laboratory: 15 % Term Project: 10 % Final Exam: 35 %
COURSE OBJECTIVES 6/27/2015
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To introduce students to the mostly used numerical methods in the different engineering fields. The course is not theorem-oriented one. The emphasis will be on understanding the concepts of the numerical methods and on applying these concepts for solving various problems. MATLAB and Microsoft Excel will be used as tools to solve the problems using the different numerical methods.
COURSE LEARNING OUTCOMES (CLO) 6/27/2015
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1. Be aware of the mathematical background for the different numerical methods introduced in the course. 2. Understand the different numerical methods to solve the algebraic equations and to solve system of linear and non linear equations. 3. Understand the different numerical methods for interpolation, differentiation, integration and solving set of ordinary differential equations.
COURSE LEARNING OUTCOMES (CLO)… 6/27/2015
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4. Understand how numerical methods afford a mean to generate solutions in a manner that can be implemented on digital computers. 5. Use the built in functions in MATLAB and EXCEL. 6. Create MATLAB functions for solving numerical engineering problems. 7. Work on group projects.
COURSE GROUND RULES 6/27/2015
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1. Use of programmable calculator: not allowed 2. Class Attendance:
a) taken at 10 minutes; after 10 minutes can attend but recorded absent
b) no makeup except with a valid (medical) reason 3. Plagiarism:
a) will be reported to HoD for formal actions b) borrowing and sharing of resources are not
allowed 4. Cell Phones:
a) during lectures "Silent" or "OFF“, during exams "OFF".
b) not allowed to use as a calculator
INTRODUCTION
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Lecture 1: Chapter 1 & 4
Modeling, Computers and Error Analysis
TOPICS COVERED FROM CHAPTER 1 & 4 6/27/2015
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1. Why numerical methods? 2. Mathematical Modelling concept 3. Error Analysis:
a. Significant figures b. Accuracy and precision c. Error definitions
i. For known true value ii. For approximations
d. Major errors i. Round off ii. Truncation
1. WHY NUMERICAL METHODS? 6/27/2015
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1. WHY NUMERICAL METHODS? 6/27/2015
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2. MATHEMATICAL MODELLING CONCEPT 6/27/2015
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Why do we do modelling?
To understand physical phenomena and describe it mathematically To use the mathematically formulated model as a tool to predict the behaviour of a system for the new conditions In order for a model to be used as a simulation tool the model should represent the reality as close as possible. Therefore, validation of a model with experimental observation is an absolute necessity to gain confidence in a model before using it in a simulation processes.
MATHEMATICAL MODEL… 6/27/2015
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Complex example: MATHEMATICAL MODEL…
Model for falling parachute:
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MATHEMATICAL MODEL…
Dependent variable
Forcing function
Parameters
Independent variable
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Conservation laws are fundamental laws that are used in engineering.
If no accumulation (i.e. no change over time), in and out must be balanced.
MATHEMATICAL MODEL…
Conservation Laws and Engineering:
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Flow in = Flow out
MATHEMATICAL MODEL…
For steady-state fluid flow in pipe:
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MATHEMATICAL MODEL…
For steady-state condition in falling parachute:
Thus,
3. ERROR ANALYSIS 6/27/2015
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Why errors are concerned ? Numerical methods yield approximate results, results that are close to the exact solution. The question is “How much error is present in our calculation and is it tolerable?” Motivating example: How long does it take to reach a 11 carat uncut diamond located 20 meter away, if you travel by a car that runs at 10m/s on the 1st second, 5 m/s on the 2nd second, 2.5 m/s on the 3rd second and so on i.e. speed decreases by half on the next seconds?
Answer by a mathematician: Answer by an Engineer: Answer by a lawyer:
3A. SIGNIFICANT FIGURES 6/27/2015
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3A. SIGNIFICANT FIGURES (SF) 6/27/2015
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3A. SIGNIFICANT FIGURES (SF) 6/27/2015
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Significant figures also are indication of the precision with which the quantity is known.
The more significant figures, the more precise is the value.
Generally, if you report the value of a measured quantity with 3 SF, you indicate that the value of the third of these figures may be off by as much as a half-unit.
Example: 8.3 g, means the mass lies between 8.25 and 8.35 g.
3A. SIGNIFICANT FIGURES (SF) 6/27/2015
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3B. ACCURACY AND PRECISION 6/27/2015
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Accuracy: How close is a computed or measured value to the true value Precision (or reproducibility): How close is a computed or measured value to previously computed or measured values. Inaccuracy (or bias): A systematic deviation from the actual value. Imprecision (or uncertainty): Magnitude of scatter
3B. ACCURACY AND PRECISION 6/27/2015
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3C. ERROR DEFINITIONS 6/27/2015
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3C. ERROR…TOLERANCE 6/27/2015
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3DI. MAJOR ERRORS- ROUND-OFF ERROR 6/27/2015
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3DI. MAJOR ERRORS- ROUND-OFF ERROR 6/27/2015
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3DII. MAJOR ERRORS- TRUNCATION ERROR 6/27/2015
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Truncation errors are those that result from using an approximation in place of an exact mathematical procedure. Non-elementary functions such as trigonometric, exponential, and others are expressed in an approximate fashion using Taylor series when their values, derivatives, and integrals are computed Any smooth function can be approximated as a polynomial. Taylor series provides a means to predict the value of a function at one point in terms of the function value and its derivatives at another point. Taylor Series (TS) is built term by term, started with zero-order approximation. The higher the order of approximation applied, the lower the truncation error.
3DII. TRUNCATION ERROR- TAYLOR SERIES 6/27/2015
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3DII. TAYLOR SERIES… 6/27/2015
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For exact solution infinite number of terms are required In most cases, only a few terms will result in an approximation that is close enough to the true value for practical purposes
3DII. TAYLOR SERIES… 6/27/2015
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3DII. TAYLOR SERIES… 6/27/2015
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Derivatives used in Taylor series can be approximated using numerical differentiation with:
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2.125.05.015.01.0)( 234 +−−−−= xxxxxf
at x =0.25, the derivative can be calculated directly as:
25.00.114.04.0)(' 23 −−−−= xxxxf
Given
9125.0)25.0(' −==xf
TRY FOR YOURSELF …
SUMMARY 6/27/2015
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1. Why numerical methods? 2. Mathematical Modelling concept 3. Error Analysis:
a. Significant figures b. Accuracy and precision c. Error definitions
i. For known true value ii. For approximations
d. Major errors i. Round off ii. Truncation