Chapter 7 Enotes

8/13/2019 Chapter 7 Enotes

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Chapter 7

Ordinary DifferentialEquations (ODE)


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2VcQcFdtdc

Given a differential equation;

Such equation plays important role in

engineering because it express therate of changeof a variable as afunction of variables and parameters.

When the function involves oneindependent variable, the equation iscalled ordinary differential equation

(ODE).


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OrdinaryDifferential

Equations

Runge-KuttaMethods

Multistepmethods

EigenvaluesProblems

- Eigenvalues- Non-self-Starting

Heun Method

- Eulers method- Improvement of

Eulers method-Runge-Kutta method-Sytems of Equations


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Runge-Kutta Method

This section will solve ordinary differentialequations of the form;

),( yxf

dx

dy

The method in general form;

sizestepslopevalueoldvalueNew

hyyii

1

The slope, is used to extrapolate from anold value to a new value over a distance, h.


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The first derivative provides a directestimate of the slope at xi;

where f(xi,yi)is the differential equationevaluated at xiand yi. This estimate can be

substituted into the equation 25.1

>>> Equation 25.2A new value ofyis predicted using the slopeto extrapolate linearly over the step size h.

),( ii yxf

Eulers method

Try out Example 25.1


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Comparison of the true solution withnumerical solution using Eulers method.


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Improvement of Eulersmethod

A source of error in Eulers method is

the derivative at the beginning of the

interval is assumed to apply across

the entire interval.

Two simple modifications are

available to avoid this limitation:1) Heuns Method

2) The Midpoint Method


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Heuns method

To improve the estimate of the slope,

its involves the determination of two

derivatives at the initial point and

at the end point.

The two derivatives are then averaged

to obtain an improved estimate of the

slope for the entire interval using

predictor>>>equation 25.15 and

corrector>>>equation 25.16


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1. (slope at (x0, y0)

2. (estimate of y at x1)

3. (slope at end of interval)

4. y (average slope)

5. y1

(value of y at x1

using averageslope) >>(Eqn 25.15)

6. y1 (value of y at x1using corrector)>>(Eqn 25.16)

01y

'

0y

'

1y



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Runge-Kutta Method

Runge-Kutta methods achieve the accuracyof a Taylor series approach without requiringthe calculation of higher derivatives.

The method in general form;

hhyxyyiiii

),,(1

where is called an incrementfunction;

hyx ii ,,

nnkakaka

....2211


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),(

),(

),(

),(

11,122,1111

22212133

11112

1

hkqhkqhkqyhpxfk

hkqhkqyhpxfk

hkqyhpxfk

yxfk

nnnnninin

ii

ii

ii

as, ps and qs are constants


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2nd-order Runge-Kutta method

where there are 3 versions of 2nd-order

Runge-Kutta method;

a) Heun method >>>equation 25.36

b) Midpoint method >>>equation 25.37c) Ralstons method >>>equation 25.38

The method in general form:>>>equation 25.30 to 25.30b


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3rd-order Runge-Kutta method

>>> equation 25.39 to 25.39c

4th-order Runge-Kutta method

The most commonly used. Called asthe classical fourth-order Runge-Kutta.

>>> equation 25.40 to 25.40d



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Systems of Equations

Many practical problems in engineering andscience require the solution of a system ofordinary differential equations rather than asingle equation:

),,,,(

),,,,(

),,,,(

21

2122

2111

nnn

n

n

yyyxfdx

dy

yyyxfdx

dy

yyyxfdxdy

Try out Example 25.9 & 25.10


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Multi-step method

The familiar simple second-order methodused is non-self-starting Heun method:


>>> Figure 26.5


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Boundary-Value andEigenvalue Problems

An ODE is accompanied by 2 supportiveconditions. These conditions are used toevaluate the integral that result during thesolution of the equation.

There are:a)Initial-value problem.

b)Boundary-value problem.


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Conditions are specifiedat the same value of the

independent variable,then we have an initial-value problem.

Conditions are specifiedat different value of theindependent variable,

then we have anboundary-valueproblem.


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Given a noninsulated uniform rod positionedbetween bodies of constant but differenttemperature. Where T1Ta.

Boundary-valueProblems


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200)(

40)0(

01.0

10

20

0)(

2

1

2

2

2

TLT

TT

mh

mL

T

TThdx

Td

a

a

(Boundary Conditions)

(Heat transfer coefficient)

If the rod is not insulated along its length andthe system is at a steady-state, heat balance;

(Temperature of surrounding)


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Finite-Difference Methods

Finite-divided differencesare substitute forthe derivativesin the original equation.

2

11

2

2

2

2

2

0)(

x

TTT

dx

Td

TThdx

Td

iii

a


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aiii

ai

iii

TxhTTxhT

TTh

x

TTT

2

1

2

1

2

11

2

02

This equation is applied for each of theinterior nodes of the rod. The first and lastnodes,Ti-1and Ti+1, respectively, resultingset of linear algebraic equation.

Try out Example 27.3 and Problem 27.3


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Eigenvalue Problems

Eigenvalue problems are special class ofboundary-value problems involving vibrations,elasticity and oscillation systems.

0 XIA

Eigenvalue problems are typically solve bythe general form of linear algebraic equation;

where is an unknown parameter calledeigenvalue.


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0)(

0)(0)(

2211

2222121

1212111

nnnnn

nn

nn

xaxaxa

xaxaxa

xaxaxa


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Tutorial

Problem 25.11

Problem 27.28(b)

Problem 28.10

Chapter 7 Enotes

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