204 C H A P T E R 5 Numerical Solution ofInitial -Value Problems
5.c.y'=— (y+l)(.y+3),for0</<3,with y(0)=-2;actual solution y(f)=-3-1-2(1+e~2l)~l.
d.y'=(t+2/3)y3-ty,for0<t<2,with y(0)=\;actual solution y(t)=(3-fIt1+6er)",/2.
LeiP{t)bethenumber ofindividuals inapopulation atlime t,measured inyears .Iftheaverage
birth ratebisconstant andtheaverage death ratedisproportional tothesizeofthepopulation (due
toovercrowding ),then thegrowth rateofthepopulation isgiven bythelogistic equation
d-ljp -=bP(t)-k[P(t)}1
dt
where d=kP(t).Suppose F(0)=50,976,b=2.9x10"2,and£=1.4x107.Find thepopulation
after 5years .
5.6Adaptive Techniques
Youmight liketoreview the
Adaptive Quadrature material
beginning onpage 138before
considering ibismaterial .Theappropriate useofvarying stepsizewasseen inSection 4.6toproduce integral approx -
imating methods thatareefficient intheamount ofcomputation required .This might notbe
sufficient tofavor these methods duetotheincreased complication ofapplying them ,but
they have another important feature .Thestep-sizeselection procedure produces anestimate
ofthelocal error thatdoes notrequire theapproximation ofthehigher derivatives ofthe
function .These methods arecalled adaptive because they adapt thenumber andposition
ofthenodes used intheapproximation tokeep thelocal error within aspecified bound .
There isaclose connection between theproblem ofapproximating thevalue ofa
definite integral andthatofapproximating thesolution toaninitial -value problem .Itisnot
surprising ,then,thatthere areadaptive methods forapproximating thesolutions toinitial -
value problems ,andthatthese methods arenotonly efficient butincorporate thecontrol of
error.
Anyone-stepmethod forapproximating thesolution ,y(t),oftheinitial -value problem
y=/(r,y), f o T a <t<b,withy (a)=of
canbeexpressed intheform
w.+i=W i+hi<J>(titwithi),fori=0,1,...,N-1,
forsome function 0.Anideal difference -equation method would have theproperty that
given atolerance e>0,theminimal number ofmesh points would beused toensure
thattheglobal error,|y(/,)— w,|,would notexceed eforanyi=0,1,....N.Having a
minimal number ofmesh points andalsocontrolling theglobal error ofadifference method
is,notsurprisingly ,inconsistent with thepoints being equally spaced intheinterval .Inthis
section weexamine techniques used tocontrol theerror ofadifference -equation method in
anefficient manner bytheappropriate choice ofmesh points .
Although wecannot generally determine theglobal error ofamethod ,there isoften a
close connection between thelocal error andtheglobal error.Ifamethod haslocal error
0(hn+]),then theglobal error ofthemethod is0(hn).Byusing methods ofdiffering order
wecanpredict thelocal error and,using thisprediction ,choose astep size thatwillkeep
theglobal error incheck.
Toillustrate thetechnique ,suppose thatwehave twoapproximation techniques .The
firstisannth-order method obtained from annth-order Taylor method oftheform
y(»i+i)=y«i)+ ,yfo).h)+0(h"+i).
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This method produces approximations
w0=a
w,+i=w,+h<t>(ti,with)fori>0,
which satisfy ,forsomekandallrelevant handi,
Ingeneral ,themethod isgenerated byapplying aRungc -Kutta modification totheTaylor
method ,butthespecific derivation isunimportant .
Thesecond method issimilar butofhigher order .Forexample ,letussuppose itcomes
from an(n4-l)st-order Taylor method oftheform
y(tl+1)=y(t,)+Aftfe,yfe).h)+0(h"+2),
producing approximations
wo=cr
w,+1=vv,+ wnh )for i>0,
which satisfy ,forsome Randallrelevant hand i.
Weassume now thatatthepoint tjwehave
w,=W i=z(/,),
where z(r)isthesolution tothedifferential equation thatdoes notsatisfy theoriginal initial
condition butinstead satisfies thecondition z(r,)=w,.Thetypical difference between y(t)
andz(t)isshown inFigure 5.3.
Figure 5.3
y,
i)-
Global>/error
zCO-1
-w,
aVLocal
error
a U*i+1t
Applying thetwomethods tothedifferential equation with thefixed stepsizehproduces
twoapproximations ,w,-+jandw,-+i,whose differences from y(f,+/i)represent global errors
butwhose differences from +h)represent local errors .
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Now consider
z(ti+h)-wl+1=(w,-+1-wM )+(z(/4+h)-wi+l).
Theterm ontheleftsideofthisequation is0(h n+l),thelocal error ofthenthorder method ,
butthesecond term ontheright sideisO(h n Jr l ),thelocal error ofthe(n+l)storder method .
This implies thatthedominant portion ontheright sidecomes from thefirst term;thatis,
zifi+h)-W|+I%wl+1-wi+i=0(hn~]).
So,aconstant Kexists with
Kh n~{=|z(h+h)-wi+\|%|w,+i-wi+i|#
and Kcanbeapproximated by
„ IVv.+i-wi+i|
A^:hn+\(5.3)
Letusnow return totheglobal error associated with theproblem wereally want to
solve ,
y'=/(f,y),ioxa <t<b,withy (a)=a,
andconsider theadjustment tothestep size needed ifthisglobal error isexpected tobe
bounded bythetolerance e.Using amultiple qoftheoriginal stepsizeimplies thatweneed
toensure that
|y(f,+q h)— Wj+i(using thenew stepsizeqh)\<K(q h)n<e.
This implies ,from Eq.(5.3),thatfortheapproximations wi+\andw*+iusing theoriginal
stepsizehwehave
|Wj+,-WI+l| q"\wj+l—W;+,|Kqh*~ff+l 9hi<
Solving thisinequality forqtells ushow weshould choose thenewstepsize,qh,toensure
thattheglobal error iswithin thebound e:
Q<ehl/n
.|W/+1“ W/+i|
Erwin Fchlbcrg developed this
andother error control techniques
while working fortheNASA
facility inHuntsville ,Alabama
during the1960s.In1969 he
received NASA ’sExceptional
Scientific Achievement Medal for
hiswork .TheRunge -Kutta -Fehlberg Method
One popular technique thatuses this inequality forerror control istheRunge -Kutta
Fehlberg method (see[Fe]).ItusesaRunge -Kutta method oforder 5,
16f 6656 , 28561 , 9f 2f+135*'+12825*3+56430*4"50*5+55*6’
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toestimate thelocal error inaRunge -Kutta method oforder 4,
25, 1408, 2197 , 1,
Wl+'=Wi+2i6k'+X65ki+4mk*-5ks'
where thecoefficient equations are
*i=hf(thWj ),
k2=hf
k3=hf
kA=hf
kS=hf
ki=hf(n+
^.w,+!**),
/ 3/i 3, 9,\
Vi+T’H’'+32il+32*2J’
(\2h+IT’
(
( h 8
V 2’Wi_
271932 , 7200 , 7296 ,
W,+2197*‘2197*2+2197*3) 
, 439, o, 3680 , 845 ,
*+* "'+216*"S*2*
, 3544, 1859 ,*1H-2*2—__._*3+77777*44104) 
-SM -
Anadvantage ofthismethod isthatonly sixevaluations of/arerequired perstep,
whereas arbitrary Runge -Kutta methods oforder 4and5used together would require (see
Table 5.9onpage 188)atleast four evaluations of/forthefourth -order method andan
additional sixforthefifth-order method .
Inthetheory oferror control ,aninitial value ofhattheithstepwas used tofind the
first values ofwi+\and wl+1,which ledtothedetermination ofqforthatstep.Then the
calculations were repeated with thestep sizehreplaced byqh.This procedure requires
twice thenumber offunctional evaluations perstepaswithout error control .Inpractice ,
thevalue ofqtobeused ischosen somewhat differently inorder tomake theincreased
functional -evaluation costworthwhile .Thevalue ofqdetermined attheithstepisused for
twopurposes :
 When q<1,toreject theinitial choice ofhattheithstep andrepeat thecalculations
using qh,and
 When q>1,toaccept thecomputed value attheithstep using thestep sizehandto
increase thestepsizetoqhforthe(i+l)ststep.
Because ofthepenalty interms offunctional evaluations that must bepaid ifmany
steps arerepeated ,qtends tobechosen conservatively .Infact,fortheRunge -Kutta -Fehlberg
method withn=4,theusual choice is
q=(«»Y/4»OM(«r.\2|wi+i-wi+\\J \|w,+i-wj+il/
Theprogram RKFVSM 55
implements the
Runge -Kutta-Fehlberg
method .Program RKFVSM 55incorporates atechnique toeliminate large modifications instep
size.This isdone toavoid spending toomuch time with very small step sizes inregions
with irregularities inthederivatives ofy,andtoavoid large stepsizes ,which might result
inskipping sensitive regions nearby .Insome instances thestep-size-increase procedure
isomitted completely andthestep-size-decrease procedure ismodified tobeincorporated
only when needed tobring theerror under control .
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Example 1UsetheRunge -Kutta-Fehlberg method withatolerance TOL=10-5,amaximum stepsize
hmax =0.25,andaminimum step sizehmin=0.01 toapproximate thesolution tothe
initial -value problem
y=y-t2+l,0<t<2,y(0)=0.5,
andcompare theresults with theexact solution y(/)=(/+l)2— 0.5e'.
Solution Wewillwork through thefirststepofthecalculations andthen apply theprogram
RKFVSM 55todetermine theremaining results .The initial condition gives to=0and
wo=0.5.Todetermine wjusing h=0.25,themaximum allowable stepsize ,wecompute
*i=hf(t0,w0)=0.25 (0.5-02+1)=0.375 ;
Jt2=hf
^r0+ w0+ =0.25/^0.25,0.5+
^0.375
^=0.3974609 ;
3, 9\
32*'+32*2) *3=hf(to+^,w0+
0.25/(0.09375 ,0.5+^0.375 +^0.3974609
^=0.4095383 ;
), / 12, 1932 , 7200 , 7296 ,\
*4=*/(«o+Eh,wo+2^*,-^+^*3J
=0.25/(o.2307692 ,0.5+^p0.375-^0.3974609 +^0.4095383
=0.4584971 ;
i ur(> 439, 3680, 845,*5=hf(«,+ft.WO+— -8*2+_*3-— 
( 439 3680 845=0.25/(0.25,0.5+— 0.375-8(0.3974609 )+— 0.4095383 -— 0.4584971 il
=0.4658452 ;
1859 / 1 8 3544k6=h f(t o+-h,wo--ft,+2k2-— k}+^*4-JM
=0.25/(0.125 ,0.5-^;0.375 +2(0.3974609 )-^0.4095383\ 27 2565
1859 11 \+ 0.4584971 0.46584524104 40 J
=0.4204789 .
Thetwoapproximations toy(0.25)arethen found tobe
16, 6656 , 28561 , 9, 2,*'=w o+j35*.+12825*3+56430^-50*5+55**
16 6656 28561 9=0.5+T^O.375+ 0.4095383 +- 0.4584971 - 0.4658452135 12825 56430 50
+^0.4204789
*/ /
=0.9204870 ,
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and
25, 1408 , 2197 , 1,
W,=W0+2l6*1+2565*3+4l04b"5te
25 1408 2197 1=0.5+— 0.375 +^7^0.4095383 +— — 0.4584971 --0.4658452216 2565 4104 5
=0.9204886 .
Theapproximations with h=0.25 ande=10"5give
q=(-~7)'=0.9462100 .\2lvv,-Wily
Since q<1,weneed torecompute thevalues ofwjandwiwith thenewstepsize
h=0.9462100 (0.05 )=0.2365525 .
Recalculating with thisstepsizegives usq=0.9986023 which isagain lessthan1.Redoing
thecalculations with thestepsize
h=(0.9986023 )(0.2365525 )=0.2362221 .
gives avalue ofq=0.9999643 which isstilllessthan one.This requires achange instep
sizeto
h=(0.9999643 )(0.236222 )=0.2362137 .
Forthisstepsizewegetq=1.000002 >1.So,weaccept that,with thestepsize0.2362137 ,
theapproximation w\issufficiently accurate approximation toy(0.2362137 ).
Theresults from program RKFVSM 55areshown inTable 5.13.Notice thatthestep
sizeonthelastlinehasbeen adjusted sothatanapproximation isgiven fory(2).Keep in
mind thattheerrors listed areglobal errors ,andtheRunge -Kutta -Fehlberg method controls
local errors .
Table 5.13
ti yW hiRKF-4
w, ly(f«)-wi\RKF-5
Wi 1
0.2362137 0.8950018 0.2362137 0.8950028 1.0x10'60.8950016 2.0xlO'1
0.4724278 1.3661020 0.2362142 1.3661042 2.2x10"61.3661031 1.1x10"6
0.7147675 1.9185718 0.2423397 1.9185755 3.7x10"61.9185745 2.7x10"6
0.9647675 2.5482226 0.2500000 2.5482282 5.6xlO62.5482272 4.6x10"6
1.2147675 3.2204398 0.2500000 3.2204475 7.7xlO"63.2204469 7.1x10"6
1.4647675 3.9118102 0.2500000 3.9118204 1.02x10“ 53.9118202 1.00 x10"s
1.7147675 4.5922707 0.2500000 4.5922836 1.29 x10“ 54.5922839 1.32 x105
1.9647675 5.2232194 0.2500000 5.2232351 1.57 xlO55.2232362 1.68 x10-5
2.0000000 5.3054720 0.0352325 5.3054883 1.63x10'55.3054883 1.63x10s
Anadaptive Runge -Kutta method isavailable inMATLAB using thecommand ode45.
Tousethiscommand foroursample initial value problem ,wefirst need todefine the
function /(/,y)using anM-filewhich wename odefunction 2.m.
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function YY=odefunction 2(t,y)
YY=y-t“ 2+l
Wethen need tomake thefunction known toMATLAB using
f=Qodefunction 2
Wedefine arange oftvalues forwhich wewant toapproximate they(f)values beginning
attheinitial value a=0andending atthevalue b— 2.Therange oftvalues areplaced in
thestructure called tspan
tspan =[0.00.20.4 0.60.81.01.21.41.61.8 2.0]
andwedefine theinitial value y(0)=0.5with
y0=0.5
Weinvoke thefunction ode45using thefollowing command
[T,YY]=ode45(f,tspan ,yO)
which produces thevalues oftandcorresponding approximations to>(/)inthetwol l x l
arrays
0 0.500000000000000
0.200000000000000 0.829298806102213
0.400000000000000 1.214087852350423
0.600000000000000 1.648940818452941
0.800000000000000 2.127229773242783
1.000000000000000 and YY= 2.640859343211129
1.200000000000000 3.179941816703143
1.400000000000000 3.732400315281681
1.600000000000000 4.283484106130412
1.800000000000000 4.815176603278762
2.000000000000000 5.305472351203669
Variable Step -Size Predictor -Corrector Methods
The Runge -Kutta -Fehlberg method ispopular forerror control because ateach step it
provides ,atlittle additional cost,twoapproximations thatcanbecompared andrelated to
thelocal error.Predictor -corrector techniques always generate twoapproximations ateach
step,sotheyarenatural candidates forerror-control adaptation .
Todemonstrate theprocedure ,weconstruct avariable -step-sizepredictor -corrector
method using theexplicit Adams -Bashforth Four-Step method aspredictor andtheimplicit
Adams -Moulton Three -Step method ascorrector .
TheAdams -Bashforth Four-Step method comes from theequation
yft+i)=ytt)+zr[55/fe,yft))-59/tt_
i,y(fi»i))24
251+37/ft_
2,yfc-j))-9/ft_j,y(f,-3»]+^/5)(A.)*5
forsome in f,+i).Suppose weassume thattheapproximations wo,w j,...,w,are
allexact and,asinthecase oftheone-step methods ,weletzrepresent thesolution tothe
differential equation satisfying theinitial condition z(f,)=w,.Then thedifference between
theactual solution andthepredicted solution w,+itPis
251z(f>+i)-W/+1.P= Z<5)(£;)/>5forsome A;in(r,_
3,r,). (5.4)
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Asimilar analysis oftheAdams -Moulton Three -Step corrector method leads tothelocal
error
19zft+t)-W/41= z(5)(Ai)*5forsome A,in(r,_
2,tM). (5.5)
Toproceed further ,wemust make theassumption thatforsmall values ofh,
Z(5,(A-)*z(5,(A/),
andtheeffectiveness oftheerror-control technique depends directly onthisassumption .
Ifwesubtract Eq.(5.5)from Eq.(5.4),andassume thatz(5,(£,)«z(5)(£, )*wehave
W,+1-Wj+t.p=^[251 Z<5)(AI)+19Z(5)(A,)]»|/»5z<5)(Af),
SO
Z(5,(A.)**^(wi+i-w'+i.p)-
Using thisresult toeliminate theterm involving /I5Z(5)(£,)from Eq.(5.5)gives the
approximation totheerror
Izfo+i)“ w.+il^19h5
7208— |wl+,-w,+ltP|19|w,+i-Wj+\tp\
270
This expression was derived assuming thatwo,w\,...,w,-aretheexact values of
y(fo),y(fi),   ,y(ti),respectively ,which means thatthisisanapproximation tothelocal
error.Asinthecase oftheone-step methods ,theglobal error isoforder onedegree less,
soforthefunction ythatisthesolution totheoriginal initial -value problem ,
y'=/(/,y),fora<t<b,withy (a)=cr,
theglobal error canbeestimated by
|y(r,+1)-w<+1|%Izfo+i)-w.+il%19|w,+i-Wi+i,p\
h 270/i
Suppose thatwenow reconsider thesituation with anewstepsizeqhgenerating new
approximations vv,+\>pand Tocontrol theglobal error towithin e,wewant tochoose
qsothat
|z(f,-fqh)— wl+i(using thestepsizeqh)\
i ^ qh
Butfrom Eq.(5.5),
|z(r,H-qh)-Wj+i(using qh)^ qh19
720z<5)(A.)|?V %19
7208
3/i5lwi+l“ w/+l.p
soweneed tochoose qwith
29
7208
3h5K+i-w.+i.pl4,4 1 9 W/+1“ Wi+i,p 4qh=—— — q<£.H270 h *
Consequently ,weneed thechange instepsizefrom htoqh,where qsatisfies
q< — \'%2f )1.
V19K+i-wl+1</,|y \K+|-w,+liP|/
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deemed Cutany Mjppic-cdcontent do»notmaterial yalTcce theoverall learning experience .('engage [.camon reserve*theright lorenxwe additional contort atanytime ifcutoeqjcni nghto mtrictinn*require It212 C H A P T E R 5 Numerical Solution ofInitial -Value Problems
Anumber ofapproximation assumptions have been made inthisdevelopment ,soin
practice qischosen conservatively ,usually as
QVK +i-Wi+i.pl/
Achange instep sizeforamultistep method ismore costly interms offunctional
evaluations than foraone-step method because new equally -spaced starting values must
becomputed .Asaconsequence ,itisalsocommon practice toignore thestep-sizechange
whenever theglobal error isbetween e/10ande;thatis,when
e
10<lyfo +i)-w«+il_
h19
270hK-+i-Wi+].P\<e.
Theprogram VPRCOR 56
implements theAdams
Variable Predictor -Corrector
method .Inaddition ,qisgenerally given anupper bound toensure thatasingle unusually accu-
rateapproximation does notresult intoolarge astepsize.Program VPRCOR 56incorporates
thissafeguard with anupper bound offour times theprevious step size.
Itshould beemphasized thatbecause themultistep methods require equal step sizes
forthestarting values ,anychange instepsizenecessitates recalculating newstarting values
atthatpoint.InVPRCOR 56,thisisdone byincorporating asasubroutine theprogram
RK04M53,which istheRunge -Kutta method oforder 4.
Example 2Use theAdams Variable Step-Size Predictor -Corrector method with maximum step size
hmax =0.2,minimum stepsizehmin=0.01,andtolerance TOL=10“ 5toapproximate
thesolution oftheinitial-value problem
y'=y-t2+1.0<»<2,y(0)=0.5.
Solution Webegin with h=hmax =0.2,andobtain wo,wi,w2,andw3using Runge -
Kutta ,then findw4pandw4byapplying thepredictor -corrector method .These calculations
were done inExample 2ofSection 5.3where itwasdetermined that theRunge -Kutta
approximations are
y(0)=w0=0.5,y(0.2)%w,=0.8292933 ,y(0.4)%w2=1.2140762 ,and
y(0.6)%w3=1.6489220 .
Thepredictor andcorrector method inExample 2ofSection 5.4gave
y(0.8)%w4,=w3+^(55/(0.6,w3)-59/(0.4,w2)+37/(0.2,w,)-9/(0,w0))
=2.1272892 ,
and
y(0.8)«w4=w3+^(9/(0.8,w4p)+19/(0.6,w3)-5/(0.42,w2)H-/(0.2,w,))
=2.1272056 .
Wenow need todetermine ifthese approximations aresufficiently accurate orifthere needs
tobeachange inthestepsize.First wefind
"-5SS""--fsa12-1272056-11272852'-2'941->r’-
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Because thisexceeds thetolerance of105anew step sizeisneeded andthenew step
sizeis
/10-V4( 10-5\1/4
q h=\*T)=(2(2.941 x1(H)J(0-2)=0.642 (0.2)«0.128 .
Asaconsequence ,weneed tobegin theprocedure again computing theRunge -Kutta values
with thisstep size,andthen usethepredictor -corrector method with thissame stepsizeto
compute thenew values ofw4/,andw4.Wethen need toruntheaccuracy check onthese
approximations toseethatwehave been successful .Table 5.14 shows thatthissecond run
issuccessful andlistsalltheresults obtained using theAdams Variable Predictor -Corrector
method .
IBDieS .IH r y(fi) hi sTI1
0 0.5 0.5
0.1257017 0.7002323 0.7002318 0.1257017 4.051 x1060.0000005
0.2514033 0.9230960 0.9230949 0.1257017 4.051 x10"60.0000011
0.3771050 1.1673894 1.1673877 0.1257017 4.051 x10~60.0000017
0.5028066 1.4317502 1.4317480 0.1257017 4.051 x10“ 60.0000022
0.6285083 1.7146334 1.7146306 0.1257017 4.610 x10“ 60.0000028
0.7542100 2.0142869 2.0142834 0.1257017 5.210 x10~60.0000035
0.8799116 2.3287244 2.3287200 0.1257017 5.913 x10“ 60.0000043
1.0056133 2.6556930 2.6556877 0.1257017 6.706 x10"60.0000054
1.1313149 2.9926385 2.9926319 0.1257017 7.604 xlO"60.0000066
1.2570166 3.3366642 3.3366562 0.1257017 8.622 x10-60.0000080
1.3827183 3.6844857 3.6844761 0.1257017 9.777 x10"60.0000097
1.4857283 3.9697541 3.9697433 0.1030100 7.029 x1060.0000108
1.5887383 4.2527830 4.2527711 0.1030100 7.029 xlO60.0000120
1.6917483 4.5310269 4.5310137 0.1030100 7.029 x10-60.0000133
1.7947583 4.8016639 4.8016488 0.1030100 7.029 x10-6 0.0000151
1.8977683 5.0615660 5.0615488 0.1030100 7.760 xlO60.0000172
1.9233262 5.1239941 5.1239764 0.0255579 3.918 x1080.0000177
1.9488841 5.1854932 5.1854751 0.0255579 3.918 x10“ 80.0000181
1.9744421 5.2460056 5.2459870 0.0255579 3.918 x10"80.0000186
2.0000000 5.3054720 5.3054529 0.0255579 3.918 x10” 80.0000191
MATLAB uses thecommand odell 3toimplement avariable -stepsizeandvariable -
order Adams -Bashforth -Moulton method .Itcanbemore efficient than ode45forsmaller
error tolerances because itcanusehigher -order methods ifneeded .
E X E R C I S E S E T 5 . 6
1. Theinitial -value problem
y'=y/2— yV,for0<t<0.8,with y(0)=0
hastheexact solution y(t)=V2sin(e'— 1).
a.Use theRunge -Kutta-Fehlberg method with tolerance TOL=104tofind w,.Compare the
approximate solution totheactual solution .
b.UsetheAdams Variable -Step-Size Predictor -Corrector method with tolerance TOL=10-4and
starting values from theRunge -Kutta method oforder 4tofind W4.Compare theapproximate
solution totheactual solution .
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2. Theinitial -value problem
y=—y+1— j,for1<t<2,with y(l)=1
hastheexact solution y(f)=14-(el~‘— l)/-1.
a.UsetheRunge -Kutta-Fehlberg method with tolerance TOL=10~3tofind wiandw^.Compare
theapproximate solutions totheactual values .
b.Use theAdams Variable -Step-Size Predictor -Corrector method with tolerance TOL=0.002
andstarting values from theRunge -Kutta method oforder 4tofind*v4andw5.Compare the
approximate solutions totheactual values.
3. UsetheRunge -Kutta -Fehlberg method with tolerance TOL=104toapproximate thesolution tothe
following initial-value problems .
a.y'= for1<t<1.2,with y(l)=1,hmax =0.05,andhmin=0.02.
b.y=sinf+e~\for0<t<1,with y(0)=0,hmax =0.25 ,andhmin=0.02.
c.y'— (y2+y)t~l,for1<t<3,with y(l)=— 2,hmax— 0.5,andhmin— 0.02.
d.y=— ty+4ty~\for0<t<1,with y(0)=1,hmax=0.2,andhmin=0.01.
4. UsetheRunge -Kutta -Fehlberg method with tolerance TOL=106,hmax =0.5,andhmin =0.05
toapproximate thesolutions tothefollowing initial -value problems .Compare theresults totheactual
values .
a.y=j- for1<t<4,with y(l)=1;actual solution y(t)=f/(l4-Inf).
b.y=1+j+ ,for1<t<3,with y(l)=0;actual solution y{t)=ttan(lnf).
c.y=— l)(>-l-3),for0<f<3,with>(0)=-2;actual solution .y(/)=— 3+2(1+e-2,)_
l.
d.y=(f+2/3)y3-f)\for0<t<2,with y(0)= actual solution y(t)=(3+2f2+6«/*)“ ,/2.
5. UsetheAdams Variable -Step-Size Predictor -Corrector method with TOL=104toapproximate the
solutions totheinitial-value problems inExercise 3.
6. UsetheAdams Variable -Step-Size Predictor -Corrector method with tolerance TOL=10~6,hmax =
0.5,andhmin=0.02 toapproximate thesolutions totheinitial-value problems inExercise 4.
7. Anelectrical circuit consists ofacapacitor ofconstant capacitance C=1.1farads inseries with a
resistor ofconstant resistance Ro=2.1ohms .Avoltage E(t)=110sin/isapplied attime t=0.
When theresistor heats up,theresistance becomes afunction ofthecurrent i,
R(t)=R0-fki,where k=0.9,
andthedifferential equation foribecomes
/1+2*.\d«+_1__.=\dS
y R{))dt RQC R0Cdt
Find thecurrent iafter 2s,assuming i(0)=0.
5.7Methods forSystems ofEquations
The most common application ofnumerical methods forapproximating thesolution of
initial -value problems concerns notasingle problem ,butalinked system ofdifferential
equations .Why,then,have wespent themajority ofthischapter considering thesolution
ofasingle equation ?Theanswer issimple :toapproximate thesolution ofasystem of
initial -value problems ,wesuccessively apply thetechniques thatweused tosolve problems
involving asingle equation .Asissooften thecase inmathematics ,thekeytothemethods
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