Week10&11 Differential Equations

8/13/2019 Week10&11 Differential Equations

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WEEK 10 & 11

Ordinary Differential EquationsEulersmethodsImprovements of Eulersmethods

Runge-Kutta methods

Adaptive Runge-Kutta methods 2


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At the end of this topic, the students will be able:

To identify and apply the methods outlined to

solve ordinary differential equations (ODE)

LESSON OUTCOMES


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DIFFERENTIAL EQUATION

Equations which are composed of an unknown functionand its derivatives are called differential equations.

When a function involves one independent variable, the

equation is called an ordinary differential equation (or

ODE). A partial differential equation (or PDE) involves twoor more independent variables.

Differential equations are also classified as to their order.

A first order equation includes a first derivative as

its highest derivative.

A second order equation includes a second

derivative.


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Differential equations play a fundamental role in

engineering because many physical phenomena are

best formulated mathematically in terms of their rateof change.

v ~ dependent variable

t ~ independent variable

vm

cg

dt

dv


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This chapter is devoted to solving ordinary differential

equations of the form

The estimated slope is used to predict the new value yi+1

using the old value yi over a distance h.

The procedure can be implemented step by step to compute aset of new values ysand hence, trace out the trajectory of the

solution.

New value = old value + slope x step size

hyy ii 1

),( yxfdx

dy


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The first derivative provides a direct estimate of the

slope atxiwheref(xi, yi)is the differential equation evaluated atxiand yi.

This estimate can be substituted into the equation:

A new value of yis predicted using the slope (equal to the first

derivative at the original value of x) to extrapolate linearly

over the step size h.

),( ii yxf

hyxfyy iiii ),(1

Eulers Method


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Example

Use Eulersmethod to numerically integrate ODE:

From x = 0 to x = 2 with a step size of 0.5. The

initial condition at x = 0 is y = 1.The exact solution is:

y =0.5x4+ 4x310x2+ 8.5x + 1

5.820122 23 xxxdx

dy


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x yEuler ytrue t (%)

0 1.000 1.000 -

0.5 5.250 3.219 -63.11.0 5.875 3.000 -95.8

1.5 5.125 2.219 -131

2.0 4.500 2.000 -125


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Error Analysis for Eulers Method

Numerical solutions of ODEs involves two types of

error: Truncation error (error cause by the nature of thetechniques employed to approximate values of y)

Local truncation error (result from application of the method in

question over a single step)

Propagated truncation error (result from the approximationsproduced during previous steps)

The sum of the two is thetotal or global truncation error

Round-off errors (caused by limited numbers of significant

figures)

)(

!2

),(

2

2

hOE

hyxf

E

a

iia


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If the function has continuous derivatives, TS can be

expanded about a starting point (xi, yi),

and Rnis the remainder defined as

Now, if we substitute the derivative function f(xi,yi) then

obtain,

Comparison of , we conclude that the

truncation error is due to the remaining terms in the TSE.

hyxfyy iiii ),(1


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Therefore, the true local truncation error is given by

For sufficiently small h, higher-order terms would

decrease appreciably.

The approximate local truncation error can then be

written as


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The Taylor series provides a means of quantifying theerror in Eulersmethod. However;

The Taylor series provides only an estimate of the localtruncation error-that is, the error created during a singlestep of the method.

In actual problems, the functions are more complicated

than simple polynomials. Consequently, the derivativesneeded to evaluate the Taylor series expansion would notalways be easy to obtain.

In conclusion,

the error can be reduced by reducing the step size If the solution of the differential equation is linear, the

method will provide error free predictions because for astraight line, the 2ndderivative would be zero.


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a) Comparison of two numerical solutions withEulersmethod using step size of 0.5 and 0.25

b) Comparison of true and estimated local truncation

error for the case where the step size is 0.5.


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ExampleEstimate the true and the approximate local truncation

error associated with the Eulersmethod in the first step tosolve the following initial-value problem

from x = 0 to x = 3 with h = 0.5.

252 23 xxxdx

dy


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Improvements of Eulers method

A fundamental source of error in Eulers method isthat the derivative at the beginning of the interval is

assumed to apply across the entire interval.

Two simple modifications are available to circumvent

this shortcoming:

Heunsmethod

The Midpoint (or Improved Polygon) method


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One method to improve the estimate of the slope

involves the determination of two derivatives for theinterval:

At the initial point

At the end point

The two derivatives are then averaged to obtain an

improved estimate of the slope for the entire interval.

hyxfyxf

yy

hyxfyy

iiiiii

iiii

2

),(),(:Corrector

),(:Predictor

0

111

0

1

Heuns Method


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Note that Corrector equation is

in an iterative form. So, we can

iteratively obtain the improved

estimate of yi+1until it converge

to prespecified tolerance.

Note also, however, that the

convergence is not necessarilyto the true solution.

The termination criterion of the

corrector equation is given by

Refer Text Book for explanation


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Comparison of the true solution with a numericalsolution using Eulers and Heuns method


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The slope at the mid-point of the interval is used toextrapolate the solution at the end of the interval.

The value of the unknown function at the mid-point of the

interval is

This value is used to estimate the value of the slope over the

interval,

This slope is then used to extrapolate linearly fromxito xi+1

21

Midpoint (Improved Polygon) Method

hyxfyy iiii ),( 2/12/11


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The Runge-Kutta methods have the accuracy of the

Taylor series approach without having to evaluate

the higher derivatives.

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Runge-Kutta Method

),(

),(

),(

),(

constants'

),,(

11,122,1111

22212133

11112

1

2211

1

hkqhkqhkqyhpxfk

hkqhkqyhpxfk

hkqyhpxfk

yxfk

sa

kakaka

hhyxyy

nnnnninin

ii

ii

ii

nn

iiii

Increment function

ps and qs are constants


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ks are recurrence functions. Because each k is a

functional evaluation, this recurrence makes RK methods

efficient for computer calculations.

Various types of RK methods can be devised by

employing different number of terms in the increment

function as specified by n.

First order RK method with n=1is in fact Eulersmethod.

Once nis chosen, values of as,ps,and qsare evaluated

by setting general equation equal to terms in a Taylor

series expansion. Thus, at least for the lower-older version, the number of

terms, n usually represent the order of approach.


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The second-order version is

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Second-Order Runge-Kutta Methods

),(),(

)(

11112

1

22111

hkqyhpxfkyxfk

where

hkakayy

ii

ii

ii

Values of a1, a2, p1, and q11 are evaluated by setting the

second order equation to Taylor series expansion to thesecond order term.


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2

12

1

1

112

12

21

qa

pa

aa

Three equations to evaluate four unknowns constants are

derived.

Three of the most commonly used methods are:

Huen Method with a Single Corrector (a2=1/2)

The Midpoint Method (a2=1)

RaltsonsMethod (a2=2/3)

A value is assumed for one of the

unknowns to solve for the other three.


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Heun Method with a Single Corrector (a2=1/2)


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The Midpoint Method (a2=1)


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Fourth-Order Runge-Kutta Methods

hkkkkyy ii )22(61 43211


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For an ODE with an abrupt

changing solution, a constantstep size can represent a

serious limitation.

Adaptive Runge-Kutta Method


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Step-Size Control

The strategy is to increase the step size if the error is too small

and decrease it if the error is too large. Press et al. (1992) havesuggested the following criterion to accomplish this:

Dpresent= computed present accuracy

Dnew = desired accuracy

a = a constant power that is equal to 0.2 when step size

increased and 0.25 when step size is decreased

Implementation of adaptive methods requires an estimate of

the local truncation error at each step.

The error estimate can then serve as a basis for either

lengthening or decreasing step size.

a

present

newpresentnew hh

D

D

Week10&11 Differential Equations

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