lecture11 - initial value problem eulers method

8/9/2019 Lecture11 - Initial Value Problem Eulers Method

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Numerical Solutions of Ordinary

Differential Equations

Lecture 11:

Initial Value Problem: Eulers Method

MTH2212 Computational Methods and Statistics


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Dr. M. HrairiDr. M. Hrairi MTH2212MTH2212 -- Computational Methods and StatisticsComputational Methods and Statistics 22

Objectives

Introduction

Eulers Method

Convergence analysis

Error analysis

Error estimate


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Introduction

Differential equations: equations composed of an unknown

function and its derivatives.

Differential equations play a fundamental role in engineeringbecause many physical phenomena are best formulated

mathematically in terms of their rate of change.

)()( tvmcg

dttdv

! v- dependent variable

t- independent variable


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Dr. M. HrairiDr. M. Hrairi MTHMTH22122212 -- Computational Methods and StatisticsComputational Methods and Statistics 44

Introduction

Fundamental laws of physics, mechanics, electricity andthermodynamics are written in terms of the rate of changeof variables (t = time and x = position):

Newtons second law of motion

Fouriers heat law

Lapalce equation (steady state)

Heat conduction

etc.

mF

dd !

dx

dTkq '

02

2

2

2

x

x

x

x

y

T

x

T

2

2

'

x

Tk

t

T

x

x!

x

x


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Introduction

One independent variable Ordinary Differential Equation

(ODE)

Two or more independent variables Partial DifferentialEquation (PDE)

Order of a differential equation is determined by the highest

derivative.

Higher order equations can be reduced to a system of first

order equations, by redefining a variable.


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Introduction

An ODE is accompanied by auxiliary conditions. Theseconditions are used to evaluate the integral that resultduring the solution of the equation. An nth order equationrequires n conditions.

If all conditions are specified at the same value of theindependent variable, then we have an initial-value problem.

If the conditions are specified at different values of theindependent variable, usually at extreme points orboundaries of a system, then we have a boundary-value

problem.


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Eulers Method

Solve ordinary

differential equations

of the form

With initial condition

The estimate of thesolution is

),( yxfdx

dy!

ii yxy !)(

hyy ii J1


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is the

estimate of slope at xi

is thedifferential equation

evaluated at xi and yi

The estimate is given by

This is Eulers method

(Euler-Cauchy, Point-slope)

),( ii yxf!J

),( ii yxf

hyxfyy iiii ),(1 !

Eulers Method


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Error Analysis for Eulers Method

The numerical solution ofODE involves 2 types of error:

1. Round-off errors: this caused by limited numbers of

significant digits that can be retained by a computer.

2. Truncation errors:a) Local truncation error: due to the application of the

method over a single step.

b) Propagated truncation error: due to approximation

produced during previous steps.

Total or global truncation error = local truncation error +

propagated truncation error.


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Error Estimate for Eulers Method

The general form of the differential equation being integrated:

Taylors series expansion of y about a starting value (xi

,yi

):

(1) into (2) gives

Subtract Eulers method equation from (3) gives the truncationerror

),(' yxfydx

dy!!

)(!

...!2

1)(

2''

'

1

!

nnn

iiiii hOh

n

yh

yhyyy

)(!

),(

...!

),(

),(

)('

!

nnii

n

ii

iiii n

yxfyxf

yxfyy

)(!

),(...

!2

),(1

)1(

2

'

!nnii

nii

t hOhn

yxfh

yxfE

(1)

(2)

(3)


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Example 1

1. Use Eulers method to integrate numerically the ODE:

from x = 0 to x = 4 with a step size of 0.5 and an initialcondition at x = 0 is y = 1?

2. Estimate and tabulate the errors

5.82012223

! xxxdx

dy


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Example 1 - Solution

1. Using Eulers method

Evaluate the first prediction y(0.5)

Evaluate the second prediction y(1)

hyxfyy iiii ),(! 1)0( !y 5.0!h

5.258.5)(0.5)(1)()5.0( !!y

5.85.8)0(20)0(12)0(2)1,0( 23 !!f

0.5)()1,0()0()5.0( fyy

875.5)( .5)(5.25)()( y

!! 5.8)5.(2)5.(2)5.(2),(23f

.5)()25.5,5.()5.()( fyy !


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Evaluate the third prediction

5.125-1.5)(0.5)(5.875)5.1( !!y

0.5)()875.5,()()5.( fyy x yEuler

0.0 1.000

0.5 5.2501.0 5.875

1.5 5.125

2.0 4.500

2.5 4.750

3.0 5.875

3.5 7.125

4.0 7.000


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2. Estimate the truncation error

For the first step

4)3(

3'''

!4

)(

!3

)(

!

)(h

yxfh

yxfh

yxfE iiiiiit !

12)1,012),

2)1,0212),

20)1,02026),

)3()3(

''''

'2'

!!

!!

!!

fyxf

fxyxf

fxxyxf

ii

ii

ii

03 5.)5.0(4

)5.0(6

4)5.0(

0 43

t


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x yEuler Et

0.00000 1.00000 0.00000

0.50000 5.25000 -2.03125

1.00000 5.87500 -2.87500

1.50000 5.12500 -2.90625

2.00000 4.50000 -2.50000

2.50000 4.75000 -2.03125

3.00000 5.87500 -1.87500

3.50000 7.12500 -2.40625

4.00000 7.00000 -4.00000


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Remarks about Eulers Method

The Taylor series provides only an estimate of the localtruncation error. It does not give a measure of the globaltruncation error.

The global truncation error is O(h), that is, it is proportionalto the step size.

The error can be reduced by decreasing the step size.

If the underlying function, y(x), is linear, the method provideerror-free predictions. This is because for a straight line thesecond derivative would be zero.


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Effect of step size - example 1

lecture11 - initial value problem eulers method

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