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ORDINARY DIFFERENTIAL EQUATIONS

ENGR 351

Numerical Methods for Engineers

Southern Illinois University Carbondale

College of Engineering

Dr. L.R. ChevalierDr. B.A. DeVantier

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Ordinary DifferentialEquationswhere to use them

The dissolution (solubilization) of a contaminant into

groundwater is governed by the equation:

where kl is a lumped mass transfer coefficient and Csis the maximum solubility of the contaminant into the

water (a constant). Given C(0)=2 mg/L, Cs= 500

mg/L and kl= 0.1 day-1, estimate C(0.5) and C(1.0)using a numerical method for ODEs.

CCkdtdC

sl

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A mass balance for a chemical in a completely mixed

reactor can be written as:

where Vis the volume (10 m3), cis concentration (g/m3), Fis the feed rate (200 g/min), Qis the flow rate (1 m3/min),and kis reaction rate (0.1 m3/g/min). If c(0)=0, solve theODE forc(0.5) and c(1.0)

2

kVcQcFdt

dcV

Ordinary DifferentialEquationswhere to use them

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Before coming to an exam Friday afternoon, Mr.Bringer forgot to place 24 cans of a refreshing

beverage in the refrigerator. His guest are arriving in5 minutes. So, of course he puts the beverage in therefrigerator immediately. The cans are initially at75, and the refrigerator is at a constant temperatureof 40.

Ordinary DifferentialEquationswhere to use them

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The rate of cooling is proportional to the difference inthe temperature between the beverage and thesurrounding air, as expressed by the following equation

with k = 0.1/min.

Use a numerical method to determine the temperatureof the beverage after 5 minutes and 10 minutes.

airTTkdt

dT

Ordinary DifferentialEquationswhere to use them

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Ordinary Differential Equations

A differential equation defines a relationshipbetween an unknown function and one or

more of its derivatives Physical problems using differential equations

electrical circuits

heat transfer

motion

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Ordinary Differential Equations

The derivatives are of the dependentvariable with respect to the

independent variable First order differential equation with y

as the dependent variable and x as the

independent variable would be: dy/dx = f(x,y)

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Ordinary Differential Equations

A second order differential equationwould have the form:

d y

dxf x y

dy

dx

2

2

, ,

}

does not necessarily have to include

all of these variables

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Ordinary Differential Equations

An ordinary differential equation is onewith a single independent variable.

Thus, the previous two equations areordinary differential equations

The following is not:

dy

dxf x x y

1

1 2 , ,

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Ordinary Differential Equations

The analytical solution of ordinarydifferential equation as well as partial

differential equations is called theclosed form solution

This solution requires that the constants

of integration be evaluated usingprescribed values of the independentvariable(s).

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Ordinary Differential Equations

An ordinary differential equation of order nrequires that nconditions be specified.

Boundary conditions Initial conditions

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Ordinary Differential Equations

An ordinary differential equation of order nrequires that nconditions be specified.

Boundary conditions Initial conditions

consider this beam where thedeflection is zero at the boundariesx= 0 and x = LThese are boundary conditions

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consider this beam where thedeflection is zero at the boundaries

x= 0 and x = LThese are boundary conditions

a

yo

P

In some cases, the specific behavior of a system(s)

is known at a particular time. Consider how the deflectionof a beam at x = ais shown at time t=0 to be equal to yo.Being interested in the response fort> 0, this is calledthe initial condition.

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Ordinary Differential Equations

At best, only a few differentialequations can be solved analytically in a

closed form. Solutions of most practical engineering

problems involving differential

equations require the use of numericalmethods.

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Scope of Lectures on ODE

One Step Methods

Eulers Method

Heuns Method Improved Polygon

Runge Kutta

Systems of ODEAdaptive step size control

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Boundary value problems

Case studies

Scope of Lectures on ODE

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Specific Study Objectives

Understand the visual representation ofEulers, Heuns and the improved polygonmethods.

Understand the difference between local andglobal truncation errors

Know the general form of the Runge-Kuttamethods.

Understand the derivation of the second-order RK method and how it relates to theTaylor series expansion.

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Specific Study Objectives

Realize that there are an infinite number ofpossible versions for second- and higher-order RK methods

Know how to apply any of the RK methods tosystems of equations

Understand the difference between initial

value and boundary value problems

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Review of Analytical Solution

dy

dxx

dy x dx

yx

C

4

4

4

3

2

2

3

At this point lets consider

initial conditions.

y(0)=1

and

y(0)=2

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yx

C

for y

C

then C

for y

C

and C

4

3

0 1

14 0

31

0 2

2

4 0

3

2

3

3

3

What we see are differentvalues ofCfor the twodifferent initial conditions.

The resulting equations

are:

y

x

yx

4

3 1

4

32

3

3

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0

4

8

1 2

1 6

0 0.5 1 1 .5 2 2 .5x

y

y (0)=1

y (0)=2

y (0)=3

y (0)=4

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One Step Methods

Focus is on solving ODE in the form

dydx

f x y

y y hi i

,

1

y

x

slope = yi

yi+1

h

This is the same as saying:new value = old value + slope x step size

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Eulers Method

The first derivative provides a directestimate of the slope at xi

The equation is applied iteratively, orone step at a time, over small distancein order to reduce the error

Hence this is often referred to as EulersOne-Step Method

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Example

24x

dx

dy

For the initial condition y(1)=1, determine yforh= 0.1 analytically and using Eulers

method given:

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Error Analysis of EulersMethod

Truncation error- caused by the nature ofthe techniques employed to approximatevalues ofy

local truncation error (from Taylor Series) propagated truncation error

sum of the two = global truncation error

Round off error- caused by the limited

number of significant digits that can beretained by a computer or calculator

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....end of example

Example

0

2

4

6

8

1 0

1 2

0 0.5 1 1 .5 2 2 .5

x

y

A n a ly t ica l

Solution

Numerical

Solution

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Higher Order Taylor SeriesMethods

This is simple enough to implement withpolynomials

Not so trivial with more complicated ODE

In particular, ODE that are functions of both

dependent and independent variables requirechain-rule differentiation

Alternative one-step methods are needed

y y f x y hf x y

hi i i ii i

12

2,

' ,

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Modification of EulersMethodsA fundamental error in Eulers method

is that the derivative at the beginning ofthe interval is assumed to apply across

the entire interval Two simple modifications will be

demonstrated

These modification actually belong to alarger class of solution techniques calledRunge-Kutta which we will explorelater.

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Heuns Method

Determine the derivative for the interval the initial point

end point Use the average to obtain an improved

estimate of the slope for the entireinterval

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y

xi xi+1

Use this average slope

to predict yi+1

h

yxfyxfyy iiiiii

2

,,11

1

{

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y

xi xi+1

y

x

xi xi+1

hyxfyxfyy iiiiii2

,, 111

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y

x

xi xi+1

h

yxfyxfyy iiiiii

2

,,11

1

hyy ii 1

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Improved Polygon Method

Another modification of Eulers Method

Uses Eulers to predict a value of y at

the midpoint of the interval This predicted value is used to estimate

the slope at the midpoint

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We then assume that this slope represents avalid approximation of the average slope forthe entire interval

Use this slope to extrapolate linearly from xito xi+1 using Eulers algorithm

Improved Polygon Method

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Both Heuns and the Improved Polygon

Method have been introduced graphically.

However, the algorithms used are not as

straight forward as they can be.

Lets review the Runge-Kutta Methods.

Choices in values of variable will give us

these methods and more. It is recommendthat you use this algorithm on your homework

and/or programming assignments.

Runge-Kutta Methods

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Runge-Kutta Methods

RK methods achieve the accuracy of a Taylorseries approach without requiring thecalculation of a higher derivative

Many variations exist but all can be cast inthe generalized form:

y y x y h hi i i i 1 , ,

{

is called the incremental function

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, Incremental Function

can be interpreted as a representativeslope over the interval

a k a k a k

where the a s are constant and the k s are

k f x y

k f x p h y q k h

k f x p h y q k h q k h

k f x p h y q k h q k h q k h

n n

i i

i i

i i

n i n i n n n n n

1 1 2 2

1

2 1 11 1

3 2 21 1 22 2

1 1 1 1 2 2 1 1 1

' ' :

,

,

,

,, , ,

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a k a k a k

where the a s are constant and the k s are

k f x yk f x p h y q k h

k f x p h y q k h q k h

k f x p h y q k h q k h q k h

n n

i i

i i

i i

n i n i n n n n n

1 1 2 2

1

2 1 11 1

3 2 21 1 22 2

1 1 1 1 2 2 1 1 1

' ' :

,

,

,

,, , ,

NOTE:

ks are recurrence relationships,that is k1 appears in the equation for k2

which appears in the equation for k3This recurrence makes RK methods efficient for

computer calculations

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Second Order RK Methods

y y a k a k h

where

k f x y

k f x p h y q k h

i i

i i

i i

1 1 1 2 2

1

2 1 11 1

,

,

a k a k a k

where the a s are constant and the k s are

k f x y

k f x p h y q k h

k f x p h y q k h q k h

k f x p h y q k h q k h q k h

n n

i i

i i

i i

n i n i n n n n n

1 1 2 2

1

2 1 11 1

3 2 21 1 22 2

1 1 1 1 2 2 1 1 1

' ' :

,

,

,

,, , ,

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f x yf

x

f

y

dy

dx

substituting

y y f x y hf

x

f

y

dy

dx

h

i i

i i i i

' ,

,

1

2

2

Now, f(xi , yi ) must be determined by the

chain rule for differentiation

The basic strategy underlying Runge-Kutta methodsis to use algebraic manipulations to solve for values

ofa1, a2, p1 and q11

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a a

a p

a q

1 2

2 1

2 11

1

1

2

1

2

Because we have three equations with four unknowns,

we must assume a value of one of the unknowns.

Suppose we specify a value for a2.

What would the equations be?

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a a

p q a

1 2

1 11

2

1

1

2

Because we can choose an infinite number of values

for a2 there are an infinite number of second orderRK methods.

Every solution would yield exactly the same result

if the solution to the ODE were quadratic, linear or aconstant.

Lets review three of the most commonly used and

preferred versions.

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y y a k a k h

where

k f x y

k f x p h y q k ha a

a p

a q

i i

i i

i i

1 1 1 2 2

1

2 1 11 1

1 2

2 1

2 11

1

1

2

1

2

,

,

Consider the following:

Case 1: a2 = 1/2

Case 2: a2 = 1

These two methods

have been previously

studied.

What are they?

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a a

a p

a q

p qa

y y k k h

where

k f x y

k f x h y k h

i i

i i

i i

1 2

2 1

2 11

1 11

2

1 1 2

1

2 1

1 1 1 2 1 2

1

2

1

2

1

21

1

2

1

2

/ /

,

,

Case 1: a2 = 1/2

This is Heuns Method with

a single corrector.

Note that k1 is the slope at

the beginning of the interval

and k2 is the slope at theend of the interval.

y y a k a k h

wherek f x y

k f x p h y q k h

i i

i i

i i

1 1 1 2 2

1

2 1 11 1

,

,

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a a

a p

a q

p q a

y y k h

where

k f x y

k f x h y k h

i i

i i

i i

1 2

2 1

2 11

1 112

1 2

1

2 1

1 1 1 0

1

2

1

2

1

2

1

2

1

2

1

2

,

,

y y a k a k h

where

k f x y

k f x p h y q k h

i i

i i

i i

1 1 1 2 2

1

2 1 11 1

,

,

Case 2: a2 = 1

This is the ImprovedPolygon Method.

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Ralstons Method

Ralston (1962) and Ralston and Rabinowitiz (1978)

determined that choosing a2 = 2/3 provides a minimum

bound on the truncation error for the second order RK

algorithms.

This results in a1 = 1/3 and p1 = q11 = 3/4

y y k k h

wherek f x y

k f x h y k h

i i

i i

i i

1 1 2

1

2 1

1

3

2

3

3

4

3

4

,

,

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Example

1.0

11..11..

42

hsizestep

yeixatyCI

yxdx

dy Evaluate the following

ODE using Heuns

Methods

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Third Order Runge-Kutta Methods

Derivation is similar to the one for the second-order

Results in six equations and eight unknowns.

One common version results in the following

y y k k k h

where

k f x y

k f x h y k h

k f x h y hk hk

i i

i i

i i

i i

1 1 2 3

1

2 1

3 1 2

16

4

1

2

1

2

2

,

,

,

Note the third term

NOTE: if the derivative is a function of x only, this reduces to Simpsons 1/3 Rule

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Fourth Order Runge Kutta

The most popular

The following is sometimes called theclassical fourth-order RK method

y y k k k k h

where

k f x y

k f x h y k h

k f x h y hk

k f x h y hk

i i

i i

i i

i i

i i

1 1 2 3 4

1

2 1

3 2

4 3

16

2 2

1

2

1

2

1

2

1

2

,

,

,

,

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Note that for ODE that are a function of x alone thatthis is also the equivalent of Simpsons 1/3 Rule

34

23

12

1

43211

,

2

1,

2

1

2

1,

2

1,

226

1

hkyhxfk

hkyhxfk

hkyhxfk

yxfk

where

hkkkkyy

ii

ii

ii

ii

ii

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Example

Use 4th Order RK to solve the following differential equation:

dy

dx

xy

xI C y 1

1 12 . .

using an interval ofh= 0.1

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Higher Order RK Methods

When more accurate results arerequired, Buchers (1964) fifth order RK

method is recommended There is a similarity to Booles Rule

The gain in accuracy is offset by added

computational effort and complexity

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Systems of Equations

Many practical problems in engineering andscience require the solution of a system ofsimultaneous differential equations

dydx

f x y y y

dy

dxf x y y y

dy

dxf x y y y

n

n

n

n n

1

1 1 2

2

2 1 2

1 2

, , , ,

, , , ,

, , , ,

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Solution requires ninitial conditions

All the methods for single equations can be

used The procedure involves applying the one-step

technique for every equation at each step

before proceeding to the next step

dy

dxf x y y y

dy

dx

f x y y y

dy

dxf x y y y

n

n

n

n n

1

1 1 2

2

2 1 2

1 2

, , , ,

, , , ,

, , , ,

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Boundary Value Problems

Recall that the solution to an nthorder ODErequires nconditions

If all the conditions are specified at the samevalue of the independent variable, then weare dealing with an initial value problem

Problems so far have been devoted to this

type of problem

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Boundary Value Problems

In contrast, we may also have conditions adifferent value of the independent variable.

These are often specified at the extreme

point or boundaries of as system andcustomarily referred to as boundary valueproblems

To approaches to the solution

shooting method

finite difference approach

G l M th d f B d

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General Methods for BoundaryValue Problems

The conservation of heat can be used to develop a heat

balance for a long, thin rod. If the rod is not insulated

along its length and the system is at steady state. The

equation that results is:

d T

dxh T Ta

2

20 '

T1 T2

Ta

Ta

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T1 T2

Ta

Ta

d T

dx

h T Ta

2

20 '

Clearly this second order

ODE needs 2 conditions.

This can be satisfied by

knowing the temperatureat the boundaries,

i.e. T1 and T2

T(0) = T1

T(L) = T2

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Shooting Method

d T

dxh T T

dTdx

z

dz

dxh T T

a

a

2

20

'

'

Given:

We need an initial value

of z.

For the shooting method, guess

an initial value.

Guessing z(0) = 10

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dz

dxh T Ta '

Using a fourth-order RK method with a step size

of 2, T(10) = 168.38

This differs from the BC T(10) = 200

Making another guess, z(0) = 20

T(10) = 285.90

Because the original ODE is linear, the estimates

of z(0) are linearly related.

Guessing z(0) = 10

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Using a linear interpolation formula between the values

of z(0), determine a new value of z(0)

Recall:

first estimate z(0) = 10 T(20) = 168.38

second estimate z(0)=20 T(20) = 285.90

What is z(0) that would give us T(20)=200?

150

200

250

300

0 5 10 15 20 25

z(0)

T(20)

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z 0 10

20 10

285 90 168 38200 168 38 12 69

. .. .

150

200

250

300

0 5 10 15 20 25

z(0)

T(20)

d T

dxh T T

dT

dx z

dz

dxh T T

a

a

2

20

'

'

We can now use

this to solve the firstorder ODE

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For nonlinear boundary value problems, linear

interpolation will not necessarily result in an accurate

estimation. One alternative is to apply three

applications of the shooting method and use quadratic

interpolation..

0

5 0

1 00

1 5 0

2 00

2 5 0

0 5 1 0

distance (m )

T

A n a ly t ic a l

Solution

Shooting

Method

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Finite Difference Methods

The finite divided difference approximation for

the 2nd derivative can be substituted into the

governing equation.

d T

dx

T T T

x

d T

dxh T T

T T T

xh T T

i i i

a

i i ia i

2

2

1 1

2

2

2

1 1

2

2

0

20

'

'

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T T T

x

h T T

T h x T T h x T

i i ia i

i i i a

1 1

2

1

2

1

2

20

2

'

' '

Collect terms

We can now apply this equation to each interior nodeon the rod.

Divide the rod into a grid, and consider a node to be

at each division. i.e.. x = 2m

T1 T2L = 10 m

x = 2 m

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aiii TxhTTxhT2

1

2

1''2

T(0) T(10)L = 10 m

x = 2 m

Consider the previous problem:

L = 10 mTa = 20

T(0) = 40

T(10) = 200

h = 0.01

We need to solve for the

temperature at the interior

nodes (4 unknowns).

Apply the governingequation at these nodes (4

equations).

What is the matrix?

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aiii TxhTTxhT2

1

2

1''2

T(0) T(10)

x=0 2 4 6 8 10

i=0 1 2 3 4 5

Notice the labeling for numbering xand i

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aiii TxhTTxhT2

1

2

1''2

T(0) T(10)

x=0 2 4 6 8 10

i=0 1 2 3 4 5

40 200

Note also that the dependent values are

known at the boundaries (hence the termboundary value problem)

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aiii TxhTTxhT2

1

2

1''2

T(0) T(10)

x=0 2 4 6 8 10

i=0 1 2 3 4 5

40 200

Apply the governing equation at node 1

8.4004.2

8.004.240

''2

21

21

2

21

2

0

TT

TT

TxhTTxhT a

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aiii TxhTTxhT2

1

2

1''2

T(0) T(10)

x=0 2 4 6 8 10

i=0 1 2 3 4 5

40 200

We get a similar equation at node 3

8.004.2

''2

432

2

43

2

2

TTT

TxhTTxhT a

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aiii TxhTTxhT2

1

2

1''2

T(0) T(10)

x=0 2 4 6 8 10

i=0 1 2 3 4 5

40 200

8.20004.2

8.020004.2

''2

43

43

2

54

2

3

TT

TT

TxhTTxhT a

At node 4, we consider the

boundary at the right.

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For the four interior nodes, we get the following

4 x 4 matrix

48.15954.12478.9397.65

8.200

8.0

8.0

8.40

04.2100

104.210

0104.21

00104.2

4

3

2

1

TT

T

T

T

T

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0

5 0

1 00

1 5 0

2 00

2 5 0

0 5 1 0

distan ce (m )

T

A n a ly t ica l

Solution

ShootingMethod

Finite

Difference

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Example

Consider the previous example, but

with x=1. What is the matrix?

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Specific Study Objectives

Understand the visual representation ofEulers, Heuns and the improved polygonmethods.

Understand the difference between local andglobal truncation errors

Know the general form of the Runge-Kuttamethods.

Understand the derivation of the second-order RK method and how it relates to theTaylor series expansion.

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Specific Study Objectives

Realize that there are an infinite number ofpossible versions for second- and higher-order RK methods

Know how to apply any of the RK methods tosystems of equations

Understand the difference between initialvalue and boundary value problems

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end of lecture on ODE