MECN 3500 Inter - Bayamon Lecture 11111111 Numerical Methods for Engineering MECN 3500 Professor: Dr. Omar E. Meza Castillo [email protected]

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LectureLecture

1111

Numerical Methods for EngineeringNumerical Methods for Engineering MECN 3500 MECN 3500

Professor: Dr. Omar E. Meza CastilloProfessor: Dr. Omar E. Meza [email protected]

http://www.bc.inter.edu/facultad/omeza

Department of Mechanical EngineeringDepartment of Mechanical Engineering

Inter American University of Puerto RicoInter American University of Puerto Rico

Bayamon CampusBayamon Campus

Lecture 11Lecture 11MEC

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Tentative Lectures ScheduleTentative Lectures Schedule

TopicTopic LectureLecture

Mathematical Modeling and Engineering Problem SolvingMathematical Modeling and Engineering Problem Solving 11

Introduction to MatlabIntroduction to Matlab 22

Numerical ErrorNumerical Error 33

Root FindingRoot Finding 4-5-64-5-6

System of Linear EquationsSystem of Linear Equations 7-87-8

Least Square Curve FittingLeast Square Curve Fitting 99

Numerical IntegrationNumerical Integration 1010

Numerical DerivationNumerical Derivation 1111


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Derivation FormulasDerivation Formulas

Numerical DerivationNumerical Derivation

33


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To solve numerical problems and To solve numerical problems and appreciate their applications for appreciate their applications for engineering problem solving.engineering problem solving.

44

Course ObjectivesCourse Objectives

Lecture 11Lecture 11

•We like to estimate the value of We like to estimate the value of f f '('(xx) for a ) for a given function given function ff((xx).).

•The derivative represents the rate of change The derivative represents the rate of change of a dependent variable with respect to an of a dependent variable with respect to an independent variable.independent variable.

•The difference approximation isThe difference approximation is

•If If xx is allowed to approach zero, the is allowed to approach zero, the difference becomes a derivative:difference becomes a derivative:

x

xfxxf

dx

dy

x

xfxxf

x

y

ii

x

ii

)()(lim

)()(

0



Numerical DifferentiationNumerical Differentiation

•The Taylor series expansion of The Taylor series expansion of ff((xx) about ) about xxii is is

•From this:From this:

•This formula is called the This formula is called the first forward divided first forward divided difference difference formula and the error is of order formula and the error is of order OO((hh).).

h

xfxf

xx

xfxfxf

xxxfxfxf

ii

ii

iii

iiiii

)()()()()(

))(()()(

1

1

1

11


•Or equivalently, the Taylor series expansion of Or equivalently, the Taylor series expansion of ff((xx) ) about about xxii can be written as can be written as

•From this:From this:

•This formula is called the This formula is called the first backward divided first backward divided difference difference formula and the error is of order formula and the error is of order OO((hh).).

h

xfxf

xx

xfxfxf

xxxfxfxf

ii

ii

iii

iiiii

)()()()()(

))(()()(

1

1

1

11


•A third way to approximate the first derivative is to A third way to approximate the first derivative is to subtract the backward from the forward Taylor series subtract the backward from the forward Taylor series expansions:expansions:

•This yields toThis yields to

•This formula is called the This formula is called the centered divided centered divided difference difference formula and the error is of order formula and the error is of order OO((hh22).).

h

xfxfxf

hxfxfxf

hxfxfxf

hxfxfxf

iii

iii

iii

iii

2

)()()(

)(2)()(

_________________________

)()()(

)()()(

11

11

1

1


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Forward

Backward

Centered


•The forward Taylor series expansion for The forward Taylor series expansion for ff((xxii+2+2) in ) in terms of terms of ff((xxii) is) is

•Combine equations:Combine equations:

212

21

22

22

)()()(2)(

_______________________________________

2

)()()()(2

)2(2

)()2)(()()(

)2(2

)()2)(()()(

hxfxfxfxf

hxf

hxfxfxf

hxf

hxfxfxf

hxf

hxfxfxf

iiii

iiii

iiii

iiii

Finite Difference Approximation of Higher Finite Difference Approximation of Higher DerivativesDerivatives


•Solve for Solve for f f ''(''(xxii):):

•This formula is called the This formula is called the second forward finite second forward finite divided differencedivided difference and the error of order and the error of order OO((hh).).

•The The second backward finite divided difference second backward finite divided difference which has an error of order which has an error of order OO((hh) is) is

221

212

)()(2)()(

)()(2)()(

h

xfxfxfxf

h

xfxfxfxf

iiii

iiii


•The The second centered finite divided difference second centered finite divided difference which which has an error of order has an error of order OO((hh22) is) is

211 )()(2)(

)(h

xfxfxfxf iiii


•High accurate estimates can be obtained by High accurate estimates can be obtained by retaining more terms of the Taylor series.retaining more terms of the Taylor series.

hxf

h

xfxfxf

hxf

xxxfxfxf

iiii

iiiiii

2

)('')()()(

2

)(''))(()()(

1

211

•The forward Taylor series expansion is:The forward Taylor series expansion is:

•From this, we can writeFrom this, we can write

High-Accuracy Differentiation FormulasHigh-Accuracy Differentiation Formulas


•Substitute the second derivative approximation Substitute the second derivative approximation into the formula to yield:into the formula to yield:

•By collecting terms:By collecting terms:

•Inclusion of the 2Inclusion of the 2ndnd derivative term has improved derivative term has improved the accuracy to the accuracy to OO((hh22).).

•This is the This is the forward divided difference formula forward divided difference formula for for the first derivative.the first derivative.

h

xfxfxfxf

hhxfxfxf

h

xfxfxf

iiii

iii

iii

2

)(3)(4)()(

2

)()(2)()()(

)(

12

212

1


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Forward FormulasForward Formulas


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Backward FormulasBackward Formulas


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Centered FormulasCentered Formulas


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1919


ExampleExample

Estimate Estimate f f '(1) for '(1) for ff((xx) = ) = eexx + + xx using the centered formula using the centered formula of of OO((hh44) with ) with hh = 0.25. = 0.25.

SolutionSolution

5.15.012

25.125.01

1

75.025.01

5.05.012

12

)()(8)(8)()(

2

1

1

2

2112

hxx

hxx

x

hxx

hxx

h

xfxfxfxfxf

ii

ii

i

ii

ii

iiiii

•From Table 23.3:From Table 23.3:


717.33

)149.2()867.2(8)740.4(8982.5

)25.0(12

)5.0()75.0(8)25.1(8)5.1()(

ffff

xf i

•In substituting the values:In substituting the values:


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www.bc.inter.edu/facultad/omeza

Omar E. Meza Castillo Ph.D.Omar E. Meza Castillo Ph.D.

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Documents

MECN 3500 Inter - Bayamon Lecture 11111111 Numerical Methods for Engineering MECN 3500 Professor: Dr. Omar E. Meza Castillo [email protected]