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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
Try out Example 25.5
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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
Try out Example 25.7
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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:
Try out Example 26.2
>>> 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
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