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Applied Numerical Methods
with MATLAB®
for Engineers and Scientists
4th Edition
Applied Numerical Methods
with MATLAB®
for Engineers and Scientists
4th Edition
Steven C. Chapra PowerPoints organized by Dr. Michael R. Gustafson II, Duke University andProf. Steve Chapra, Tufts University ©McGraw‐Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of McGraw‐Hill Education.
©McGraw‐Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of McGraw‐Hill Education.
Part 1
Chapter 1
Mathematical Modeling,Numerical Methods, and
Problem Solving
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A Simple Mathematical Model A mathematical model can be broadly definedas a formulation or equation that expressesthe essential features of a physical system orprocess in mathematical terms.Models can be represented by a functionalrelationship between dependent variables,independent variables, parameters, andforcing functions.
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Model Function
Dependent
variable
ൌ
𝑓^
independent
variables
,^
parameters,
forcingfunctions
Dependent variable
- a characteristic that usually reflects
the behavior or state of the system Independent variables
- dimensions, such as time and
space, along which the system’s behavior is beingdetermined Parameters
- constants reflective of the system’s properties
or composition Forcing functions
- external influences acting upon the
system
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Model Results
Using a computer (or a calculator), the model can be used togenerate a graphical representation of the system. Forexample, the graph below represents the velocity of a 68.1kilogram jumper, assuming a drag coefficient of 0.25kilograms per mile
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Numerical Modeling
Some system models will be given as implicit functionsor as differential equations - these can be solved eitherusing analytical methods or numerical methods.Example - the bungee jumper velocity equation frombefore is the analytical solution to the differentialequation
𝑑^
2
where the change in velocity is determined by thegravitational forces acting on the jumper versus thedrag force.
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Euler’s Method
Substituting the finite difference into thedifferential equation gives
𝑑^
2
i
𝑖
i
𝑖
𝑑^
2
Solve for
i
𝑖
𝑑
𝑖^
2
i
𝑖
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Numerical Results
Applying Euler’s method in 2 s intervals yields: How do we improve the solution?•
Smaller steps
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Summary of Numerical Methods
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Part 1
Chapter 2
MATLAB Fundamentals
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Calculator Mode
The MATLAB command widow can be usedas a calculator where you can type incommands line by line. Whenever acalculation is performed, MATLAB will assignthe result to the built-in variable
ans
Example:
ans
MATLAB Syntax
- Syntax is the rules that must be followed
when developing MATLAB programs orsimply using the Command Window as acalculator.
MATLAB Syntax-Variables
- The range of floating point numbers is
between -1.7x
308
to 1.7x
308
(Version
R2014a).
‘>>whos’
used in the
Command Window will list the definedvariables along with their characteristics.
‘>>clear’
removes all
variables from the Command Window.
Special Variables
- Inf
- when a variable exceeds the largest
or smallest number available.
- NaN
- Not a number, usually the result of
an undefined mathematical operation, e.g.,0/0.
to 15 significant figures.
