Initial Value Problemsfor ODE'sTEupWe want numericalmethods for

solving ODES of the form 3

IF f toy

Iastebwite.gey3T

i e we want numerical methods to

approxiethe solution yLt of

Moregenerally3 gnth order ODE

andSystem of ODES

One basic idea we will follow is toapproximate the solin at certainpts Ct Ctm

and use interpolation to get thefunction jangleBefouwedineinweneedsomembackgroundtheorgs

Definm Ht g is Lietz in y

i

Exampleme Hey Hy D EDxE3Bis Lipschitz bee

I fl t y Ht g I Itt Ily I YallE 21 y y 1

Lipschitz Inst TB

VI E fo D

wIn It

set

ic

Sufficient cand for a ft tobe Lipschitz

VPExistenc.vn EenessofSotnsiTheoremIo Existence

i

a a

y f singglo 3

has a sokn on te E bi

Solinmi The function f singis conti's on R g i

ytsf.geThis is true for all 4 0

andffgyz.pefLtiyXEECdtD2sguTfo

Chooseµ themlaterso the IVP has a Solh for

Ittf mink y Pick 2 1f 4

IVP has a Solh for ITI ElBy Theorem 1

y

Theorem 2 uniqueness2

If f andy

are cont's in

i

has a unique Solin for

It tote mink n

MEE.gg fH a

Note that the interval maybesmaller than the base ofthe rectangle TLHere is a different theorem

Theoremge Let D a b x R andlet f be continuous on D If

has a unique Solin ylt for teca.bzExamp1emo Use the theorem to show

that F a unique solution toYEE It t silty 0 52

yesSolum The function 8ft g Htsityis continuous on 6,2 xDhas f yf costly l E 4 on teCory

f is Lipschitz with const 4

so by the theorem the IVP has a

unique sofa

Well PosednessMtn

The JVP oo daff fly t EEG b

year 2is well posed if

i

also has a unique solution2 Ct with IZLE YG CLE

V continuous Sct with 1843ICEECE

C La b

Why well pose dresswe

Modeling errorsRound off errorsMeasurement errors

theorem If f is cont on Dwhere D La b xD and if

isis well posed

Euler's Methodmmmm

IV P e deft fl t g EE Casb

yeas L

y

Pick Ntl equispaced pointsto e o Ev C mesh points

withte at Ci Dh for h b

Tstepsizeand note that ectitia

ytti d yttisthytt.at fuzzy

I d b bWi ti Wi h f ti Wi approx

tE

Difference Eq'm

Now we have approx values offat Eo stir so we can

interpolate to find an approx

gig

Example y y Eti te Lo DY o 0.5

Pick he 1 use Euler's methodsto _o t e l Ez 2

Wo O S

w Wot h f to Wo015 t l o 8 02 1 2

Wz W t hf ti Wi2 t Ix 2 12 11 4

Error Bounds

Theory If f is Lipschitz continuouson D Casts xR with const Land if

i w

then lyctis we.ch Ce4ti a2

Nighecorder ayloemethodinMr

Local Truncation error2

3

In Euler's method we computed

wit Wi t h fCti Wi

MoU we maybe an

iteration of the form

ihhTwith0tiwDEweFfose.Thelocal truncation error can be

defiedpredicted value at tixylti DJFI.IT utESi

Titi 6 Fit fyi hottie

Example s Euler's method has

C it h Yin Zi h8Hi yytti

HE esest titis

Ochix

This suggests improvingupon Euler'smethod by using higher orderTaylor approximations

gait D y Lte 1 E h y ti

Ln 11 y si

g fE Egin fee Ct ykD

D

Git yCte t tf f ti yLtiD

ih f Gi yesDCtoget our numericalmeth

If we ignore the error term we have

Taylormethodsofordnnof.no

IIwi hTMti w

fCti wD t Ez f Cti w En ft Ctiw

Euler Taylor of order I

Exampling DevineTaylor's method oforder 2 with h 1 for the IVP

y y Eti tear

Y o Oo 8

Solin we have flt g g Etland we need to computefifty adz yet Eti

alwaysgilt 2Tremember that

Y is a of TEDt y Eti 2T

So Tatti wif fltiswD hzftti.io

Wi Eiti the Wi Ei 2 titi4th Wi titi Lte

no as iWiti Wi t the Wi Ex Etf

theorem Taylor's method of order n

hfron.hn nFca9ffi

Advantages high order local

truncation error

Disadvantage Must computeevaluate derivatives

of 8 Lt g