0 alOOl, the: wluhOOlcu,,'e Throoghoullhe: hook we ""II _ many uamplu thal illullnlle lhe: de~mlence (e,the, .. n"ti,·. or on.. n!ili'-e) of the: ",IU1l0n 10 an ,n,hal \'Ilue problem on the: inillal "alllO
5,4
DifTcnmt Types or Symbolic Solutions
In S«t"'" S, I. ". showed how to "';1Iu.lte and plot I 'ymbohc ",Iutioo ,"~n by 4101v. on lilt idr.I."", th", n outpulS an explicit ...Iuh"" ,n"elv,ng unly function, bu,11 into MATLAIl A. we mcnl,o .... d then:. many Olhc:,'y~, of OUtpuI In: JlO'lI'bk. and.".. mull "".1 w,lh lhem differenlly, Here we gi". add",o"~I ..ampl .. illu,mumg other JlO'l,ibilniet_ "~llml)le ~.4
Cons,der the diffcrcnlt.1 "'Iu'l!On
Thil equal'on 'I .... ,ther /O .... OJ nor .. parabl~. and 'Ii WIUlIOR) do no! ha'~ • formul. 'n 'ennl of elementlry funcl""". HQ\\'e"cr. 4101 v. t in find • fom>ula for the: IOlution,. To getlhe: ... Iptton .... nh V(Oj -3.,YP" » .01lt .. dlolv.('Dy" • + y " lt', ' y (O) . . . ) ' , ' . ' ) / n pretty(lolll)
",1/
10_,21),
, '" , • ,'" , , '",,' ., ,'" ,
IhyA111,
( ~ ...... (1I)1
-o~U!l1
, A! rvB1U.
"
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, ,'"
"
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,'" , • 2 pi
A!,-yAl(-tl , Al,-y8 1l· tJ
,'"
"
(b.opIcr _~ Soluli"". of D.tr~",oriaJ EquarioflJ
die IIDIutiOn is so 1001. ,, ~ ulCd pretty 10 prinl ,I -
-
fuJIChoo" iii -
II lJIdE>. ampl.51 ,,,","umes
a """"' A_ h Y Al . and not; this key
n.
In
,,!he
_ _ fUIICllon, 1$ Ii Toolbo~ are balCd on softwa", called Maple MATLA B al/emph 10 Maple OUt· M.pk (U"'(;OO' R"!IlJln ulICon,-.ned. In th,~ (.) and Alrylli (x) are nul co,w.n.d !U th." and airy (2 •• ) ,\ s a ",~ult> moSl MATLAB cum_ l"I0II- MAn.A B
.",,,,.n
Howe.. r. SlOe' . uh . 's part of the Ml p1c nprcllion' .ueh •• • 012 Thu,. we rely E umple ~_l 't' 11 ~imply luh,hlute, 1 for t ,n Iix formula for .012 . ;;;.., nu:n.ncally. You can foo:e the outpUt 10 be numeric with
't', 111
• 010.0101, IT'bh(~.{.o12,
' t', tv.h )l l .....y 10 plo!. family of solulions As 10 b.llI1lple ~ I. SW1 by defining.
wld....... nc ,rnuJI condillOn .,(0)
c:
. . .d21. Ilaol. . {,Dy. t . y '2'. ' yI O) • Cl', 't·) 1b pIoIlbe lIOIulion. ...,th .,(0) - ,3.· 2,~, -2, _ .,0 on lhe ,nler..a] 0 ., I bold OIl
•••• ,ft..,...., . .1• • 0,0.01,1,
< I. type
.....1• • ~'I.o12., 't·. tv.1a ) , •• foz a.al • ,],0.5,0
...
plot(t.... 1&. 4ovl>h {.uha l yvd a.
' C' , eval))
•• hold off Hae .." firsl . uh, titu(ed Ihe "~lOr of I ,.,.1".. inro th. w lutiOft.llccau .. these ~aluc' are ,he MIIIII fOleac h cu""" , (We did not u'" doubl. Qt ,hi. point. because nu",.ricalevaluauon IIlIOI ~'YI)'. (lornen )IOSlibk, unt,l a .,.Iue of r ,. ~ified.) The resul! i •• veelor of 1)'IDbo1h: npn- .. ion. w. tailed yval . lncn " .. slICCeui.,.ly iubmtured clICh of Iix
th.,
j.4 Different '/}p<'. of Symbolic SoIlIt,on,
61
R.mark S.J ... // oflho comllWld. IH u~ ,n tI".c.amr~ 11>0 ....,rl for E..mp~ S.I, and m any O!htr ~ • ..:' 4aoh'a ~relch an UpliClI ",Iullon !O. f,n' order Njl/.rion.
F. ... mpl. S.S The diff... nlial NjWllion d~
~'+I
dt
~"
i • ..:'parubl., Bnd M ...TL ...B can do 1M nc"c\l:UY inlc.IPal.o whc 'I' ~~ ao13 • 4aolv.('Dy • (y" • lI/y '5'1 ao 13
1/2 'y-2,l/2'atan(y-21'Cl
t
0
1I0"'."or, MATLAS cannOl J,o/,.. the ••"u/l'n, "l1I&I,on for ~ ,0 .. nn. of I, ..... i. QUI""" an ,mph,'I.olulIon, Wilhou. an npli<:i. formul. for ~ In I.rm, of mu,1 131, & d,fferenl aPflro;och 10 plull,n, and cI·III1.un, "",lUon •. hrll, "'C npre" 1M CODsWII of .nlegnlhon, ",111<11 M...n. ... 8 call. Cl. as. fUlIClIon I and II' » ao13.xpr • &oh'&(&oll. 'Cl') 8ol) expr
I,,,,,
or
_t.l / 2·y-2 _1/2·atan(y-2) The ",Iulion cu," •• a.c Ihe le,'el cun' •• oflhi. c~ptt,,;on, "h'th .. c male inlo an onony, mo"' funellOn., fo//ow" ao13flloo • • It. y) .vallvaotor1 •• taoll.xpr)) Then wo cln u\<: OOOtOl1O 10 1'101 $<:1'0,,11 solulionl. "Y for 0 '. I . ~ and 0 ~ ~ S :1: » IT YI • _abl1ri410,O.05.5, 0,0.05,3), .~ cootou olT, Y• • ollf .... cIT. TIl » dabel t. ylabel Y l1H: .O\U/I '\ ~'n ,0 Fillln: S5. Unfonunalcly, wlltn 4aol v. cannol fond an e,pl"'l1 ""Iullon. ,I I"'" an error me ... ge when you 11)1!O spec,fy an IIl11i.1 <'OIId'lion None1Mkll. you c... C:l\Lly find the ron,tan( of inl~gr.l1iOf\ fOf a Ii"," inirial <0,0(1111011 by pl ugtn' the "r.... opo ... ""h... of 1 and V inlO .011ft,lDC So, 10 plot the ",lullOII .. ,ill ,ml ..1 COf\o:hllOll ,( 0) Ion 1M ,nl,,,.1
>.
0< 1 ~ 5. lype: 0 • ao13flloc(O, 11/ » IT Y) . . . .bgdd{0,0_05.5, 0,0,05,'11 » oootou r(T, Y• •013funo(T, Yl, (e OIl The Je5uII IS shown in I~,"re 5.6. Remember th." " Iltn m.lln8 contour 1'1011, you ",UII ehOOI. ho1h the hori~o"lal and "erli<:al rang.. In lhe . .ahl1r 14 command; some Iri.1 and clTOl ma~ be n....' 31)110 find a 1·.I1ICI I nlnge 11..1" appropriale.o. panic"l •• wlurio" Th plOl I f.m,ly of soluhonl. say with 10,/..1 COnlllllOll' V(O) I >I ,2, I.-I , _ , 2, o,'cr lilt u
same ."""" Iype::
ChapIN.~ Sul"u..",J uf /)iJTrrcnual 1i'q"3Iions
•
•
I •
..
I
.. •
•
•
. . •
,
•
•• U. ' • • ~ll!UDC(O. 1,o.a ,all •• uua'..rC~. y. ;oI3!UDC(T. Y), eval;)
'lb evalualt. ptiticulOlr ",1"lioo 31 a pwicular "alue of I, ~ou ell! ~ubstl1"IC the appre.. . .' vallIe of l,nlO .01 UIlDC, ~I i. equal.o lhe cOO\I~ot of Intcgralioo you fouod from lilt IalIlII condJboo. and "'he IOf U. One way 10 do Ihls is as follows'
•• .,.. YI ;o1.-C;o13!IlDCI', y) - e , y) 1116689668Sl623181628896?2S841 2 11 689668S)62318l628896?288412 HeN u \, thf \'&l1lC .... prniou,ly compu.ed for .he: m;tial Contl"'OO 1/(0) I, and we I'Id for thf nIl>< ,,/ Y( ~)_ 11",,"'.,-••, a, 'H d,,,,u\~ In S«'IOO 3,1. solving algchraic
ha,..,
......101\.1. nolll ..·.) • .u-aighlf....ward. In Ibi' else, lhere are lWO solution s 10 .he cquallon rot, 4) BI>C'd on "'Kure 5 6. ,I I. d.ar lila. rhe POS'lrl'C soluliO!! IS the one we ,,'ani ,n IbiI cue Bul hoar.n mmtllllat !>QffitlimeJ iolv. docs IlOI fInd all the .\.OIUlIOOS of lI!
equIIIon. aftd .. !wn II ,II.' a single aM".', .hat may IlOI be: lhe solutIOn you ,.,anl II is lifer 1m ,..MnJt" ,'''ph lho soIutioo )'011 want ROO tllen usc ! nco 10 .\.01,.., numerically rot I/llIInl'" Inlll.l , ..." """don lho gNpl.•• ' •• roC'Cy) .ollfllDcl a. y) - c. 3) ~
3.1117
R_rk 5.4
The
LAB Iynla. "hen d•• ling wi.h Iymbolic npl~'$lons. WhIle II may h,,-e """n mo", IIIW' IIlWIOI)pt _olvtrt.oll!llDc(a, y) _ e, y) ,u\III,,,"equab,illnhe.e,,,sulls ln IDClTO! mt' ..... B. In'lead, ,,·c I)'p" an .'1"","00 .ol lfunc(a. y) - c for .olv. 10 let equal I" 1<1" Wh,le e " .Iready dcfinrd, y 15 undcfinrd.' IIIls PO'OI, SO ,,'e m~s' cltellll' It h'l>c: • ,),moolrc ,,,"able 10 a,'old an ffTOf mr'~gr. The funclion !a.ro
am
"
,
. " .. "
, "
" I~gu,e
S 6 SOI""OI, or d~/d!
"
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1
f".,
"''1<>m:5 Ih input 3fj1umen,!O ~ a funmon, nO! a ,~mbohc n,,,.,\;on, II) "'. mlk lhc: ,.'p'< S>ion 10 be ..110 urn inlu an anonymou. funcllOll In (h,\ co .. , y ;1 I dummy """able In the: funchon lkfinilion. loo"...d not be dttllml IImboF", bc(
•:umplc 5,6 The d,rt.",nh. , .qu~lIon
" "
... + II
" "'parab le. lIS in lh. P""' IOUS c>ample, bU I ,hlS l,n", d .o l y . can'l do the II inlcgral: H . 0 14 .. ,S-" lv. i' Dr .. . >le for '''''h equation" Hr ma rk 5.5 Althoogh ",e ha,.., focuw:d on flr~ order cqUJuon~ In IhOl _ lion , .s. 01.,. al
Ch2p1rr .~. SoM,"'" o( DiffcrcnuaJ Equalio.u MATL'o D. lIl c an) J.Ofl,,·~'" syMcm. can occ._ rtSull, As the foll""mg n amrle shows,. good the of wluhon~ '0 d,ffc ... nl,al «j .... lions is a ",Iuable (O!IIJIII~r.I!C'".r.II.d ... sull •• and Ihc ... r"", in ,dtnli f)',og SIIu.lUonS in h&I ~ a mj , I. :><\mg or i~1 ",'Ull
na''''''
,
/ 7'" Coaudr, the initial ,000oe I'rotolr m
d.
,
II(O)
" .... MIl . hol_ C'Dy . t. - 2 . y - 2', 'yIO) . 1 ' , 't ' )1 .. . abtl_15. 1_1 11 1, t. i tl. " ... 4 1 die IIIPb ta FiIU'" 5,1_ The solu\iO/l ,hould be conlin_II . bul the grnph has a ._1 0 A«m1i"l 10 the in,lial coodnion, lhe solUlion \hould be I 31 j O. bul s1Ml5. ' t', 0) gi" es the ans ...... NaN. for ''nnI a number"_ The ",ason asa .... MATtAB 1ub 0, nnd wh ile ils solo lio" soh'.' dlldlIfIiualilll equation bollt for! > 0 and 1 <: O. •1 IS diKonun llOUS "1 t O. .,... .
2
•
I .. . . .. .... . .. .... .
PIJure ',7: Tho "Solution"
of dU/ d!
,
"
,' + ,'. , (0)
' . A""ord ",g to " .olv.
Chapter 6 A Qualitative Approach to Differential Equations In (hi~ chapler. we di~"I' a q~~I"~"'... OppnlO< h (0 (he \u,",)' (If diff.rential «IUII;<'>I1', &lid o/II:un quahlll"'" infOf11Utonn Ibo
m. d,fferential "'IUllion.
Con,iM' the g.nero] fll'll onk. d'rr.... nl'al.qualion
ffr, ~).
(6 I)
'I<'" ,n,
We can oblllin qUllilllt". ,nfor'lt1.l'ion about the «>I .. noos I/(t) by (6.1) gcomoruically. Specifically... c can obtain (hIS ,"fonttation from the d,rt<;Mn hdd of (6.1). Re<.11 ,halm. d"":lion fIeld is obtained by draWlnS IhlU\lp ... h point in the (I. III-plane a .hon lino Ofgmtnl wl1h .lope I(r .1/). Solution •. or intcgnol corv... of (6. I) h.,'clhc prop:ny 1Il.1 a,each orlhe" points thcy .... IIngenl to the d"cuioo field althll point. and thc",f" .. thc SCnc!'2l q""'IIIIII>" na'uft of the wlUlion, ClfI he dc,cfTflil>«l from \be wm:uon field. DI' rection held, can be dni..'" by h;and for >Or1l<' IImrle d,ff.renua] cqll.lllOns. l:oJ( MATl.AB can dnlw lhem for MY firsl ordor "4U,'iOfl. w. ,IIUllr.lle 1M qu.hllh . . . ppro<>eh w,th I",,, example •.
6.1
Direction Field for a ."jrst Order Linear Equation
Cons,d.r the equa"on
"
"
(6.2)
MATLAB'I command for plo"'n, d,rncliOfl lidd . is qv.1.., .... U\N ,n <:orljun
"que"". of <'''OlOlaool:
C'JJIpkr 6, A Qu;1/,r~!"'C APr""'.,)' /0 D;lTc~nnal Equa,;ons _ltO.2 ,2)I
I), ... il tillht by 16 arid of unifOflTlly ipxW po,nl< in the r"Cctangle_ !he rigill-Iwtd "ok of 1M d,fkI'Cn"oI equl1ion al.he &n d f(r, 11)' al .ilt5C poinl~ The ,'<'(;1(,rs ( 1,3) have •• lilt d,ffeI'Cnl grid po'"Ii. The q.,o1. ...." • ,equiI'C~ foul i"ruI~: lilt array T of I_value •. IWO coml"'nenli of lhe di=lion field .,me .i/.e, Ihe C(Kie 0"" ( , i (8) ) S;/C" 8 In lhe la" command . ...1., "'''0" fIeld TIle ",suI! looks hke 'hi.:
z,
II I,
,
,
I
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I !< I , .. I "
I " I "
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FIJW1' /1.\. Direclion Field for P.qual,on \6.1)
i.
•
"rofTtt,: il " """,,whal hard tu ,.ad because Qf the /IC! thai many ' On.e cart
g., a belle' I"'lUfe by (.snhng the DIOW$ ~ Ihal
e.ampl •. by d .. ,d,"g .."h ,-ector ( l , ~) by 11, length In ookr ' 0 do thIS In MA1' ..... R VI." mod,fy lhe precedinl ~""nc. of
'II ..
".
_,bgridt-2,0.2 ,). _1,0.2,2), -.1-'1') - 2·'1, 1¢11 • S. "2 ),
. . cYi.-zI'l'. T. 1./L. S./L, 0.51. axi l tight .. II1U1oil 't', ylabll 'y' ." ~itl. 'DlrlctlO1l rll14 for dy/dt .. IXP(-tl - 2y' n..1IIh enlf)'. nA in the quiv,r comm.nd ..doc" . ,lie lenlllh of .he .'«.oes by half ... p!WWnllo !he am'" Mad, from ' ''' anow,"g up ,he 13,1, of "" ..by .'...:,on_ The ""ul! i, 4 &1110 f\suft to 2
6, I. /)i,ec/ioo h eld (or • Firs, Order Linear Ilqu~''''''
"
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",;~
""~ 'I/'~
..
, ., , , " ",
-,, ~: \\\i1:iIiii;;;;"",:;n~ "" " "" 1 " " ' . " . "",1 ..." ., ....... " ' " . ,1/,
".. ' ",.,
- .. .. ,
.
"",,~.
••
"~
1111'
"
., i.L.LLLLLL1Il ,.11111,11,1,.,1,:
Fi gu", 6.2: Imprm'ed [)"tcll<>n h . ld fOf Equl tino 16.2>
Not. LIt)t .... hll" .... ,,'~ at It .... C Iu" .• Iaboh," at """" point, Llren " '" III(: .. asln8 aL!hill poinL . ... d Iho, all ",lull",",' ol'fll"'1«h uro &1 I - ",-, ",. g"n"al wlu,ion of equation (6 .21 il< _r + 0, ". ~l gUrt 6.3 ~how~ .."o ral solution c u("\',,~ ,upmm~ on LIre drrecLiOct fiold. 1'1 01< frorn equation (6 ,2) IL ..][ Llta, rI(l) 0 "hen ~ : .. '. and Lltu ; lit" cu,,·. " ,,1>< .. lhe muimum poin" oc;<;ur on ,he solul,on ru("\ c; In h gurt 6.1. LIte Cu("\" U :' ., .. ind,cated with a d."hed bn •.
, , ; ~ -., ·,·.,.l '~l t ' l \ 1 ,11),1 r, , , ,;-"" , ~"\'\"\\"""'\'\"" \ \\. ",II .. " " ,, I'~-" "" " \\ " '"" """ """ '\\\ ,, I"· ,, ,'~""" , .. " "" ""'" ' ," '~--, ' , .. , " , " , " """'_--0.; " .,' \,
..., , " '!:' .. .. - , ..
-~ "
• "JII ( I ' #I//~/""
,~~.- '
1111' ',J J "
~
...._ .
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: UjJUiY1iL;;;;.i;~;i.i;
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.,
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...
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Flgu.. 6,), Dirtc, ion Field ar
i',nrdH Ii. l l"II",ue lhi' ,dra furthe. by d,rr. .. nl,.ung (6 2) . r
lion PO'"U 011 lhe solution cu"..,J ho .Ionll the cu"·.
~
:,.'
U
J)lreCtiOD
Field (or a Non-Linear Equation
.2 1-- t6.llcaD \It' whd c' piicilly bt<;a~<.C il IS I"",» Now lei uS COII,ilkr an ~~lLffiplc
dill rUlf \It' 1OIVft1 uphcilly in itml.l of oknltnl.., fUl>ClIO
(6.3)
Ita d1rcct1oo fte(d .. ohtainod wilh Ih. MATLAB ( onllnand! ..,. I", 'I) • __ bIIddt-210.2,2, "2,0.2,211 ,. . . . . '1,"3 • 'II L . a~t(l + $."2)1 .... CDl~t,.. 'I. l./L. S. / L, O,!). axi_ tight .... alu.l ' t', ylu..l 'y' .. , titl. 'Dl~eetloD PlaId to~ dY/4t • y "l • t'
n. ftsllll ...\>Qooon ,n F'g\ll'C 6.4
From lilt pictyre. ,loW»1 thal .ll WluliO
........... , , ,.., , ,, , ,, ,
, ,, p
,
,
/
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", , , , , , , ,• , ,,
,, , , ,, ,, ,, ,, ,, ,, ,,,, , , t ,, •, ,, ,, , • , ,,, ,,, ,, ,, "t, ,, , , , ,, ,, "• , ,• , , , , ,,, , , •, , , , ,, ,, , • ,, , , ,, ,, , • , , .:t , , ••, ,.., • • •, ,, ,,•' ",,• ,,• ,, ,,, ,, ,, ,, ,t " , ., , , ,• ,, ,, , ,, ,, ,, ,, ,, , ,, ,, ---' ,,,,, '.e ,• ,," ,, ,, ,• ,, ,, ,• ,,, ,, ,, ,,, ,, ,, , , ,, , ,• • , , , ,, u "" , , " " "
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Figure 64 Dir«I1O<1 Fotld fOf EqOllOIIO
~J ~or I
,. U ..,d Ih.1 011 ",Iuli"", ewnlually b...o me po»l'''l'. Ftg"re 6.4 oIso I"UCSIS thaI I lUll"'" IJ'I!"'O'I"+'
u' + I ..... ~I •. ;tfId can \It rewritten al
6 I. Au!""""""" Equ.ui(WI' and Ihu. can I' I mU ll blow up In finil. I,me lIy the unique""" of 'olut,oo, and ~ &llIICe al the du«I,OO field ,n Filun: 6.4. il i,clear thaI "II solut,onl mUI' blow up •• ~"rlO Ume.
F. •• rdH 6.2 U", MATLAlll0 graph """" or lhe: dn",c1ioo fidd,
your le,lbook. If yOll can sol .. e ony of Ihe .qU>I,OOI •• plrc,lly. If)' sUl"'flmpo.inl .orne of Ihr solUlion cu,ve, On lOp of II'
6.3
In
Autonomous Equations
Equat,on' of In.: form HII) .
,..sr.
",hich rio 1>01 ,"" 01\-. t ,n Illt nihl·hand arc • • Iled aUl_"" tqlWion .. If. [>hY' .nl Iy"cm fo11ow. rule, of .'OluIIOO thaI do nOl chanie "'Ith "'1'1<'. then the o>'Ol"uoo of the system is go.erned by:tll aulOnomous C<ju.uon Such C<jual;nnl ill. particularly IJlICn3blc 10 qualit.l" C 10,lysi" Considt. ~uat;OfI (65) To I'uPpolC ftu) hal 1"0 ,.ro, y, and 11, . C31led erilir a/ ".,.IIIS of the diff... nual cqu:mOfl. Funht~. ,uPf'OIC the grapIl of fell) i, a, In !-ogu •• 6.5 , 1bcn by ronlldonni the proponlu of f l u). lhe d,rtttion licld ofl6,5)
.how.
FiIU'. 6,j; Righl.h.od S,d< of Equalion (6 5) can be Clslly uno:lcrstood . nd l polllb (I. 1/) w,th II :> 1/, e
ChJpI<16 A Q<J.III~I"'" AI~>W.1
poslllve and I/If;rta>n from 0 tu ""~, II ,ncrt.": ' f. om 11110 ""; finally, the ~Iope. F,,, horiwntalliM an: COfl~llIlt ("nc~ f ~~ no! dcptnd 011 1)TbI dirKI_ field chu. h.a~ th< Be,.,.".L al'P"~".ntt ,hown In F'Ku.e 6,6. NOIc 11»1 Apu 6.5 dtpi(t. f(lI) H_ II. ",hilc hKurt 6,6 lhovo~ II" !. Sc,..,,,,1 fOClS au suggts~d
+
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. , .,oJ" " , ,, ,," , "
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tlncLlon lick!. n..... ly. cha" Ii) t~r~ a.~ 11'1'0 coo"aOl ..,LuhOll. 1/ • II, aod II 11'.: ..... IIIIt1I1i u/>oYt" II, I. nd to 00; and (IIi) aLL OIhc. M.)\ulioo. tcnd 10 II, as I - 00. W. fkrt ve IhcK prt>JXll i.. of the ~oL"'i"n. ~(I) dlf«,Ly from lhe graph of f(lI) a1
""""
I 'I'hI ron.I ...1 funcl;""' 1/(1} II, and 1/(1) I/j arc ",Iuhuns. caLLed the ~q"jJjbr",m 1III1IIj0fU' Tllty...,!he: uOlque ",Iullon. thaI UI;\fy 11", ,o;t;al cond, hons II{O) II, udll(O)
1I' . ...·I""
Con ...... . ",lull,>o 1/(1) .. "h 11(0) < II,. Tlltn. b«au~ of lbe uoiqueness of sol .. II""
"mtI I hr Ih.u 1/(1)'
1/. for I)) I and II'{I)
f{W) > ()
",11<'1' 11(1)" ... 'I>ing fu ....,lIoo S ...«;I" allo boundlOd obo>'o. ,lila, . hm,1 al
lnft ft'ly hm 1/(1) ,-, ('oold " MPI'I'" ,hot ~, 1/, ~ The '''".... "
1/'(1)
b < 1/,. 1\0. httlU":
f(l/e/)) ... f(b) > 0
(6.6)
,f b < 1/,. Ihen (6.7)
Bu! nt"a""o' (b,6 ) and (6,7) lay Ihat I/{ I) "WroachU a hu"wnlal ltSyrnplolc 01 1M ...... I" .... mil Il' ,top< i. "ptnll:",rnl1y" blHarr than a p"'1l"'e number, The ~ ..111111i """loa.,1idl(>tl guarnn'«1 Ih.t
lim /I{I) ,-,
/I,.
). If 1/(1) 1\ • ..,Iution ""Ih 1/(0) > Ill. liltn 1/(1) > y, for aliI, Also
/MI)) > O.
1/'(1)
lIoJICO 1/(1) is one. ",;un inerusml In fat!. 1/(1) _ "'" TIu, can bt """'" by u... d in pat'! fbI. Somoti""', .•, .. r lhall iN from the an argu"",nt ,im,lar to eumple Iltlow. I/(t) act ..lIy rra.cbtl.",. ,n hmtetime,
th.,
w. "-r mml 4 Now con"tlt. 1/, < 10\, < 1/" Then if 1/(1) i, I .oluliOIl ,,',th 1/(0) hll'el/' < I/(t) < 1/, for aliI AI",. 11'(1) /(1/(1) < 0 lienee 1/(1) is. docrr",,", fUJlChOO A. In pan 2. n" no( d,fficulllO ,how that lim, .,1/(1) 1/,. N",o that In \h" analym tilt propen;',
or 1/(1) .... tktomunod wltly from the ",0 of
/(1/). We c.U ~'"
,t, mlrumum
Ettrd~ Ii.J I..,t 1/ be the ,aluo bot"~n ~I and 1/, "herr / lIle, on that if /I < 1/(0) < Ill. thtn 1/(1) h" an mft«11"" POint "here 1/(1) 1/.
Show
63. 1 .:"lIl1lples of Autonom ous EqUlltions Now We nom,ne. 'f"'C,fic ... mrlo. tho t<j""tll'ln
,, ,
d,
"'
I.. dir«hOO fidd has tho ,oneral "f'I"'ar.>nco of I-'~urt 66 ... ,th 1/1 0 and 1/. I. Il ut ,,-••an actuaUy don'. "" nrhcn formul. soiUIIOll If II .,. 0 and 1/ .,. I ... -0 can sol,,, by sc~lmB
,-.. iablo,: /+('
ld< / /111/
1(," D
dl/
II
,,
lu lll I"
II , I
In II h..pooenllatm, t>oth "do, ~" •• I
I
, '. I
~,
In 11/:
±t' Sol..-ing for 11. ". find thai ~(I)
LI( L
h'), Noun, lhal ill)
Ai). we hoyt
~(O)
(69)
11(1)
(11111),'+1/11 AJdIoU&h formula 16,91 "u d.ri>w undrr tilt a>lump"on Ihat 1/11 ;. 0.1.
It
i. ca"ly"...,n
lO be Qiid for t il >'alu .. of LN, , MATLAB~ ... ,,,h" Ihi! .quatlon and pl()! 11>0 $Oluuoo cu,"c" Hrre;\ a $tqurnc. of
COIII/IIIIIdI lllat dot' w. " . UVlln, hoi'" 0", ,.,. .yu t 0 ,.,. '01 . dIlP1Ur (4I o lv,('Dr· r " l - r', pp I~
'riO) • c',
't'»,
a.al . _1. 0 . 1 5,2
'Ulot(IIIbI(l ol. c, c v d), p p JIlu.l 't·, rl aM l ' v '
(0.3 ) 1. ,Old
p p titl' ' aolutiool of ~ /"'t • r-2 - r' Filii" 6,1 tilt aclual $01u\,on cu,,'ol for lhe d,ff.... ntl.1 equauon-as Wa"'n by
• •
•
• "
"•
hgure 6.7: SoluIIOII CUm'! for l:qu.>t_ 16.8)
MATLAB N.... Ih., tllt $Olu""" eun'.! rom'!poodon8 to ~(O)
,-
;> I
go to inC. ",I)' on fonn.
!uh1 7 .... (.1 U .. formul. (6,9) 10 verify tho f'I'OI'O,,,cs of 1I(1) obta;,,,,d .I:>ove by lhe quahlah>'c mrlbod. Drt.nm". wh
Unuta u 1 ~ '" Ib) SUppoK 1/1, 'I. u ... formula (6.9) 10 show Ihat lI{t} _ "" on fiolle hOle hnd Ill" time ' " 11... 11,,11 lIIi' happr"., leI Show 1IIa1 1«1) 0 and vet) I arc lhe <'I1',hbnum solu tion., Dor:s formula (6,9) ylrld lilt.. W)U\;onl'
n Here is ~bcr cLlmpl. Tho Itfl paph '0 Fig.,", 6 8"",", •• rullC~"" f(JI) "'Ill ~
<
•
,
• < •
•
<
",gure 6 8: An AUl<,!DOmotIl equatIOn ,..ah 1 Cnuull'
""1'0<; Ihu, the aulonornou, diffcrtnlial rquation
look altbe crillc.1 poinl,n the
modd~_
JI
f(1I) hlI, thrtt cm.al point,
~l".
Solullons sl.lr1,nl helow b "",,. a posit;,. slop< . ...,
they IlX'fUSC' iov.:tnl b. SoIullon. I.an,ng above h ha,·. a tIC,.m". 'Iope. "" they dtc .. _ to.... ard b. Therefor •. "'C Up«l b 10 be :on asymptotically .table rqu,hbrium ooIulion. A wnila. analysi, of the ';lIm of f(1/1 .01lK•• t. IIlll the OIll
Th. ,ight graph in Hgore 6 8
,"(}W, a few of the ""luI ion
~.
,1i.Slno!c, the conclusion. Ih., "-e drew from OUt analy.is of the gnoph of /(11)-
thIS
,,,,ph
Chapter 7 Numerical Methods 10, lhi' book ) 00 ha,. ~" lh., moor diffe",ntial "'luII,"n. Ca/1be ",hN "pileili)' on I
of lh. elemenlMy fUl\ChonS of clk.iu. W. hno '«n in C'·, ' , . .olv. \0 I\0' ''''pcr ' 'uol "':lily equat,on; on lenm of 11\(\.1' fo"
(UDo;.
, "
I/(t)
i" 'd'+C-
but " is tn.....n thaI w\.<' inlcK •• 11 canoot be Ol'"',.aj on 'erms of tilt fUnc.hQfl' of ukulu, (MATlAS dou "'nl<: lh,\ in, •• ,,1 m dlMd form in ,..,111 of. 1"""lmll u f o 001 th" (unchon. called (he "error furltuon:' il u""~J by tho formula ~rf(z) To ",II. and cannot"" ,ew,;u.n in lerml of ',ponen'illo, Ingl, Ingo"""",tric (O""lIonl. tic- Th,," the: limplell nample of • Jp" ",/ f"n< IjOll :u d,!t
J.:,
.d.,
We ",II 10 solution! tha. Cltl be .'ploc"ly .. rin,," In lerm of clemenlV)' fOIKhun! Of ~f"Xlal fu,,",IIons u fo .... ula Of P,• ..bo/I<" "heth<. Ibty " ... obUI"",J , •• hand-c •.kuI3t1on Of ',a d.olve When ". ran/lOl soh., • d,rr... nhal ftjllillion ,n !,h,. "3y, Or if ,he fonnul. " '. find '1100 romph ••1
.o/w'"''''
!""
"
(lI~pl'"
7. Numcncal MCl/lod,<
Numerleal Solullons Using MATLAB 1"
. . _iiIIUulCd ,D &ndlna the ",Iulion 10 the imllal ,."Iut rroblem / (1. y).
b cooLlioiog t". ",,1.1 IUPJ>O'C Ihal we do not have _ f()Tioula for
[a.bI. Tho sub~nJll a on y. he found with
Su
I·
P W:muWtU 1ht u"" of
od.n,
slaoo. MATLAII·,
,
ode.' by consl
I
yeO)
" ,
I.
(1.1 )
bt_1O/ubOII11 ralily fouoo 10 he
yet)
,It' + i
sa. we haoe an U pllCit fonnul. for \hi . loOluI;on, ...t will he .ble 10 compare lhe .prrox,
I
lUIutlwl/. (I)a.ndthenKI1(Itul;on ~ ( t).
tbe lnl *p II to ddin<. MAnAB fuoct;on "'lualto the nalll·hand ,ide of the dlf. . . LUi'" eqn_ Tk~ an: two way. 10 do thi~ One way is to drfin< an """"J'mO
f •
_c~.
yl t ,/y
OIul. 1be dol iD fn;LQI or \be di,ision " in; the fu nclion mUll be abl. 10 ope."le on ,'«101"$.) 'lilt odta way ], ~II • funcbon M .m•. as In Chapte. ~ For OUr problem, the M ·file could. ""WI"', brulltd fune. m. and would be , ~laa,. tuAclt . y )
..
~ , /'"
WI nc++hIlcf1d thol you usc anonymous fuoclion, wM novcr pm,ibk:, ) lowever. M_file • ...... JOIIlo use mort !han on<}in< 10 Jmale w luuOI1 I/o (t) OI1lhe inlf.vaI [O. 2), lyp"
;r '
"' ,'dSU, 10 2), 1) Itf It liD liDooymou. funclion. o. " al,UC_tuAc, 10 2], 1 )
If func , 1lI jOin M ,tile.
n.c", IS sllll lOOlbe. p01llb,lIiy. namely 10 1""
lhe ddiolllOl' of
lbI_ymou. full
J'dSC_Ct, y ) t. / y , 1021. 1 J J\ ' II beIe 01\ ~ WIll ,",um~ lh.l you ha,'e drfin~d the nlhl.MOO "de of the d,fJucnllal squatiOIl lS IllllIDoymOU! funclion f. The M:cond Ifgulnc nl lo 0«., 5 sp«ifLes lhe inlnml (II whlcb 10 MLloe, and th~ Ihird ILfIlumeol give, the milial value of the rolutloo .1 Ibe lint poull ill Ihi. ,olrr,'al. Tho "'lull os shIM·o io Fiaull! 1.1. "The oi",1« indicale lhe
1./. NIl,"n;"'1II Soillr,ool 1111D1 MA11..A8 poinu ~r which "". 45 cumput.d tilt a"""", . rr' connected by 10"" segmo ml_ V
..
.. /' , """"
-
"
n., W I01>On; tho ""nl .... of ."_u",,,, errd.! an: '"w,
/
~
.
.
"
"
.
Figu", 71 NUm
EnN"is~ 7. 1 I)j'piay both V.(I) and the .... '" lol"hon V(t ) Can you dlltinguilh btr..,«n the 1"'0 curve\1
I
j1' +T "" Ihe ~m< ,",ph.
You can ~I'IO plot a faJl1lly of Ippro1"""" s.oiur'''''1 by u"n," ,.«'!Of of innw '21 .... as !he third argument 10 ""e' 5 r..". .~amrl •. 10 rial Ihe ",10"(1111 of dv/dt I/v .. ,ill 1/(0) 1,1.2,1.1, . _,3.lnle »"". ' 5 (t, (0 l], 1,0.2,3) This procedu", for plallin, D fannly of s.oiolOonI ..'oo t ; as 10011 II 001l<' of Iht 101ull"". h.. a singulanly m lhe lpc<:ifi1 ,'Il ... problem 0.21 btl"",' and Ihe following E>.<",i ... Stt Sttlloo 862 for 1 infoomalloo on 00. poinl. By defaulr. od. ' 5 ploohtc<s " plot of . n 'ppnl>lrmle IOlurl<)fl To <,1111<1 "0m<1icaI
add'''.....
,al u.. r ~pc
»
(t , ye l • ode'SU, (0 l], 111 The I ,·oJ ... , II "hich ode ' 5 CQlllPUles lhe "1'P1'I.I"m>l. s.oiuhon _ \torod in lhe "OXlor t . lAd lite ~~pond'ng IJII"'O,,,,,,,lelOlulloo .~I,," art IrorN in ilIo ,oxror ye llH: "'micolon IUpprel ... S prinling of the "("ClotS. Wilt" o>ph. It I'" pl o t ( t, Y. ) Nale WI pl o t (t, y.) prod~l • graph wilhoUI cireles To di'pl"y rile ....1... ' of r and I/. (r) in columnl lide.by-si"'. 1)'1'0 [t yet , Nolic( ilIOi lhe I , ... Iue~ are ~"""'n hy 04.4 5 and m>y rndllllr _ , .. Iuel. dcd I f)'ou~", 'nr~",,1cd onl)' ,n ilIo 11111< or 1Io{l) II rho.1III of tho In" .."lIl. ,n this ca", I 2. )'ou can rype )'& (.nd ), "hllmare >olo ilon 011 0,0,2,0 .4, ' ,2.
",.
It. ,.1 • l'dSlf. 0.0 .3,2. 11, . , . . . . . . . 1Ibk ohhe ,'allie' of I ~nd y.( I ). alonll with tho cOI"'ll"'no1'''1I ,'al~' of ............... ,U)-Jil t l .• )~ .... ~. ~1I:.-2 • 1), .... II .1: 1 _
.
-~.~ w. ..... fOZ 71:
1 _ ,0 ob.ain IS ·o1illi. ou,pu, in order.o:> 'co 11",
,,
y.(I) 10000000iXXXXJ00
----
, ,""""""'" 0.2(l(l(X)OO(lOOOO
,""""""'" 1.00000000000000 , lOOOOOOOOOOOOO
1 """""""" 1. " " " " " " " """""'" ,1 " OOOOOOOOOOOOOO
1,01980390030003 I 07703295800 176 1.16619037435052 128062484 16S690 IAI42J3S 5S9924 2 1.56204992884201 1.72046504740386 1,88619622083269 2.05912601:30438S 2.23606797273725
y(IL __ I,..............................
-~
1.0198039027 I856 1.0770]2%142690 1.16619037896906 1.1806248414l16!i7 1,41 4213S6237310 1.562049935181 B 1.12046505340853 1,886796226<11132 2.059126028 19740 2.23606791749919
Tabl. 1, I, Appro.imlle and 10""" Soluhon s of dyl dl
'Iy
Jzuteiptdipto.n &II e_l. The 04et~ command III. mpls ,o prodllee an appro"!Nte ..,lmicw""Ib""" I... \ban or tquallo 10-', thouJlo oomellintS th<: K<:uracy !NY be ben., or .... lit. !Ire SKtion~ 7,] IUId 1.4 for addi1ion ~\ ,nform.>ll{III on Kcurocy. 1lMI oc.tS conut\lmd w,1I only compu,e a solution on an inl.rv~1 wnh the inol,al con diIIoa Jiven II one of the endpoinn. ' 0, w. hay. nlready descn bo:d how 10 use 5 on .1mavIi ollhr fO!Tll (I" . b]; 10 Ule il on &/I im . ...... 1 of th<: foon In, 10]. cnl.r ,he ''\lIlies vlt Ill .... "in, ordc,. For u&mf'k. 10 pIol an appro~unalt J.OIUlIOI1 of 1hc: ,",1,,1 'alII<' pJ' m(7.J)on1hc: in ...... I ,O]. I~pc ..:. l17lSlf. 10 ,11. 11 1b.,ec1fy YalIIe. for t. use a nrB"'i>. inc.. mem , II 11\ O. ~ O. 21 ~ 1 , NOlI &uJIPO'C: you wanl 10 plO! lhe fam,l y of J.Olullon. of d~/dl t/~ ""th ",,"al nt 'lI(O) 1.I.2.U .... ,3onlhrinlC ....... I (··1,2] U,""g o<'l• ., ,,,,h 1- 121 u,he II uuf qtI/I1Itnl ... ould produce .... ;1IC0Iltt1 gnph. "",,au1C 1hc: ,n,".1 rond,"on ' WOllkl .. £f,1led zl t ·1 ,n\le~ of I O. H. R., . 1Cnc, of commallll. lila. produce, 1hc: ...... FiC'liFiIjl/I by compu,ing !he appro~'m~'e sol"lion. sepataltly on the 'nle ..... I. (- I, 0]
0<'1.'
a1:
1IIII10,2J.
.... [I:b d l • Pakl .0000."(f. ..:. ,1otCI:b..'. ybak )
(0 -11. 1,0.1,3),
7,2. Some Numern:aJ Mc/hods
» » » »
ho14 on ltfor , yforl ·04.'5tf, plotttfor, yfor } ho14 off 1l>e "'lull IS Ihown In Fllu,e 7.2.
" to 21, 110.2'])1
,
,
."
II:
•
,,'
.. -
"
"
Figu." 7.2: A Family of Num
n.
We hl". Illustnilcd 04.n willi the ml~a1 •.,,11Ie problem I), .. h"h c;\o <;l"I~ to< sol,.d e.pllellly. We C;\Jl )uII as rudily .Wly 04d5 10 any fiJ')I order inluJI \.1 ... pr0blem. Conlider. for eumple, dV . I + V'
"hleh cannoc be wh.. d
"
v{O)· I,
021
terms of elellWn,uy f."""on •. 1lIc commands » II' .. _tt. y) t • y. ·2 1 » 04e'5tll'. [0 I}, 1 11 » .... h({O 1 0 10011 will ,"'ph ~n appro"nuleloOlutlOll . In
•~ ....ds. 7.11l>e solutiQn
vet) 10 (7.21 II 11Il'''''''In' for t po"" ••. ;>nd the ... " . fini,e .,' vet) . 00: lbe Intc,,'al of u;,l, 1').
value /. ,ur ll Ih.1 Ill", Execul. lhe p",vioul Ihl'« commonds, Doe< 04.' 5 .., ...1 ,be ,'ab", of t "
7.2
Somc NUlIlcrical Methods
In order 10 8"•• onle idea abo"1 how 04.' 5 nkulDlcS IW(\).im.'e , .. Iocs, .. c art going 10 d,w:u," Ie'''''') nume.kal nlclhods. We "'111 apply each melhod 10 problom 0.1).
Chapl~r 1.
,,
Nu",..';"", M..1hOOs
; ; =j(t,I/),
1o < ', < IJ<'
.:~:':~'!:':'~. ~
b.
part Ml the u rne wIdth, or sup siu, /, I,., I,' AI
TIlt Baler Method iJ dOlt 10 Euln and il b.>~ 00 I"'" Wl&<,nl hne Govea the initial , al .. ~, ....., d.fi .... ~, Ittursi,..,ly by
1/1.+1 = II,
.
,)
+ h / (1,,11,),
1·0,1 , . . ,n
\.
11(1,) + hll'(t, l,
by Ihe tange nt hne apprOxl matlOll
11(' , ) + h/(!" II(I, }), II. + h/ (t " lI,j ,
u~lng
the d,ffell'nl i.lequnl lon
since 1/(/,) 'II,
1/1+1'
11IIq. wol"",ioru
... ..u.. 7
c'
~(I ,) '" II, should become he lle • • 00 hel\., as h is taken snun.r
7.1 CoIuidc. !he initial "al"" probkm (1.1): d,
,
" ,
11(0)
I
w. wiIh 10 .ppnnimatc 1/(0.3) uling !he Eul.r Method "'''h MOp .i", h dept.
~lInd
'••" " " "
,
0,
"
. 0.1,
"
Yo + h/( ta,~) . I/IJ 10\ 1/, + ht,/1I1
1.02911.
0.2,
"
+ hlo/flo
03
,
0.1 and thltt
7.1 . .'>0",,, Numerical Method$
Smcc 1I10.:!)
J'j ()'j
93
\.Q.lI030 __ ,"C fiod Ih.ot 1~(03) -11,1
Error Nc~1
U'" /.
001~2.
0.05 and (j "cps:
.,
I
"II>
IOI)2~,
V,
I 001~
V.
10119
1/.
I02l~
U.
103iO
Error . 11/(0_3)
-~,
00070.
If lho: ,n,1Il1 , ...1... problem is su fficicnlly sl1>l.1Ollo", then (Of the Eule. Method one Can Ulow .h,t.M error in "epp,ng from I" 10 any I, ,n the inknillu < I <: b ... li,~ u
Error
r
Ch,
.'i,·",
,t.
,,'hen: 'I a constanl tha' depends on f(I, V), part ..1 den> the ,",ll.' C'OI'\dJ~on. and the mten1l1. hul n01 on h. MOftO'lc'. ,t can be ~hown that the error il actually prop:lftiuoal to 10_ llecao>e of this, the Eul •• Method i, calkd a[orJI rm/"rn"/wd. Note that on Ihe example, clliling the step lite 10 half hld the effecl of CUIt,ng the error appro'i .... tely in half, a, "'rc:cl.d for a firsl order method. It is 8110 u\(!fullll knuw th.l""a/ ,T",T. SupJlOiC u(l) " the ",lu,ion of the d,rr... nuoJ '(lual'On r.alisfying u(I, ) V,. Tlltn lho: local error in It"W,nl! from I, to I, .. is ""fi... d
,,'"
,, . ,
1l{I,.d
V,,,:
i r ., lho: e"or made in ~ ".p assuming lhe w lul,OII , .. I.. at IJ
"1/,.
Usinl! Tayl<>f\
formula II i. ea. ,ly . hown thaI. fOf the Eulc. Melhod,
2I \I ", I, ,.' ' for lome I J € (I" I, .1l. Thus the local mOf " pruportlOllalto h'. B«au~ local c,:,"", pl1)vidu a " mplc C{Jmparison Ilf mtthods and ,I used ,n the """gn of numtr>od we d, IoCUI'. TlIe errOr d,~u,.w ," lite pn:yi;,os l'"rnll",plt is called g/,,/>III , nUT, ,n order 11> d''''"Bu"h it from loc~1
'v(I,+d"~'"
(w(I,.d
~(I,.,»)+(\I(I,.,) ~1")'
(bopre, 7, Numerical M erhods
side of Ihi. ~qualion is the local NTI)r. which is p"'1lOr' 1M doff. ",...:.. . 1 I, +1 of III< IWO $Olulion~ of lhe drffcrt nlLal nIuc$ lIll , ) UId lI, . 1 I, 1lIe ,ize of Ihi . Imn depend. on (he l i« of IIId 011 1M dllf~",nhal equalLQII. 'p« ificall y on .... lI
1,1 Icad. a
r'
UiCWI 10
. . . . . COIIlWIt
10 I,. ",e
m
.1TQf$.
rune. ,,/o J
b_10 I" 10'
_(b - lo} h ,
b , !be &lobaIfrror i. bounded by a eon, lanl hmes /, _ Nnl, .U~ the d!ffe",nual equallon" u/lltablt • .o Ihal ll(I , ~l ) II,., IDly be IarJer !han 11(1, I II, . 1lIen the global error al I}~ I "the Slim of Ihi . llUJCf quantity .00 1M '()Cal ~rror , I ,,., . Thu •. ,n Ihi' , iIIiOlion, tile llooal crror dtpt"\lb 00 the , tabilny of 1M dilf.",nllal .qualion . , ",ell as on lilt numerical melhod, and C""nOI be completely ,,"ued b)o the . implt cak ul.lIOn sl .rchcd I bm.. . II can .,,11 be pro' cd (h'l, as . Ialed OIItia'. IM llobalerror i, proportional (0 h. In Ib .. rf,ult !ilab,hly ,nnr>enrc, the COO"'anl of propIlI1ionabty: if the din ..onli al ~qU'I'OIl ;5 stable, lhe cOIl, ranl "of modernle li« , if IIn'lahle, 1M eon, r.arrt nuy he large. The d fee! of 5tab,\t(y and ," ' tabiir(y on numencal lOIutiOllt will be di..:u... d furth. r in Soclion 7 A 'I1It EUtrMrrtbod '" MATI ..... a . Tht Eult. M~thod can be implc .... n1~d usinB the fol · function M·llle. namc:d my "ule •. m
..wm,
faDatiaa It, yl •
~l.r ef.
t ini t , yini t , b , n l
Il1o aal_ q»zwl='ti _ f o .. i .. ithl value " .. obl_. Il1o tr/4t • fIt, y l , Yltini tl • yinit. Il1o Ai;' '".tlo... y el ) , .. . , y en ~ 1) . ... c . lc ul.t." . t Il1o ~ • • 1 point. t i l ) •...• te . . . 11 i n the int. ~. l III 1t:iaJ.t. b). ftII ript_ b ..... aid. of t lw dltf . ... ntial \II 1 eel_ 1. "eUDed •• at! anonyJaOU. f unc tion f.
* "'.-l.tloo
of
b
f .. o. ti .. i t.
b • • nd
...
t.1atti/D' t. aDd y •• l_lIth . . . i c ol"""" vec t o ....
t
"
. . . .r o a(D • 1. 1),
y .. . . roa ( n + I , 1 ) , , Ca l cul a t i o n of !)Ollie. tli ) aDd tIM eornapOAdlDg '" .pprox ~ t. valu •• yli ) frca the KUla ....thod f o r.ula. tI l )
.. t lnt t/
10'(1) .. r i nl t,
tor1. l , n
tli •
0.'
1 ) .. t l i ) • hI
y li • 1 ) .. Yli) + h*f(t(11. yei)),
To run Ihis M·file. def,ne an anonymous rUntl/on f oolTtlpondmg lo the riK/lUw'KI ,ide of Iht d,ffe,"n!lal ~uallon. choose appro",... ,...I"", r.,.. I~ la\) four "'1"""'''15. ond type
It . yl .. ~." l.r ( f. to, y O, b, D). lf lllcnJlu.lwld.;do"tk§n.ed <wI' Ii.. methods., ..... of ",h ... h l rc d,K U\\n!
ne~l_
'l"h. ,« lQr t w,lI ronlain the po,n" lu, I, . . .I ~ . IIId the '"tC\(lf Y ,,,,,II CIlrIlaln the roI'"" poodmll arrro~,mal'oos !/O, 10'" _ , 11. - TlIrw con he d"pby
,1 ",,',\
The 1l11liali~alion of t ~nd y as ~olumn '"«(OB Ii nol .".nllal. bu( tu'O P" " P""" l·i"l. {t yl ,\ in lobu lar form beel.le t and Y "f<' columnl, Sorond. in,\r l lilmg t and y " ~,,,e ~ the rrogram 10 ",n n\Orr ,melen(ly. To appn)\imDlC (he wl uliOli al a poinl I th.r " r1QI It,. "t would h3'" 10 in,upuldir. A SI mple "ay (0 Ihos is 10 "Conn«l1M do"" by r\r.I.wing 11ral&h,Ii... SCi"",nl, from .ach point (I" ~.) 10 1M 1lt~1 poinl (I. +,,1/.,,); tha, ~(I) woulC'rcd ""Y II> inrcrpolal< " dt5",ihed in S«lion 11.6.1 ,) To gtaIlIl ~( I ). Iyp< p lot (t. y l.
7.2.2
Thc Impro\'cd Euler !\Iclhod
We as" m ,(nM"nh lhe I""semline: "ppro,mlilion. rrpl..,mg the ~Iopc V(I.) by 1M a,'c ra," of die I\\'O >IOPOi V(t,) and y'(t ,+ I)' n", yidd,
y(t .. d "
", ,)+ h
V_(~'li,V(',.d f(t" ,(1.» + f (I,. ,,1/(1,. ill
y(I,)+h -
-2
(7.3)
, , _ _ 1bI.,.......imlhon 11(1,) - y, and Iht I'r.. iou~ liu k, apl'",,,m.:IUOIIII(I,. ,) .
• +11/(1 ,.) IO ~I
J(I " I/, ) + J(I ,+l.lI, + hJ(I" y,ll - - - - 2 - ~~ - - -
" y,+h - II,
+ h i + 1(1, >;:11, + IIi)
y" , . wbae v,'
IU ,. 1/,). Oi,'. n tho initi . 1...lut rJu. Ihi. analym
lead! to .he ",c ursi ,·. formula
, '0, 1, .. ,11
l.
II.. I
'''''1', 7.1 Wf ~ n ",""dr, I'robkm (7.1) and find lUI app",..,m. t;on to y(0 .3) with /I
0.1 We obWn
,
~1+1, i,+J(t"fJ\l+ hll'o)
"
fJ\l+~'2
('U+ +1,_) ~>
I/J
'''"
lit
1.Q..l ~o&l
~,
hlu/II<>
l.OI9~2~
0.000033. Note thut the erro< on the IonpfO" e<.I Euler Method 0.1 ;1 , uR, idr"Lbly thlUl the mo, in the I!uler Method with" 0.05 . (11(03) _ 1/,1
I."
Tht local rlTUf rm the Iml"0,.d Eu k r M.thod i. p",portionl ' 0 h'.
'7.1.3 The Runlf-Kulla Melhod
....
k,
J(I" 1/,)
l-,
1(1, +
k,
J(I , + II2,11.
k,
J(I"" y. + " ~'l)'
"
/, 2.1/, + 2kd
+ "2k,)
~:xlI"'l'l~ 7.3 ConsIder !'foblem 17.11 and .pin fiml an '. 03 and ,ale ju~ 0 ... ',ep, Thrn ljlpO,una~on'o //(0,3).
le,"
k,
1(0,1)
kl k, k,
1(0+015,1+0.15[0))
~I
1+ i.;-(O
0
1(015,1+0,15(0,15)) /(0,3,1
03
" bu,_
011(;699
+ 0,3(01-11>1>')1)))
+ 2(O,I~
OIS U21173!j,1
t D,J llJ(,'J'J)
+ 0,21;735,J)
\.044038"" U(0,3), !//(03) ~II ,0,000007 ,Weo« ilia, threrror in lh< RUDge,KullaMelhod wnll /, 1I,3 IS much Ie" !han Iho enm in !bo Eukl MetOOd. or e'en in thr Imp"""<'d I:uler Method. wllh II 'O.O~. 1bolocal elTOf for ,ho Runge-Ku".. Method 1\ proportionlilO h'.
So EIIOI
HrmMrk 7.1 Tho ,""Ihods we Iu."e dl!<:u.>(tl can ho appliN 'u lO)' fin, OI'drr .qu.mOll, l·urlhorn1<>r<:. !hey can be appl...:! ."ng d,ffe~nl Uep 'lrC ••• tach ...p
7.2.:1
Inside 04.45
SOfl"'D~ for thr numerical sol."on of ordinary d.irre~n1ill
O<jIUUoru ran be I»lo
",,,.ho,h w. h.;o,'. !'feson ..d. as ",ell .. on I>IlIcr .....!hod., Thr numerical sol.", <>deU combine •• fourth order ,""lhod DmI a fiflll order mr'OOd. belh of which .... imilar .0 ,ho fourlh order Runge . Ku"a MelllOO di""u\\.Od abo"e, II ,'aries the "ep size, choo"ng the "ep , il-<: 01 each ,top In Dn attempt 10 .chie"e lhc de,i~d "",uraey. All effie .. n' "'Oolem numerieol ODE '01"." use ,'ari.ble '''p ,i/'< n.. ,llOO., Tho soh'rr ooI.U IS IUI.ahl. for a WIde "3ricly of ini'ial value !'foble"," Thr ..,h'., od.4.S produ.r, .ppro.i ...... solulIOII ".1 .... al the '"'111<11« of poinl. de· lennin"d hy '" \.01«1ion of s,ep .i1.<'; ..'r 10 !bose: pom.s .. intemallime poinll, Surpo", you h.'-e Kque,led appro.i"", .. !I toI""ide ",nb lbe Intrm.11 lime poml>. aDd InlerpollllOO mus, be u~ 10 gd approum."" «>Iulion value. "' Iho po,"1> 0 I 0 • 2 12 1l\( InlrrpolallM IWIX.durr coo_IS !hr .""nUI' n""le soluMn '''lues .,!he imrmal lime poinll. tt<>I ",1lI .1m"'1 II .... (lmear poI)-oomills). IJ menlloned for !he Eule. Mrihod. bul ""Ib hlp, dtptt poI)'nomJlIl. 1l\( !\lm3I. solutioo "dues 11 lbe "'le",ollll1lC points ami !he intrrpol~"00 proce.. In ,uch a wly thallhe totalemll in bolh p'ocelS"1 doe. no1 ex"..d!he dr"rcd 1e,'.I. by def,ul, 10 'J, Fin.illy. "ben od.n i, \lied wi,houl poinl' being sprcified. ""'. for Cl omple. ""ilt oddS!f. (0 21. 11. "pptoXimnle 50lUl ion v.lucs "'" computed a. lhe mlt>n.llinlC poin'<' .• nd!hrn by in.er· pol41ion at lhru eq".lIy ~p.ced points bel"''''' e .. h p." of lhe se: PO'"'" In thr g,'''ph produced by Ihi. romOland • •hr "J'IIlfO"m.lc solu llOn .-.111<1 01 .n of these: pUlnll are Indl' c.,,,d by Circles. and.1Ie "rnlers of !he circles .... lhon COflIlCCIM "nit h ... "'pnlS (iff
,d.,
F,gu,.., 1.1 I.
,"I .....
".
. MATLAB 'I>I;lud., two mher numeric,l .oh'cI\, 04.21 and ode113 . lltat Un ~ U>N "i(h,he pnlhl.m, in lhi, book 1'0' .om" i"III,1 value problem' (.cf. lfC-d 10 OS "riff prohlrm,l. 1he ",1,<" o4e45. od.n, and odelll on: 10.5 "'li,r~l"'y 1h.n IllCll>Olh thll haH be.n "''''K'''''' 'l'<"'ificBlI)' 10 . oh '. ' liff probkm•. MAT1.A B 1II':I .....~ fout .uch ....1..... , 04e15 •. odell., ""e23. aoo ode13bt. Addlllooal inform.>hOtl on aU of lht-.e ",h." ran hc In tho online 11<'11' aoo '" 1M hrow~'. Rlr Oldtllllonal ,nfOnnalH)rl on nu meric.l me1h".h and lllti. \oft"''''o IInpicmrnl.hon', ... ref.. to 0 " aha"" •. C, Mok •. ""d S. No,b. Num.'ri<'llIM.lh",ls nnd s"nwn .... , l'relll,.o H. II, 'nc., 1989. E>1'C" 1 ~lly ....'ommtnMd arc I"" boo~' lhat (<XU! on "umcrical "",!hod s In MAT LAIl L Shamp,"". I GI.dwdl. and S Thornp""'. Sohlnl: O U&5 "hll MATLAI! . ('llIll>od6< Un"CI1iIlY Pn:". 200.1, aM (' Mok •. Numtrkal Compull"~ "Uh MATIA lI. SIAM. 2~ .•11.0 11.,lable al http I"'-'W. mathl
r"un"
7.2.5 Round·off Error
.'''B
.bo>.,
n.,.lyl'" or.rror d,,,u'lN il called diK tr';Co,,,,,, ~m:>I". and ,,'auld be preJ.Cnl 'f ono ,,,,,Id ",l;I,n "" Infin,t( number of d'giIS. In .dJnion. the:", 15 " ",oJ 'r>/! aror, "hlth am« """au,," Ih. '(lmpUlr, ulel • (.. e<J. Fmne numbe, or d' gits. I...,mn/l II. (r)
~.( I )
fll(11
~.(I})
+ (11.(1)
iI.(1})
])l\c",h,.:.lwnIOrrof " Round.off Error, Sin« """1 ",,,npule~ Ih.1 run MATI.AI1 ca"y IG d'glt l, lhe major pon ion of the enor will ho th. di'c"'I"~li"n rno<, Thul. fo, m,,,1 problem •. ,ound·off Con be lafely .gnored, I" • •1>" "'a\llll ". w,lI "'ll d'llIng""h bclwttn 11.(1) and II. (t).
."Of
7.3
Conlrolling the Error in ode45
A, indrcaled am", •. 04.'5 aU.rnpl. 10 I"ovidc a" ~ ppro.irn'le .ruurron ""Ih erro, ap· pm"""'lrl)" III " A 100... acCUr1Ile &pprox, mal• ..,1"1100 can he ob1."",d by "loIn,:ur <'f'I'oo,] f""nh aq!umenllo 04.'5 11111 "'Iumenl i. c",.1«1 usrnM 1hI' 04 •••• com, m.md ~or n ..... pl •. to com"" •• an "1'1"""01.11. IOlullOI1 to lhe rnmll 'aloe problem (7, I) . 1I "'q""I,ng 12·digit Iype U optio.al. 04 •• lt('AblTo1', l . - l:l, 'R.lTol', 1._ 1:l ), u It . YI] • od.,SU, (0 :ll, 1. optionl], ~> fo~t 101111'
1.
u
"""U'''''y.
ya(lN>d ]
1M'. ,'omnWld, rn>doe. (he OUIPUI 2 , 236067tl77,199~1. " h,eh al'l'ffl~tma't$ the: tUl,,1 ~4lut ,lr) I" 12 di~'l' Ab.Tol 'I"'"itl", ab'olul. erro,. and R.1 Tol . pec,fic$ rei "IIVe ("or, Wllh the !oC1I rng< 1.-0 for AblT01 and 1.-r fur JI.1 To1 . .. he", a and r ne pOMI"'e ifl l.¥(", od.'5 (oes
99 10 IPI'rox,m.:nc 1/(1) w,th uror less \han Ulax( 10 '.I~(llllO '). W,IbouI thr~ option, M"TI-All U ~~ the .Tollnd 1_ -3 fO/i Rd Tol, The
7.4
Reliability of Numerical Methods
ad_'
We have ~laimcd lhal 5, when uled .. ith lhe deflull opl,on., I..d. 10 apt>foxim.:l.lcly 3·,10 8il ""C Uf"dCY, MOle p",ci~ly, the defaull optoon. rn,o", thellJ
11,.,
I"'" pour .. , ulls.
!I '
3c
"
1/(0)
,
(HI
'TlM:uaclwlutiooi.y(l ) e-". Now "'. find 11M: numerical wlulIO/Il u"ng 04.45, and p~ il ("ith a dalhcd \,ne) "'g_the, ... ,Ib tilt '~lCt ...lul1on (.. ,th. soI,d I,nel: ~~ f • !J(t, yl y - 3 ° .xp(-l Ot), » [t, yall .04d5(f, (0 nl, III
» » »
plot(t, yal, '--'I hold on •• plot('axp(-l Ot)',
~~
hold ott axh( [0 12 _1 11)
» ~~
»
[01111
xlaHl ' t ' y lab_l 'y'
.~ tit l a "
f a1\d arcindlII1nK",$h3blefromOlolboul W. ~e<: f.om FiIUn: 7,3 thaI the: grapltl u y al ".I~I I! I 12, (Ot in'1aoxc, an: '-.ry O. hul lhen the cu~1 \Cparate slw»1y. ~If , _'lO 3 71 , ~ 10- "), dirr. ... nl(y&l(.nd) -1.29 . • 1\d ¥i ) ,
". •
••
•
"
FI'~'" 7 I. E..d i>olid linel ~nd Al'I'fO,i ",,,'. (d.,hod hn~) SolutIons
How ,-" III .. '''h.... hc .,pl.;""d' The w ll/llon !O our d,ff• •• nhal rquation w"h mnw CUDdlIIOII \lIU1
I I • " ~I t)
,
~'ni -."n.lI.l ,al".. I (D..S' to!>
III
'1
+ fO· I ,
H~u'" 1 ,4 ~'$ a plol of the ",Julions
10"'1, 1-(O,2~,
10 ~), I. I+ {02!':i" IO -~), I +
)
w..... Uu.l."'J>
I ""plAIC 'Cf) .Iw(IIl I,,'m 'he ,olmioo ",ilh ,n"i31 , 'o lue • • ~ lly I: ,110"" wnh imhal ,'alue, pt'ltfllh ... I "l'I'fIl""" I. nnc. on .uch a 'OI,,'iuo. the nurncrical 50lul iOll will follow it ur ...otht, I.. ,h ,,,llIun" Son,. til ... w l"lion, 'I'Pro:t\;h ""'. lhe nume.lCal soluli"" doc. I~' ..
(,,,,,1.1 ". h"", anllk following lhe: soIUlIOI1 III .ll.mrl\n~ In <.kubte . R«all aho Ihal an tquallOll d, / dl 1(1., ) is un U I Alllh" '"~~<>I' <"uII OO "h. n dealmg ""Ih unll.bl. tquallo"" CaUl ioo " alwayJ monmmtnd Sup!'""" ",' ha,'c lnadt. cak.IMio" u"og I.... e rc3son.:oble
n...
,:01""
h.,.
" "
'"
I I
•
• •
-
• •
,
•
1
••
••
•
•
• •
•
conh
thai they Be both .... ural.; bu1 If lhey d,ff•• ,"MUlltWly. WD one .t.ould ~u'fI'''' th., the: fori! cakulahOll;1 lIOI .... unl•. 1..1', do Ih» WIth our uamplc. SprcifICaily..... cakula" and plot the n"IIIt,;,,"] ooIu· lion, Y.~. 'aI .. fy;"& rnon: ,tringenl lCturIC)' rtq",rrllltnll, lOj!
» » » » » »
-- . ,.
plotft, y&2, ' ) •• plot (' .KPI _2°tl '. hold. off .... b([O
xlalMl ylalal
1711
., .,." '" .,.
»t1t1." . W. >« from I'Igur. 7,~ (hal the secood numtncal Wlullon ,utennll._ o".ra long« onl<'\'ll. OU1 10 uhou, I :1, Whcrt3~ 11K: hI'S' WU ;0«"11110 0\1\ (0 lrout U. In pM!!c"111. !he II,,, nu""''' . 6 o f ' .0., ',lnOVo'!hec,"etsoluuon, "crouldhl\c cal 'UIDUOI" differ after about . ",-cn 1 "c ~,un .. , " 0 1>111 nOl be_ond. a,pro,nn'lrly(l. cOIl
Cb.pr~r
7. Nu",.. ricaJ Melbods
:r• " •
,
• ,
•
"
•
• FI'u~
•
•
"
.. • j
7.5: Eff«'! Ilflncrru"d Accuracy
!lilt 1be ",dUllY of • nwnericIJ ",lutioo can al'" ~ chtck~ by rom""hng an/l!hcr nu T
r17' lOIut1oo willi. mo~ otringe"! ..,curacy l'nlu'~II .. nl. 8t\aalJddlIlOOll waY'
'0 u1.C oddS ~ d;\.Cu~~d in St<,;oo 8.6.
Chapter 8 Features of MATLAB 'r~'~ d,"p~r
" a COnlin~,'ion
1)( Ch~pt.'"
1 and 4. W. dcscn bc """'" of the more ad,'~"".d futu, .. of MATlAB. focu'in, QIl MATLA" «Imm>nd...d l«:hnique, lila, art Him! "",(ul for ,md)lnll diffcrtn,ial ((juahOn,
8.1
Dahl Classes
"''Cry ,-.nlbl. you define in MATLAB. as .....11 u .,'(1)' '"PU' and OUtpOJllO a command. " an "rmy of dala belonging 10 • pMticulu rlaJl. In lIois boo1 "'( Ul.t primanly fourl)l'" of dar.' flontm, poi'" numbers .• ymbolic . ' prr."on,. ! for cach Ill" of d.>ta It. da.~ (II.. g"-(I' by the command ..hoa):md bow you clll crt.,< il.
Ty"" of dal. l~o.1in8
p
('rUled by
Cla« double
.,.
char function_handle
.,w...
'lpwi' numbor ""nB .". OI lyaoo IYP'"I I min, ,uidc .ingk qllOl ••
ul,na·
Vuu CM think of "" amoy as d"orn,;ooll '"" of dn.; MATI..AS IIleM'. am)1 !O h.,.. moxe Ih:on IwO bul ,,'e " "II IlOl "'" lh" f~~r"re, A IIIIgI. number (or .ymhohe up"",ion. or fu""tion handl.) " ... ,artltd by MATIAB 0$ a I.by-\ amy. '0",..,1,",1'1 called a Jml,", A I.by·" OITllY I. ~olled a""" In,,,r. ""d ... m-by-\ ;uny " cn lled a col","" Iwlr>r. (A llIing ,. ""Iually a row , ·",, 1Or of characters,) An m·by." =~y of n,IIlIl><:fS i, co iled a m""it;~. Stelion 8 4 !>elow You can sec \be dos. and amy'''''' of .,'ery ,-arlahl. you ha'" dtfincd by Iypmg ..hoa ("". SOCI;OO J6). 'The s." of ""mble dcfml,,,m< 500.. n hy ..hoa i. calk<.l y"m m,,""""c~.
<.1;""'""00'.
'OJ
""
In order to u>C MATLAII y "",mmas; often. )Orne argument. :orr opUonJI. Wh1'0 Section 2.S) for cach com'n~11d u,u.ll) (db" hal da,,,,, of input. lilt command up« 'I, as "'ell., "I'l>l tia>. of OIIIP"1 II
..,,,,,,,mn, SQ.,,.
""urn.
\bn~ romm~nd, all.",' more than OIIC'
of mpul, though somc:Umtl only oroe !bID d", "rncnllonrd In llIe onhne hdp. Th" fl.~illlhly (all hc • cOII'cmooce m!.Orne c."", Jj,d • rurali '" III"" .... For eumpk. a, we Sa'" III SeWoII J.S. a.plot "ilow! )OU 10 '1'«11)' tM funcuon 10 be: ploUed., a ~lrmK, a symoohc np"'sslOO, or. (u"cllOn hnndle. On 1h< othr. hand, \ul'(lO«: thai IOU "",'e prcvlOO,ly defined I< - ( 5 7 J, and now you all~mpt 10 ,ale thr den'all,. of JOII:r by lIpmg H 4ift (log!l<) I
'"' 0.3365 1lIC rcawn )'OU don', gel an emil nl<,\.3g. i, thaI 4itt IS the name of I funclion Ih~1 U>II1J'UICI no ..... n,·'1 d,ffe",,," ql.ouenu .. "ell I ... ymoohc den'~I"'o, Smtt '" I, I 't<.:l'>r, W L~ log (1<', and the no .........·.1 ,rJ~;on of 41 t t "apphed In ~ne.aI, "hen the OUIJ'UI uf a cOnlmand dOCSn'llool rilh, IQ you, and MATLAB I;"OS 00 error .... "ammg "",,'>g., )'OU ,hould cOII\ i
Mf mark 8, 1 1}pml h.lp 4ift only lholos YOUlhe hdp luI for lhe nurnencal ,-enion of 1110 ~umnlalld, bul cro,.. -",fe"'lICC 10 lhe Iymbohe ,','mon ill lhe bonom, If you "~nIIO \ott Ihe help Ie" for IIie .ymooh, "e"ion '"Ilead, type help eY1l/~itt
Ii'"' •
Someli"",\ lOU nred 10 coo,.n Ollt dala dass mlo anOlIle. in ordoer 10 prepare the OUlpul of {lfIt comnulld 1<1 \(n' •• , the '"pul fOf anoIller I'or nample, ". "",. u"!d doub1. 10 ~Of".n I) UP'U'UlII' 10 f\o;ollng'poonl numbc~ and a)'1ll0 con'ert numbr .. or \In"¥IIO ,ymbolic .'p"'\lioos. The cQfllmallds "UJIIlat .. alld at r ln. . con,'.rt bel",.cn rI"mbcA and sirin g', ",h,1e cha .. COO,crts a symoolic np",ssion 10 a \lnng, You can abo U,e vectori ze 10 000""11 a symbolic e~ pre~sion 10 a "CClonlCd slnng; II adds a , before ncr), · , I, and· m 1h< c~ pI"C~,ioo
mhoI,.
8.1.1
Sl ri ng M lIuilJulalion
Oflen 11 i, ulcful10 C(JncaICnalC 1"'0 lIT more UOII, keeping In 'mild '''''I a Mnng" a "row 'eelor" of characlC~ f ..... e-
""
Here ,\ a u\Cful al'l'h~~1JOn of .lnnl (Oio~n •• ion You ma, ._ , . (> "_, . _ u< lit a mini ,an.blt contDIn'"1 an e~prc'llon , .... , talel more than ~ 1, __ ,•• , ({ . ~.~ .~ '" pc. n molt meum-
'laolC~1 you car, COI~'lnll1: your mpul ontu.t.. next hoe by ' fpml '"
follow~d by ENltN
or RbTUMN, bullhll IS not allowed In lhe IlI,ddle of a .'n"8.) Breu lilt e~""'I'iOfl onto 'malier part~, "'1'1" a '-anable 10 nch pall. IfId COllCa~n.'e the .anables. as on: lb_ • 'laft hand .id. ot -.quatiOD'/ rba .. 'right "-'>lI aid. of aquat i OD', aqn .. (lb. ' .. ' rha]
~>
» »
."" left hand .1d8 of equation K. I.2
right hand aide of equation
Symbul1c und FloatinJ,: Point Numbers
We mon"oncd aoo.'c 1113. )'011 Can ",,,",,crt btl"'~n symbolIC numbtr! and IIooIJnl point numbers with double and.~ Jl;umbulO th.,)OU ')'pC' m. by derau] •. nOllUn. pClllll Hov.'."cr. If you mix .ymbohc and nOlI!!", l'O'nl numbt", ," an antlunthc ex~'''on. lho Hoollng p<:>,"1 nurnbc:r!i a", aulOlIlJI,ully con'·clted 10 Iymbollc. This ~~pl3Jnl ... hy )'00 can Iypc aY"U x and Ihen x -2 .... ,Iimul ha_ing 10 coo"er1110 a symbolic "umlle,. Ir.", j. anOlheruample: » a .. 2
•
,
u
b .. a/'Y>a(l)
o
'"
MATtA II wu ik,iH"",J so Ihal wme noaung [lOInl "umbel'$ .'" ftltoml to lht,r <\1<1 ,alues whotn cw'cl1cd 10 symbolic. Inlegcl'$. r.l1,oooi nu~', ... ,th lItull nUlIl<'ral<>n .nd iknom"'alon. loI'Iuare root> Qf .mal1 ,nlcl(rI. the numbe, 11. and certain combin."oo, of
lhese number; are liO re.lore
2.2361
.". »
aY>alc) ~
aqrt IS)
I'
bell
S"''''' n 'S d,lhculi 10 predlCi "hen MATI..AO ",II pre!ltf\" e,XI' ues., 'I ., ",ulIl<'nl 10 ap 1». (AClo"n, II In ,ul'f'!'Cn Ihc noah"g poinl c, .. lual'OO '" a nulll<'nc '" II ""~ 1'1 . aq r t (5)') ('" "'" on I'ngle 'Iuole_ 10 male ,I 3 Ilnng. t.g·. ap . nnOlloer wily on whkh single quolC$
IU .3
Structur('S
II Ie' Anothc:r data clau Ihll )·00 ... ,11 OCCl
In
"'ATLAS is • J,rve, .....
""
" 101 _ lolveC'x. Y"lI _ 2'. 1,,1
x' Y If lOU
'y - 3· .. • " )
~Yf
~
yml 1:108 801, lou \.CC WI t~ dlt. da~. of aol " ·'.truet am) ". 1\ structure ~an ~"n\aln an) numl>cr of named '"allablt~. called '1i~lds", \~tlh<-r "uh their .aluo:s. In Ih,. ~a", .•01 ba, 1\'" field., x aoo y, and cach " II ,«to, wntai",nll'w" symbolic numl>o:"" To .... ,.~" a 1'3",<,,13' r,dd. \H~ 11>0: n3n1e of 11>0: '1"",lure fllll""ed by the name of lhe I,dd, wnh a pcnod in 1>o:',,"cco. Thus. 801.x " a ,'ecl", wnla,mng the;r ,'aluc$ of Ihe two ,,,IUIIM' of lh~ 1)" '0'" of etju'lions li,'cn III .olve, and 101 ... ( 1) is the first
')I'''' ..
",11
""milt, in Ih" ,'""I", 1\1".., ,.nerally. a \\,ue,,,", can cOIllam lllulllplc dala cla'W'I.
l'Of cnmrlc. one field ~;I!I 1>0:. 'lnn~ aod "'I>lIIt, can br an .m>y orllo>li"1 POint numb",,!;, or. fUIIClIOfI handle. A tield can nen l>o: '1K'II1>er mudu"" An unmplc of II mo ... (»uplc. \Iruet""' like 1111, IS gi,eo b) od •• 5 ... I>ro II~ OUlpUl IS I>sIgned to a sioglr ''3I1abl.; \Ce Seclion 8.6.1 ~Iow. Ahhou~h In th" I!oo». .... "ill u\C ~trllCtPn:~ only "hro!he) occur u MATLAB output, lOU ,-.n ddiO<' yoo. O\\n \UU<;' un:\ with the conunand nruct, ~ the onlonello:lp for
.",mpl¢,.
8.2 II'.
Functions and Expressions
h"" u,.d Ihe leml' rll'"nrotr andfimcrrOll ,,"hQUl carefully dl§cu
""'W""O IhrlWO. Slriclly .pea long. if we define !(z) zl I I. Ihen! (wnllen wilhoul any p;onlcourM', " C oflen blu. thIS dl~lIocllon by calhng !(I) D. zl + I a fun<"oo, 001 In MATLAll 11M: d,lT.rc""e fuocllOl" and .'p.c.,ion. is impollaol. In MAl LAll. an nrre'\loo Un belong to enl>t, II>t min, o. 'ymoohc dass of data ('oo,,,k. tbe [un"",ngrumrle » t • ,,,All . 1'1 fIJ)
"",,,un
,'"' TIt" ... ull mOl)' ht puullng If IOU are .1pe<;lIng f 10 ""I 101•• funcllOl. SInce r
,~a
string. t(31 theth"'l char.>lIn8 of onc charac,.r,) TIping r ( 5) '" ,11110.1 yield . and t ( ~ 1) woold p.oduce an eftor "'..'ngc. 0"" way [[I ,e, Ihr YI>U ha'"c 11'1" ,igncd 10 '~fI~hl~,·
»
'"
x •
3/ .".l(f)
S. I.
M,,(~
IIbour M·fjles
'"
T)'pm, f rut .... , than eval If) abo,-c " 'ould Iu,·c "mply OUtput tile e.prts~ion f w.thout ~ub\IL1UHnll 3 for .. aoo e"31uahng tIM: ~Iull 1"Qf a .I11ful;u ~a.on .... used eval .n Cl\Jpl~r ~ IU define I funclioo from tile OUlpul (If. symbolic ton'IIMnd_ You eM 11<00 \uh,tilule ••'a1"" mlO an urression uI.n. auba, II dtlCnbed In S«tion 13.1. You h~"e lu"",d Iwo ways 10 dtfiroe YIJU. O\O-'n functions: 00 \be rornrnand II .... ulm, e Ill' ,~~Ic In "anonymous function", Of bUD. (Ke Secllon 3.5); and by c:~.un, a /unclll>n M hk (see 5«1100 4.2.2)_ Anonymous &n.d 'nllne fu .... tlOOs are ~ u!.eful f(lf defining "mple funel.ons Ihal can ~ ~\pn:sJ.td In ~ hne and f(lf IUm'"l the IJUtpul of . ~yrn""li~ ~ommand inl(l a luncl,,'" Funchon M. llI01 "'e u!.elul lor dtfin'"l IUlICliool Ihal '~IUI"" sr>"cl1t l inlcnncdi~le commandllO c()mrUI~ lhe oolpul Mo.1 MATLAB comf1Wldl arc aclually M II le\, aml you can reru~ rhein for ;den 10 Ulc in )'011' ow" M. r.1e s- Io ~ Ihe M !ile fu., ' 8Y. lhe rommand _an you can .nl.1 typa . . an. Sre 1110 Seclion HJ below ,~"IlIC: c(~nn,"nd •.
loch as cx'I •• S. r'flJUIf. lhelr fiul UJu .... nt 10 be a funcllOll II you lyre an .~~I\ion ,"11.00. YOll ""II 1.1 an error .... IU~. On rhe otht. hand. rTI(lIt symbolic commands. such as .01 .... or ,boh·e . requIre their IiBI arxumonllo be .,the. a ,'ring or a symbolic uJlRl$ion. and...". luncllOO An Importanl dlffe~fICe ber,,-ecn lInn,s and I)'moohc uprtmonl II thaI MA1LAB aUlom:llluJly ~ub~lItu~~ u~.-defined funcliorll and "ari-,>lcs ,nlO ~ymbolic c>p.tlsioo thaI you I)'re. bul 001 InIO mings . (OIi l il anoth.1 ~n\C In "hlch lhe ~Ioglc '1~lllO t)')lC around I "nng ~urrre~~ C,,,,IU3Iion, ) Cooslder lhe fulk~""ni: cumplc: » h . elt) t. " 31 intl'hlt)', ·t·)
Warning: Explicit integral could not be found. >- In Bytn. inl; at 58 In "hilt .. in!;. at 9
'" int(hlt). t)
MATtA B ~anOO\ culuate the mtegol bc:cau!.e .. ,thm a Itnn,. II. IS regarded as an uni.nO'll·n
fU",,11on II you make (he 'npul to iDt IpnOObe.l.hen MAT1.AB ~ubslilule~ the 1""'-'011' defimlion of II. hefo~ rerfOfuun, tile mleg"'''OO' » _ya8 t/ int(h{t), t)
'" 1/4't·4 8.3
More about M-liles
II 1M fil., Thel"C'~ twO kind, of M-lllc, h ks C()nlUlIlIllW ~lATI.AI:I ~a(c",.nt' are ca ot rod~ 0011"'1. and ~cripl ",-filn. wh'''h M ·filn "h1Ch :>CC'<'J>l aIgo>nent, and P d . ' d'." ""'s In Ihi' , E",I":I"~ C~3 1. anu 010 "..... ~-. uec:utc a >o!rlCl of MAT1.AB lt3lCn,.n I. -I« hon "e I""ICIII IMkhu()nal ,of""""I100 00 M liles.
j,,,,,,,,,,,
,os 8.3.1 Vurinhtcs hI
~ rillt ~ l·files
W~n)oo run I Knp! M fik. the ''''';3bl~s yoo u,," and dell"" ""long 10 your Worhpace; i ~. IMy talc on any ,aluts you ass"nnl urhCI on you. MATLAB ~s;on. and lhey ptr.m af~f the scnpl finllhrs c..uut10, Conmlel 1M foll(M,ng loCrirl \I.fi~. cal1M
scriptexl.m" ,,_[123t)1
1)'1"118 ecripteKl a\~lgnl the g"'~n ,ector 10 u bill d"pIJy' 110 output N"", cOnl,der anolhrl SCflpl . called scr! ptex2 . III: l ength!u)
If you II". nOI p",vioully defined u. then 'YP"'i sc ript .x 2 .... ill prooJucc an CITO< meso sage. 1l 0J'0"evcf, if you type scriptex2 .r,c. runmnll scriptexl , then the dcfinltion 0(" from the fir)1 \Cnpl ",II be uKd ,n lh< _OM .... "1'1 ~nd the outrut an• • 'w,1I be " " played If )'00 don' , " anllhe OUtput of a !.Cop! M file to depend 011 any earlte. cornplllaliOlll In )'00' MATI.AII ""\'011. pul the 1m. c l •• r all ,ltal' lhe btginmng of the M.fil., a~ 'H JU"'~IM In S«lion" 2. I, s.om.U~1 you
IIUghl "anI your ",npt M·filc to IOC'a:pt mput! mlcract"cly, You can anani~ !hll " nh I"" command iDput kif .umr~. ,f you mdlklc lhe 1m. v a .. .. i npu t (' Input var h.ra, ' I ,n lour ~nl'l. ",h.n MATI.AH setl 10 lhal polnl ,I ",II J'Iln , "Input vllr here: " aDd paUl( ",h,1e )'00 type ,"" "aluc 10 be D\~i8""d III v a .. YII\I can 41<0 fOfmal )'OIIr 01111'' '' in a more de<.<:npt"'c manner than an" .. 40 by 1I~,"g dh" n d h ,, ('The l.ng th of II h " I , dh,,(l.ngth ( u)) The length of II i8:
•
linc. yoo Can coo,'cn lhe numeriC OUlpul 10 a ~lrinl w,lh ..ua2.t r and cone_lena", the ,mngl:
Til 1<1 lh< outpul all on
0I>e
» d h" I I 'Th. leng th of II i . The length of u 1.. 4.
8.3.2
' nual.t .. U . ngtb lll) )
' . '])
VHr lllhles in ,,' unc t ion ~ 1 ·liI cs
~ ,'lIri . hl .. UloC'd ,n a funrlioo M·fi1e ore loml, mcan,ng lhallll<=y arc unaffected I>y. and ha, . nOd fN ' QI', the "annbl« '" ) OUr WOI ~ 'P=' Con,ider Ihe rollov.-mg funcilon M.m., called uQ.m: fu.tlc t1 o n a .. . q (><) " .q (><) ... t u rn. th • •Q\lar. o f x
a .. IC . oa,
l'yp,nl . <1 t )) prod""Cl 111<= ani ..... 9 . """Iher
W", ~~. a,id dOC'~ "III chanl<' I""i, defin,"ons
Of not
x or a 'I already defined ,n )"'"
8.J M"'ricc;
8.3.3
109
Siructure of Funclion M·fiIcs
TM fil'>l 1m.. In ... fUn<;lion M·file I< tailed ~ /lmo,,,,, Ikforlllon /"'t; II defines lho fullC'ion name. U' "'fil ft • • he numMr and order of mpul and OUlpul arguments Folloo>m, lho functlO" dehnltlon Ime. the.., tan M se.'era! commc:n! hllt\ th..1 """n ,,-nh _ pem:n, s'gn I"'). TM!.C Irnes arc called /rtlp It..:, and are d".p1a)ed In "'IPOOse 10 lho Hlp CQIIIIIWId In 'he 1\1 file sq .m .lxy.e. the,e I; onlyQfIC Irllt of hell' leAl. It II dl,n the MATI.AB ".temen" tha, cakulal. the fuoclion .'alw:s. In addition. the.., un be COIrUIlrnl lines (1i1lO1 bc,inning with ') any"here in an M·file. AlIs'aIC!lk'nl! In _ function M·file Ihar DOI'fIlally plod,,,e OUlfllit Ihoulll end wilh a semicolon to SUpr"'lllhe ""'Ipur , FUI...,lion M ·m.. can hn.·. mulliple iopul nnd OIIlpul Wjumc:nl<. Here is 10 cArunplc. called polarconve rt. III, with IWO mput and IWO """pul ar,umc:ntl, funct ion [r, thata, • pola",conva"'t(", )'1 , polarconvartl", Y) ",aturns tha poll'" coo",,,,l,.ata • ... of tha point with ",..,tanlJ\lll", eoo","'illlta. Ix, y) r . aq"'t(".~2. y. ' 2)1 theta • atanl(y, "II If you Iypc pol.",conva"'t (l,
,
,
th et a
0.9213 If you a'''go lhe I)Ulpul to only one >anable. II Will rt«l\'C lhe firs. OOlput "'lIu ..... "I . ..,,ardlc.\S of ,he ~ariablc: 's name; tYI',n, tHta • po l . "'CODva "'t (l, a".l>le~ "nh lhe SlIfIlC na ..... in )'oor "orklpxe.)
8.4
Matrices
..... R nd ro/U,M .-celon; ",hiel>"~ d,~usw.J A "'illm IS a rcelangular aml) 0 r ""m""B. f1W a . . In ScCliOO 3. 4. af1' nnmpl~s of malnn:s Co.mdrr lho 3 x 1 matn>
A
",til
,
3
,
7
to II
:)
"
It can he ~ntc.-.:d In MATLAB !he ronlllund ~~ ,. • 11.1,3, " 5,5.7,81 ,,10,11,13 )
'" ,
,
,
,
, 3
•
9 10 11 12 NOI~ (hall"" ",MO' ,I,m.mt in any row nil' ~paralcd by ~Onllll~$. and tl>c row. arc .><:pa",ted hy ",mieoloo•. lllc ~ltmcnl' in a row can al
.,00' of • main. CM be found" uh .1". ~>
.
. i •• CAI
•• ,
=
",e,
If hH) m~lnc ... A and 8 tho W1lI: the" sum ,s o/Ita"",d by 1)1''"' A • 8 You can ~1j.(l3dd a ",alar (a single number) to a ",atri.; A"" Ooddl C \0 ..""h .. Jerrent I.,I.. .. ";"",A _ B rcpre..,nl\lh.,hffcrenccofAandB,nndA - c~ubtnlCl'lhcnunlberc fmm <,,,,,h oIemenl uf A. If A and Bare Illuhiphcalj,·oly COIllpalOblc; !.r .• if A " " x It! mill II j, ttl " I, lhen lheir product ,.°a 1~")( fo Tbc product of a number c and 1h.. malri~ A is
In '"
gi,en b)' c°A. and A' "'I'f"<.cnl'o (he COOJu~al" Ir.Inspo5<' of A. 1'or",Of" informatlon.!« tho unl ..... hdr for ctr..... pO • • and tranel'O •• If A and a "'" the...me ;UC'. lhen,... · . is thr .I,m.m,b, -,Itm,or product of A and B. i.'., tht main' ,,~ I, J element l\ the prod""1 of the '.J clerro::nt. of A :mel B L,k~,"'. A. la " lhe el~ment-bl ,.lemonl '11101,.nt of A and B. and A •• C 1\ the nt31.i1 fomled by ",i"og .",,10 of lhe d."...nu of A 10 lhe C Mon: gcnc:",Uy. ,f r I~ one of the bu,lt- on
1'<"''''
m3Ihem~I,,;al
fUIItnce' on' \,m,lu ".y. T)pong All, (l til y,dd. lhe ",rond and founh dement, of the .ccond rov. of A. To ",1«llhe 1eOOnd. th"d. and founh elemenls of th,s rov., 1)1'<' A (2, 2, t ) ~ , ,,btI,.u,, con,,,'mg of lhe okmcnt; ,n '0"'$ 2 and 3 and In c"lumn' 2. 3. und I "gencfllltd by A(2, J, l,t) A colon by "",If ""ooIc~ an enti'" rOW 0, <:olumn For example. A ( " l) denote. the second column of A. and A (1 , ,) y,eld' lhc Ihml fOw of A MA:ILAII has I.,·eral command, that g."cmle _.peelal mau,cc!. The comm,,,,ds •• ro. (n, III I and ona. (n, II) produce" " III mal,oce, of '.'01 and 011.5. ,elpec\1",1)'. Aho.•ya(n) "'pn:SO'na lhe" X" ,dentily ,,,,,I,,. In 5«110111.2.1. ,",'e u~ •• ro. In the I'>,-fll. myeu 1er . m 10 ,,,,hal,,e the output ' ·CClOll.
>""
8 ...1.)
"..,.1
Sui t ing Linl,':l r Systems
SUPI"'''' A " 8 nonlingular 'l x '1 maIn. and b " a column "eclO' of length n. Then t)'P'"1I x • A\b nume,ically romputn Ih. "nuluo .01""0" In A"x • b '1')'fIC halp .. ldivida for mo..... inf"",,""on.
8j Gr;ph'ci
r"
If e,ihcr I\. Or b ,I ~ymbohl rathc:. than n . •• • . nmo'le. u",n x • I\. \ b rom ."• to I\. x • b symbohcally. Tonkulll. Ilymbolic I PUle! U", wlU11Ol1 type x • _)'11 II\.) \ b .., ut,on "'Mn both inputsare numofi<"
H.4.2
Culcu llll in): Eige llHllut'S und Elg('nH't'lors
The ~tgcny~lues of I "1uate matri~ I\. Ire eakubled "nh .1 I\. . pIIt ' ~nabk>, a~ In (U, RI • • 11J(1\.) lhen MA.... A8 ~.' J 'fyou IpcclfYI" OOUI and ., , , 'L c .. culale! both the el~ . J e'l!<' n'·ec IQl"~. rr.. e'gcn'~lues ~ lhe dl~l!OOaJ demonl l ofm.
':c :r
"""
,,
2
»
ru.
U
•
"
0 0
0. 4082 0 . 81 65 0 .408 2
1.0000
, o
o
.iIJIA)
o ·1
0 . 81 6S 0.4 08 2 -0 .4 082
o 0
2 The e'gcn''«IQ1" in IIIe filS! CQlumn of u rorre;pondl 10 lhe "gemal"" In m.1,m rolumn of R . and 10 OIl These are nu"",rical ,'al""s for the cigell»>""', To g. , symbohcally eakul ... d .,gen»>,rs.l~pe
8.S
0
(U, RJ • • ilJ(eYJI(AJ I
Gn.plties
In &CUon 3.8 "C dl5<us~d the plomng commands e"plot. plot, .nd " .... tou ... '" " 'cll 3 \ lhe ""SIC rommands f'" ""JuJ;ung and labt-hng ll1e .u.1 of I srapll. no...: com· m~lId ,. olong wnh qui v ...... h,ch "c introduced III Chart .. 6 to plOl dU'NlIon fieldl. ""' IIIe maIn g!llpillci comnW>ds lOU ,.,111 n«d for thil t'IllIf$t. In this ~tion """"ribc ad"'lion~1 cotllmand. for m.nlpul"108 smph"l. In many C3KS. we ,,"Knlle how 10 do lhlJ boIh fro", ,he MATLA8 prom.,. and WIth the mOllS<' Whi~ lOU m.o.y find !lie mouse operon",,,, sllnpl~r, the le, l comm3n
w.
K.S.I
Fi)Curc WIndows
When yuu e ~eCU IC III" fit>! plolilng commJnd ,n • gi'"n MATl.A8 Iot"ion. lhe Ir3J>h appean In a new" Indo\!. labe~d "Figur-.: I" Suil6eq""n l smphic; comn\lln
Ot "'pl""~ the gr.lpll mill" '" 1IIIIoY., You ~H <;<"ell tl\;ll bold ,,,mUlUfkl, ,houloJ:w.Id 10, l1I1her \han "'1'1:;>«. the tumnl graph If inllca.! you ",ani 10 ""UI~ a ... '" ~roph in a \ep;lrolc "Ink ~ •• pmg the old lI1Iph on )'our ~tttn, 1)1'<' figure 10 oren. lit"" "mdo\<. b~1cd "1"1"'" 2" Alle""'("cly.)ou tan ~I«( a N.... " 'Jure from lhe ~1 1. ITJ('nU In eilher your =,n ~1i\:rI.AII ",IIIIoY.' or lhe firsl filure "moJ"" Sub'~""nt 111IphlC! commands "'ill aff«1 only 11111 "",IIIIoY., unl,1 )'01> change ) our, ,,,,...111 jI,~"rr a~a'n "'III mnocher figure command, Or by \I,inglllg another fi&lI,," (0 Ihc forrillound "'Ih lhe mouse, 'I)'p,"g figure aIOl" w,1I create ~ Ihird figure "mdow, wh,le I)'ping figurell) w,1I sw,lch Ihe cUITCnl "lIldow b3C~ 10 "hgur. I" You call delele h~Uf. w,,,dl"'" wnh II", command 0;:10.e; ICc the nn]m. help fnr d.lails. Eoeh "lIure "lIIdow hns J 1001 bar underneath ,I' m"nu bar "'Ih ,lIm(euts for se,'eml n..,nu ncnh. mdud,nK 00 the I.fllhe u,ual ieoos for opening, ;a,'mll. and pru'hng ftguru, N.3f lhe nmldle. there are Iwo 1t0nS with plu~ and nlilln' s'gns, for loommg in aod OUI ('h,~ 00 lho .<1111 "nh a plu; "gn. and then d,et on a (1Oonl III your j!fllph In room m ncar 11m poth hor",:onl~lIy and ,..,niclllly "'thOR tho: cUm'nl ue', More loo""ng oplion\ ale a,a,lahle toy nlbl,c1icbng in the f'gu,," ",,1llow ",h,le l.oom'"8, by u~mg lhe Oplloru ,ubmenu (,f IIlr '1",,1s meno, Of by ul"'g lhe command ....... (~e in onl,"" belp), C h<~'I1~ (he ''''I i,'on 10 the riCht of the h.nd ieon nllow$ yOll In rolale 3D grophi", Mun:][) man'l'ul"1I0ns arc ","il.bl. by right·c1ie~'"11 in lhe figure window. aod by '.k,"in~ th" Co,""", Toolb", from the Vie,,' menu ("hdmll tbe " .. I "'On to lhe nlht of Ihe rot"l" ieon enable, the Dala Cursor, which allolH you In d"rlay Ih. <:(lordmat.! of a (1Oinl on a eu,,"'e by clickmg on or neaf lhe "U" r, Ri~ht 'd'c~,nll in (lie fiaure wmtlow II".' se'ernl "l'lionS; eb:tJ\ging s"lection Slyl" 10 ~l"",e 1")ta poInt!,) Th,s Cln be u'otful especially after ~oonung. boca,,\(" d.>la nlll)' then be SpmI'ed far ap;u1 Another way 10 lei coordnla.es of a pomt (In the COlntnaod \\',...so.., 001 the ~g"n: '" ,ntJo,.,) " by 1)1'1111 gil\P\&t (1); th,s al1(l\\' )<1U 10 dltl OIl any pomt ,n the figu,," ",,1.r opens th.ee wmdo"" iurrl)und"'g the f'gure; the," an: collc~m'dy ~I\"wil a< PIOI Tools, and y"" enn "1>0 op.:n them ""lh the command p lot tooh (Th"", 1001, nre nOl B''llilablr 'n Ihe l\1,iCmtQ,h 'crOllJ mcan, "I (d,lml!- hKU"", J.O<1IC of "hieh "'e '" ,II d'Ku" below ~bny of these cap;lb,hl,eJ ~rc al'" a,~,I~hlc in the luSt'r' and Tool s ITJ('nUS, and from lhe 1'101 bl'l Toolbar m the
,,,,ilii0\0
p
'"
Vlewmcnu
YOli C~n use .ubplot !(I draw ITIOI'< !h prepar~ sidc-by· ..de gnlp/l~ for a i'
,,!w
You c~n aha (",ale muiliple graphs wnhlD lhe .,.me fi u'" w,n ' • menu In lhe Figure I'ale!!e. Then sel~ wllh Iht I Iht dow U""g!he "t",Subplo!s plO!!o appear_ ffiOU§e Uti In "hlCh you "'3n11ht nUl
:r •
0
0
0
.
of "
•
-
.
o
•
o
o
Figur. 8, I' Two Graphl in one FiBurc WindOW
K.S.2
Editing "'igures
hi Sec hon 3.8,2 we ,'howed how!o U~ xlabel . ylabe1 , IUId ttth!o bt-.el a, •• and pul a !l1le on a g"'ph /I.", ...., drscnbc ""..,1111 OIhc'r "'3)'i to modif)' and lllnWI. ,raph' Wm,1I p)ouill8 mor~ Ihan One cu" •. i!" oflen unportan! to d'~llngl"'h amoog tlltm by ma~rng them h'"'' d,fferen! roIors or ~ylcs (r g .• sohd ,... ~u, dulled). One<: )·ou',.. pJOIled n cu,,'c. you C.n change its appellJ1lrrcc In Ihe ~8u", WID" OW a. foU"",. Fir"" di<~ on tilt ICon Wllh!". a,row, JUS! to!he ngh! ofl"" pnnl ... icon, In the ~llIure Tooll>ar lnc-n,oghlcll" k 0/1 Ihe eu, I'e; 1m, cu"·. " "II be highlighted with black and l!lIf nu .. tIlII'JlCM_ ( If some!l"n, fl'iC geI< hlghhghled. l1)' Ig1m) You can clun~ the m lor, width. OJ' ~tyk of thi s e"" .... 8nlO"lI 0111<" Ihin". from Ih" menu AII.mati"tly, tf lOU /u,e opened the PIoc Toob "tndol\ ~ lurroundmg the fiau ...., a~ tkscnbrd a!Kn' •• lOU Clll1 ",1«111\( ~U"-. by lefl·clickmg on II. and modtfy it In Ihel'r'openy Edttor, Yoo can also U\.O Iht PIOI Drowsr-r 10 sel«1 a cu,ve. FI ... lly. "hen plallln, "'th plot or contour, )'OU can cboose 1110 Ii ...
5<)""""
'"
style 01 col.,.. In IlIhlmCe; for uampk. 10 plot clio: data '" ~KIOD X and Y 105 Bdashed CU!,\'C, t)""JX p l ot iX. Y. • --' J See !he onlme help (Of plot for OIller o~ions, When you ha"~ made ~3Ch cur..: a dlffe~nt st)'k or color. )'l'1li can mal" a Irgcnd In the comtr oflhe ,"'ph "'Ill the command 1.g8""'. Stt 'l~ online help. Anolilel effe~m'e 3I'J'ffiXh .... hen rrc~nl"'g multiple tUf\C~ on 1M same graph " to labe l lhi.' elll'\'(' dlrK. I)". Vou Un add le~l. a/TO'Io'S. and OIlier amJOlaUOI1l u;mg the In sert menu or, w'lh 1'101 Tool> "nahIM. u ..,,& the F,gure I'alene hrs • •decl the: Lype of annOln·
oox.
lion yoo ""anI. then click the: spot on the gl1lph you " 'nnl II to aJll>Car. For 3 texl type .he , • • 1 ""xt. "h,le for M alfOW, click agam 10 mropeny Eduor. You can nlw add unboxcd lext by u.ing the COOlmund\ t e llt II.I\d II'te" t; ~c their online "'II' for details, Finally, to add a symbol uaclly 01 ~pocific coord,nates-s.>y, tlx: coordlnale. of a special poml you .... anl 10 b'8b liglll on a cun'~ ·you can typo hold on rollo\<~d by IlI1 al'J'fOPriale plot comnund, t-t;, pl ot ( 1 , 1 . • 0' ) to pia( a small corck: renlen:d al lbc poinl (1, 2)_ Stt tJK, online MIl' for pl o t for O1Mr l,7"Lablc symbols. You on eh.1ng~ 1""IKn... of a graph', "-'~', ~och •• tJK, fOOl aJed 10 numoo lbc ale, and 1~l lbc v~!'h u"ng tJK, Propeny &lnor. I"~I clock on lhe uc, or on a bl ...1.. ""'. w, th ,n IIKm 10 ",INI thtm, moch as you ~~l • cu ....·c for cdiung_ If tJK, un do not M C'OII>r b'ghlllhlcd, mal..~ .un: 11K .dll (arrow) lCOO ,n lhe hgun: Toolbu i. bighllghted, and if not tMn eli,'1.. on this iron firM. 'OKn click on the r'Q"ll.b In the Propcny Ed,tlJlIO t hanlle Ihc fo nt Often a font thai i-l largct than lhe defu"llnnpro"cspnntcd ootP"I. You CD n al", chnnge Ih. fOOl of a t.~t annolation you h''''j) added by right·cllcking On It ,,"d " ' lng the [KIIl·up onen u, or hy ~ I ccti"g It ami using tile I'rope, ty l!d,tor. To .re .n of lhe o"D,lohle way. to modify. curyC or other 01*£, in Ihe graph, the UC', ur the cnt",. hIU'. Window, ~lccl'lx: ooJCct you want 10 modify and click In-lp«lor in Ihc Prope n y Ed,tor, (I f tl>< Propeny Ed,tor is not open al",ady, lighl·clid on the objecl Dnd IcI.ct I'roli'frll es .... ) 10 !.rlcclt"" figure w,ndow 1I.,.,1f. did; 'n the: border OIItside the AnO\h(r ""ay to <e. and dl.nl<' propcn ... ofliK..., ob~u IS ","h lhe: comnunds get and eet 1O!.tt 11K propcnles of tht eulTtnt figure. Iype get t get) ; for the Current 3Aes. Iype get ( g c a ) ; and for t"" Curren' .,.,I«t~d CUt"" or othtr ob~I, 'l'pe get (geo) ('OK com· m3ndl g e t . g <;: a , and g<;:o \.laDd f", "1<'1 current (,lIurel .... 'sIobJCCI .. ,) To change. for eJ.. ample. 11K FontS,,~ propcny ofllK culTtnl 3Aes to 16, type e.t (gea , ' P'o DtS iKe', 16) WM.1iK tht numlKrs on lhe: au, WIll cllanl<' m B 16 pomt foOl ,mn..d'Bldy. you w,lI nNd lu retype llI1y previoo< t i t le, xlabel . or y labe l commands m enlll1J:C t"" !itle Dnd lalKh
U.',
8.6
"~elllu~s
of MATLAU's Numel'icil l ODE Solvers
In Ch3p1~r 1 ""c '''troo.loced the command 0<1. t 5. which iIQ"'cs imtial "aluc problems . He", ,, ~ duenlK ", ... eraJ adtl!tional w"ys 10 u..., thIS ConUlIlInd. 11lc1C fcature. al§<> apply 10 11K ollKr nUITK",,,1 §<>h'cl'S ITKnuone<J on Chaplet 1. soch., ade 1 S e Our d.,""u" ion il ba...:d On !he: lynl.O.\ l\a,labk: in MATLAII 6 (11.12) and later "crsKlnS, "'hiell ,s used
86 I"'~w ...s of MAnA/I's Numerical ODE 50/...,,,,
'"
only if Ille 1i"'1 Inpul llIumen! t(l oc1a ' 5 II a (U""tlOll IwIdle. ThIs " the ~ If yOll Uk' ode'5 In tho ways iUlllesled m Cha~'1, But If tbe f,,~t ,"..,.., argument of ode' ! IS III millie fU""IIO<1 or aMong conlalnlng the name of an M· file. II ""II uk' III old (MATLAB S) syntu. and mOSI of the fUlu.n de""'ibed below ""II rKM "'OIk .. we desrnbc Ihrlll .
H.6.1
EYHiu;lIioli of Numerica l Solul ions wllit deval
When ;olvlnglhc differenl ..1equahon
d,
"
with ode' 5 ..... e ha,'c k'Cn th.3t you Cll1 r.p«ify :m mle" ~I of I ,,,ltIC •. In .... hlCh case MAT. tAli compules the nume.ical ~Iulion at ,"~rmedl~le' 'altIC~ Ihal II e~l> Of yOll Can specify a "~Clor of I .alues. III whIch case MATlAR outputs the oumeneol ~lu1ion.1 Ihc I v.llle$ you chose. Anolher app
u,
,,1
'0004 5 ' ex tdat ll: Ilx lll truc t] x: (lxll double] y: ( lx U dou b l e l st" .. s : [I x l su·ue t ] i d ll t " : [l x l st r ue t ] The OU1put IS thl'n a 51tuC1U'" thaI conlalns. among DlhIOIN in .01 .... nnosc fields "'" nl"uy. called >t ami y . regardless or wh~1 Iwe .. you usc for lbe mdel"ne dell! variables). Ho..... ' ·... yoo """" 1101 cOllCf'm y "o lver:
4aval
Onc. you h.3,'. , tlnd the OU1put of 04. . 5 InIO a .UUClure. yoo can uk' 4aTd 10 .,,,luat. the numerical wlullOlI at any .-.1.....1 of I " 'IIII,n the In~"'al gi'~n to od.aU For u'll1ple: » daval(.ol,OI0.S I 2) IIn 8
1 414 2 1.8028 2.2361 1. 0000 1. 1180 . he 'cal ,0IUI;on Ilotbor You Can thcn run 4.v1I1 1S mally II111'S as yoo want to ge l 1 "um
"'
.s. v.l OOl only w,th Ihe "ariou~ lJlher MATtAH Iordln",)' in;,ill.! '1110. problcm~. bul 01>0 ""th lhe cort\mand~ bvll' g (whkh We w;]I usc In nu~r 101(1 wi,. boond:uy ,alII<: rrob1cml). Man (",hieh ",h.I delay differential cqU~IIOOI). and 04. 1 5 1 (.,.. h",h -ohu ,,"plicn 001'-1). You C:tll
H.6.2
u~
I'lottin~
Fllmilies of NU1n ~ ricnl Solutiolls nf OUEs
In ("hapler 7 we pinned a (anl,I)' or nU~""31 wluUOOI of B first OI'Iitr differential NJunuon u"n~ a ';n~le od. ' 5 COfllIt1300 and a ,~of ,nnw roOO,',on' Whl1e lIIi' " ,"h In =0)' U'<:I, II reluJ\! 10 "" 'lI(:o""l'lc" &r.Ip/I ,f anI one: of tile r.olUlion~ cannoc br
""",hod
computed O'e, lhe cnl,", 5p«,ficd 'nlc,,'al. h IS al~ un~ull3blc ,(.Ix- un~_'n funclion .. I, ,helf n "eei"r, u, ",,11 be d,e C3<e wh.n we U$C od. ' 5 In (Ulure ch~plcr$_ In the-e CaIC$, ;1
j,
1""1Ic' to u," n loop to compu,. and plot .""h curve 5<'paralely.
•
.. ..
h S"'" 8.2: SoluIK)f" of dy/d!
"
"
..
yl up 10 lhe: " mil Smgu1arotl
1'0, ~lanll'le. ,uJIPO":' lMl yoo 10'.1\1 to 1'101 nuftltrn:.1 5Olulion\ of
'"
" "
",til 11(0) 2, 1,0,1.2 on lh~ mtc,,·al 0< I < 1. You can I'roo.]ucc Figure 8 2i>y typil\& the foil"",,", command" u
u
9 _ tHt , y ) y . All It. y) _ od.4 5 (g ,
( 0 1 1. -2,1) /
W.. rnin!l: F.o11ure .. t t 4.999847e-00l Unable to meet integr .. tion tolerances without reducing the step size below the smallest value allowed (8.881784e-016) at time t. > In ode45 at 355 » pIot l t, y ) u
&>Ih(I O 1 -5 20 ) )
8,6, ,..,,, run's "r MA1'L A O's Nmllfnr~1 ODE Sokers
'"
(NOIlce .ha. e' en lhough .I>c: ligh. ~Ide of .1>c: d'ff~n:n',al cquahoo " • func!lon of V only, " c "D ,'r IH"dc II D func.ion of ""0 "ariablel. as aden rc: loOlulIon """ vIOl 21>10..$ up", InIlMy• .lnd all 5(llu"oo eun'., are lhown only for t bel " ..,n 0 Ind 0,5 e'en .houl" foor of them ~hoold eXlend fur.her .o lhe right A be lle. aPf'R*:h In .h,s CI~ ;110 dellne: 9 and then uo;" ~ loop a, fOllows » ligu r e/ h o ld on » lor yO • - 2,~
.ha,
f,,'.
yl • ad.UIII, p lo t It, y) [t,
» »
0.'
[011. yO ) /
hold o ll ... iBl [0 1 - 5 ~Ol) The r(lull il In hgurc 8,3_
"
, ,
.. .. .. .' . ..
" " " .-,
Figun: 83 Soluttons (>f dV/dt
V'
Ilumu;,;..tlly s.oh ..forhipr In Chapter' 10 and t 1 "'c d'!oCUJS how '~u~ odd5 I tho", inlllal rondlllOO eac" order equi",on, lind sYltems of c'l"allOll ' In I f~ e..~; numeric.1 s.oluhon; )'00 m"
",II
8.6.3
EH~nl IJeterllon
,I
"""',
lem th~1 II mode led by • dllfe",nlll SOlllcilmcs !he IHain que.IIOn of Inrcrell In ~,l:: I fl lbng b:ill hit rile ground? '1'110 cq~IOO I~ .. hrn a Jptelf", e,'enr /)tturs, ~,'-, ..able !he de l""rlOn of l uch "" enl l , MATLA II "umeneal 0 0 1l $01\.... hall' I fcalun: thaI. I COl,sidrr lhe fI"l uamrle frolll Ch'rl~r 7,
," , d,
'
y{O)
,
,
;md ,ul'f""~ you wanllo hnd 1M 11m" when Ih~ wlmion 1I_",atll<'\ III<' value 1 $_ S;n~ Ih( "'3,'1 wl ution of our mlll~1 .'alue problrm " U(I) .,ftT + I, "'e know Iha! lbe d. ",.,,1 'alu r " I ,(.,/2 I, III! _ 1M for ,be \n~e of IlIu511':llion, IK-re an: Ihe ~lep5 to fOllow 10 31'1'"",,,,a,e Ill" 'alue of t u,mg oda".5 h",l. )'ou mm' en-~'e an M- lik, ""y event f unc . III. tha, de"""bcs 10 od... .5 lhe e\(m )0\1 W;\/11 n 10 dela". Among Olhe, thmg •• Ih" fUllCl10n mu" \p!:(:lfy a quamny thai ch;\/1~(\ "8" when th~ eHnl ""curs, Here" an .PI~0!"'31~ e'nli M hie for lhe prOblem d."-,,bcd abo.,,: function [evant, .top, dir) • ev.ntfunc(t, YI av.nt _ Y - 1..5, atop _ 1: dir _ 0, 8of",,, ~,plamm8 lhe
".
. 118{ ,OO( TIo~ 1'1,.,1110"", mpul arxullltnl, to od •• .5 an: the usual ones The lunc IIllerv.tl [0 2) ,houtd ,Iart at III<' lillie where Ihe m,lia) cond\l\OII II{O) 1" g"'"" and end al a lime hy "h,Iop solvlnR lhe d,!fen-nllal "'luallOl1 one. lhe evenl happen!, ' " I) if ~ou wan! II 10 ~ccp 80lng and [lI),,,bly del.cl multlpk occu"."" •• of Ih. even!.
'"
1ne third a'gument, "-h,ch ....e called d11" . iohould be 1 If )'OIl "anllbe ,",'cm to Ottllfonl, L.
"Mn event .""ng" f.om negall~e 1(1 pI»'ln't - I (Of lho: tHnt 10 OCCur In •• Ihe. dJl'~'ion In th.
I
u~ .... ~ y !"XU",'. 10 ntglm"t, or 0 rot I'~ .......-c:, ... would ha,.., oilu'lIN lilt
. . saUl( an",.,. ""th d11< "' 1 ,001110 ",",'tnt " 'oold "",' ....... ,,-,•• __,I""""th .,, """n ... diro. - 1
8.7
Troubleshooting
rn Ihl' ~cllnn ..... e lilt SOme advi<:e for a...;';dtnll common ml\takes. W•• lw du.cun ...:tIn.
mil me\\ngc. and dc,"""be wme lechni'lues for "'Co.'cring from c"ors.
8.7. 1
'I'h~
Most Com mOil Mist:lkes
lIere are l common miunel on ulm. MATIAS I
U~mg · ,
I. or - 11111.00 of •• , .1. Of
• - .. ben
Itymg 10 muluply. dJ ...d., OHIpG""m,atc ''''''tors 01 matri.csc""",nt·by·demcnl UnleSllou ..;mI lO ptrform lhtit opcnu,on~ by the rules of m:>10' mull'l'bcaHOII. you should ,,,,llIdt: tilt ptriod be· fon: ll1c:..e lhn.., operlloos. In 1110>1 caws you !Ivy mdudt !be ptriod .,-cn .. lito it I. ru')I necr~'ary, lIov...,wr. )'00 !oIlould 001 includr!~ ptriod in inJlll! to JymboI ie ~011""and5,
2, USUIS 11K: wrong cia., of iOJlll! to. oommand, MOl! of !he commands ough thi' doe! not Inl<:rfcre .. "l! ,he ~or"-'C' opem1lon of MA:rtAB. "can m:l~. )'00' polll' cooUJIand., In .uch Ca!.CI. yoo can U)oC the oomm:md pn~~Y to form>' lilt OOlpul 1010 , horter hnc. thll an:
!WIll'"''''
Call.' 10
road
""
8.7.2
Error a nd Warning ~ 1 ('ssnl:cs
You ~hoold ('il ck>J,c, ~(t.nuon '0 the error ami ,,;unm, '""'''~r<'' that MATI.AB gene rat.!. Although W>C: ,"",~\.3gn ml) SO'tm cr}pt'" a! fi.~t.)ou ",11 find Ihatllley gi" _al""bk d ..... 'o 1",,1',og nu~lal.., m ,"pm For u:ulll'k. ,uJ"!X»" ,holt )OU • .,ecUI. Ille follov.in, ronUll~nd!
~~ Jl ~
_1,0.(11 ,,,,, plot lX, ,. "1 ) 11? Error using . ~ rnpower
Matrix must bl! sQUare.
u'm, "
Thi \uror " •• ,snge roinls to the Wllre. orlhe problem, InSICad of • " with" 'CClor. Se RIO arc. h",!' ......, ,hal errors lih this do not a," 3)'S produce error n1c<sagcs. For example.
Coni ilk. the follov. lOS commands: H
Ix. y) . . . . .hgridIO.O.2 .10.
0,0.1,10),
~~
cofttourix, Y • • -l • y-2) Decao"!' '' aoo y """ squ= m:lIri~ •• the PSt of' ,\ nUov>cd In lh" instan.,." lIo",.,'n. becauSt • - .. " 001 usN. the ~sulllng groph~. noc (on<'Clty ,xpICI e",,'CI of
Ie,,,'
.. ' + y' AnOJlho'. common '>011",. of CmIn" us'"g the ,",'ron& ela'i of ar:JIumenllo a command. For cUlI\f'k. ,u~!hal )'oulyl'" H _yaII . , _olv_(x '2 • • - 11 ?P syf118 Xi 801ve(x"2 x 1)
Error: The expression to the left of t.he equals sign i ... not a valid target for an a ...... ignment. Allhough II,,· lUI of the error """,age ;s nol e:"y 10 .olelp",l, Ihe ,'er1ical \lat lndicnlCS an em". "' •• firsl d
"'M'"
??? Function 'int'
is not defined for valuea of cIa ......
'double' _
Th" error me''''I" oflen an~s "hen a symbolIC fvncuoo .ecc.i<'(:, the wrong clHS of ~r' gunlt"I, In Ihll usc. ~111«' • IS • floating po,"1 number, loO I~ .i., (x) . ,,-h,le i.,t e~pccll ~
Ilnng or .)'mbolic c~prnsion, To con-ecl lhe: problem, Iyl'" _}'1U x and "'I"'al lhe com",.nd. or .ndo~ lhe argumenlln ;mgle qOOlcs- int ( • _in (.) • ) It i, imponam 10 d15lingui,h bet'" een rr"" mc:1'agos and" amillS mes5~gn_ An c ..Of """IDse Illean> Ih.1 somelhmg is really wrollg. and MATLAll probably will ,1Ol he able 10 pruducc ally output, A ",aming n...sage .. all il\lhc~110n Ihal §01l""hing might be WfOllg or that MA11.A1l un't do exaclly ",hal you'". m'lnK:ted II 10 do, Il ere;s an eumplc: ~~ x • ~51 0.01,5, "lo t(., 1. /.1
Warning: Divide
by
zero.
Although lhe ,'«tor x illCludes!he \'aIIII' O. at "h"h liz cannot be c,'alualcd, the cOIn maool OOo\c toITttll)' grarh 1/z.
Chapter 9 Using Simulink 11Jerc i, n UJ.cful au~ih3f)1 10 MATLAII. calkd Slmuhn~ S,muhnk;1 aulom,lllcally In. cluded in the MAT I-All Stude"1 Vtrsion; Wilh tile Profe~l ional Vmion, l! C&rli>t obI.ir.td
as an IIddluonalloolbo~ In 11m chaplrr. we 8"'''' Conc iiC muodoction 10 S'mulink. focllsmg on 11\ II'" 10 soh., d,ffrrrntial '"'IoallOlll. Mmy SIOOtnll IIId enlinecn 6Dd WI Simullnl. I'f""ldt, a COIl'l'nlCnl graphICal "front roo" 10 MA11.AD'1 numc""al ..,1 •.,,, 04... .5. odell, ode1 J. o ~'r_ However. one should \o:'(P ,n mind tlul aJllile commrnlS in Ihe 1~\1 cliapler about rchllh'hty of numencal ,nelhods apply cqu.>lIy ,,'cllto Simu itnk.
9.1
Constructing und RUllning a Simulink Model
We will ,liu:
!I
Ill. I ""tl! the ,munl roodnion .,(OJ
O. To launcb SlIlwlonl . d ick on tilt icon
JIi on the [)cSklop 1001'-. orciSl' Iypt .i_B ilk In w MATLAR Command W'noo..'~
.. lhe S;rnuhn~ I.'i:nry. "h"'h 1ool1 M.e F,gurr 91 on UNI.~ 'yslem< Or h~c h gure 9.2 on W",d(lll'$ rnxhlllu The firsllh'ng yoo .... '11 _
•"• §I
!§J~
IriI
••
-El
'" .:0-
I!OJ~ ....
- ---- -- --1:;1
[r.~1
--~
",sure 9.1: The SmlUilnk Lihl'3fY
'"
l2!
..
... ~' _:l-=====:::~ ....
-,
--
-_.-_ ... --,.FigURe 92: 1'ht Simuliru. Library Ilrowl<:' on ~ PC To u<.e S,muhnk to \tudy our dlfferentlal «I~tion. we n«d 10 c, cale a ",odd•• graph,cal RepReKnlalion of the «IU."on. The mod.1 =u1Is rl'Ol"thlll~1II1 of a differen.ial «IUIlIOn'; I{I.I/) as a f«db<>c~ d,agram. wheRe 1/ aod I are u!oed 10 OOIllpule I(t , 1/) am! thul rI. from which 1/ is ob.alned by 1"'.I",l1og. as in Flgu.e 9.3. To begm 10 construct a model. fir>1 click on New and .I",n Model from the Fil e lIlenu of eilhe. the MATLAll Dc~klop. lhe Sinmlin~ Libl".ory window. Q1' lhe EduorlDebuggc'_ A new """I~I ,..Iml",,, will pop up. You cRea!e!be model by drauing eenain ~Iandard elerne",' from ~ubhbr.mes of !be Sirnuhnk Library. called blocks. mtO !be model Window. and lilen conllC.'Ctmg tile", together For .ludymg orom:lr)' d,ffeRenllal «I"ollon •• mosl or all of lhe block. you n«d will be found in the Sources Llbr.uy. lile Smko L,brary. !be Conllnuous Llbnlry. and !be Math OpcralK111I L,brllry (Sec !be hst ofS,muhnl: blocl! at !be end of tile GIMIary for llle Iocallons of lile mo.l con1!llOI1ly IIKd block. for studymg ordinary d,fft.enll.1 «IUlllon S.) One hloc~ you will .11111»1 always n«d i. tile Inlegralor block from tile ContlilUOll' 1.lhr.uy_ In a f,rst·o.<.!c. dlffeRelillal n]LlallOn .uch as our 5aflll)1e ((Iuat;on'; 1/' + I. the Input to this block ",PRe~111 S rI. alld lh" \)\IlpUl to thi. block "'pre!Cnls 1/. Ir you
9. 1. CtI''''rruC/Ulg and Rwming. Simulmk MO
I
,
,
Figu,~ 9,]: lIow ro Vi.w rI
/ (1, ~J
pl ace 10 I"' t in lllI im tial condlllOtl, whkh ~rcrmlll"s the: "con~tanl f ... u~pk, " " don'l ha."(: ro change the: okf.utt IJUual condition, "'hIe: i~~~r~~ In OUr ['he mood for our cxample, once fim. hed, looks like Figu ,,, 9.4 npl.m lhe
',(~"
.- if
,-"
, 0
,
~.
)~.
t
,~
Figure 9.4: A Simu hnk Modt l fC$ rI
,; + I
mOOd, ond tho" expl. in huw to corntruct It from the bloc~. In the S,muhnk Librvy, n.. ""lid lines ( ... ilh arn.."he..r, on them) rr~;c:nl what Simullnk c&ll, sigM lr, marJt,eI!llti, caliy, the!e arr mabl • • The blocks rorTl'5pond to m3rJt,emltin.l ~tion'. Ro.wn.1bI: model51al1ing In th" uppor lerl, we 51:. tha t tho rmr~ralor Output. rh" '<1111. of y, TIle Math Function blod: ron."(:rts y 10 y2, ... hich Iddtd 10 I rommg (I'0Il1 thc: Ramp bled. (TIle Cloc k block, which OO lpuU the "nluc of t, oo" 'd h.,"(: also b«n ~d herr ) TIltn thc: "" ult, y2 + I, g~ ~k mlO the IntcgralOf u!be ioput ,/ We rrad offlh. re,"11 with 1M Sropt' block, ... hich draw. a plot of II "" funcrion of I. (Not. that Lhe Srope is I1Uocbtd 10 IbI: sis"al !I wi th 3 branch hn" ') 1\1 gi,.., another ctampk, Fi,"", 9S ~. I mo
I,
o
F;gu", !U: A Slmuhn~ Mooel for II
(sin t )y
yield good lT~ullS. b«au~ of lile fact 11\;11 num.neal dJff=nloahoo 15 much less Slable lhan numcricJ.i ,ntrllllUM. (Kc.:p In nund t~ S;muhn~ doe,.11 us calculat,ons numencal1y,) In riot., "igIUC 9.6 $hov.-s a keond model for tile sal»~ d,ff.... nual cqualloo 11' ,; + I. rewnllen., II JiI I. wIth 1100 uSC of lhe ~n''lIh''e blOC'k. tat.. we will di~.. !he diff... "". In ",~u1ts from the firsl model (F,gun: 9 4) and the 5«000 modd lFigu,," 9.6),
Figure 96: AOO!ber Simuhnk Model for"
1/1
+!
I'i!"lilet',.o back and duoCl'" In licl.,11low LO COIIwu"llhc mood In I'Igure 94. S ... rt by double_cl irk,ng on the libraries you ".«I;n the SmlUhnk l.lbrary wIndow. Within each library, YI)\I wl1l ke. 1111 of blocks "'presented by Icon, Clicking on a bloc);: Will display ,orne help lU I On whal it does. On Windows syslems, for eMmp1e, lhe Conlm uous tibn"y. w,!h the Integl1l10r k lttted.looks like Figure 97, Click on the blOCks roo "".d and dmg them "'to tl>l: IleW model Window. You don't !\fed to worry j,() much about pos.tlOOlnS 8t first, "nc~ yoo Can .rarrangc thing, later. You will llOte that most blocks ha,·c one or more input pons and one or TTlOU OOtpul ponJ. (£XUpiIO
9.1, COO.llfUclmg und Running a SUliulmt Mood
'"
-•• • ~o_ "'--.............-
•
"~.,,..
'"
.'"-'''-
...
~.
..... '\1--1<---
.-.._.-:=----a- __ _
. _I....--,"'--,.. 0 -
• 10'lure 9.1: lbt Coolm""", t ,bruy
or Oli! of Ihal JIOn. UJually you will " anI 10 dt'\lg the IIMnal SIIlI'&/I1 10 . port on lrIOIher bloc~ . bul if it '~ necessary to brnd the Jin~. you may nced 10 ,,,,ate the: Ji".. in I I.g••. lbt line w,1I .ppc~ iOloJ oor.. u', c(lnnt<1~ 11 both ends Th c",ale. brarl<:h I",•• hold down lhe
erR!. ke} . rnl Id, chcl lhe
mQU.\C II
the poont on an .."ung II"" ... 1It", )'011
" ant rhe bnlnch hnc 10 , 1:111 Yoo can ",name I blod Or pur a I.bel OJ) a signal lline) by nglll dic~mg on It. 5<'l«ting Bjork l'ropl'rtits.w or Slgnl l PmJM'rtllS-. and 1M" "'SC rlml the lIew " an"" . Some hloo1s, J..eh
a.
lhe: Sum block or the Molh function blocl .
ul uall y l'Cq'mc cusromi£llUon You do rhi1 (afler you"'" dlllWd rite: bloc k InlO )'o" r model WIndow) by doub~d dmll on the block 10 call up the Illork PIon mtrers menu For cump1c, rhe IId.uI! "crsiOll of lhe Mnlh !'Uncllf)!! bIocl COlOjl\Il<5the <.\poornlia] funclion and II latJc,INI e'u :>CCQIlhn,l y. butlherc il a IOUk s" "ch tochlngo the fUncI;OIllo 100• .!!quare. aqrt. tlr. The lkfauh wnlOrt of the Sum block ha, '"0 iopu l pottI (II the "6:00" anti ''9-00'' pullhe HllXk PlInmrltn m.nu calls up tr.. 1.*,1 of S ign! parmk'lc, _ StlUnS Ihl l lo;> + +. fo r . \ounpi<. Olp) milO t, und Ihen add the dllro (OOllom) input ~ )'ou ha,,, cu. loml « d the bIoclJ you nttd and rorul<'("lNl 1li<m. yoo CUt wok 1hem arouoo ""h the mooS" to ""'~< the IW":IfItK"C of )"00' mo.:k l as "cloUt" as pc>lllbk.
126 R~member thai to gel u;eful OU lpul OUI of )'our model , you ~d 10 ~~ room for all~uI
one OlIlpUI blorl, usuall y' SC<:, ~) IS II funcllOn of lime: t, Whe:n your model i, complele, It' . lillie for tesllng. Th,~ .. tiIC fun part! Hit Sw . e A~ ... (Will Ihe ~' lIc "lenu 10 gi\C your mood a name; Ihe fi le C~lc", inn .md l (for "S i lll"h,,~ mood") i, lIIIded aulomalICally_ W. called lhe cumple of hgure 9_4 first. ('x.mdl. ~n liIC SllIlu)hlion menu al Ibr lop of Ibr mode! .... 'ndow and "home Conlil(unoUun ...... mtlen. •. This enable; lOU to Io<'t lhe: m,(,al and final ,allk'! of t . 1"01" OlIr uampl••• nllural ChoICe: " ould he 10 l Ctp lhe dc:faull 'Iarr,ng.a! ... t () and 10 change the: "Stop TIn",," 10 t 2,0, 1'1"",' ..:1«1 S IIIr1 (agam f.om IOC S(rnul"llon ",.nu l 1>on' I br sulpl'i\C.'d .f 1h,ng' do 001 go exaclly I\)OU pl;um"d and ,f)ou gel ":niou \ error mn§,age~ . U,ually ;tI1... (r lour model .... m. lI>ppc:nslo br. Orr« your "mulalloo door, ru n succes,fully, 11 may appear rhalll(llhing 11>, II>ppcnc:d, Th"" nOl ",aUy.he ca",. l'Ilrone tlll n~, 1111 hkely Ihal one or ~ new ,.nables. C"'3IM hy Ibr ~imu!nllon. Will ~how up In you. Wo.~space, I'oruample. ""brn you run Ihe model or " ig,'re 9.4 W,lh 0 < I < 2, )'OU "'In foee U "3.iable call ed tout 'n lhe Work$pacc. an array of ~i'e Il19 x I. Thi' i. a ,'etlor of tl'" values of I wl""e the (al'1'ro~ U\\nle) ""Iulion y h.1 hren compu1ed. III Ihl~ n urnple. you """"Id nlw gel . " am' ng nlCs\.IIlIo "Sol".r Step si,.c II hreonunl I~IJ lhan ~I"'dfied nllnl","m ' lep si(.,"lhal !l\o<.<~ up '" lhe S,rnUlaIlO1l l)i .~,"""llc, "';""""'. and 101''''' OIocr " iUnrn~ mc:'!" ge!. loIylng " Warnrng- Unable 10 reduce the ,Iep ,,'" ",thoul ,;olnllnll mrn""um .rep si!e." In lhe MATI.AIl Command \\r,ndow S'II« "( kOO\Oo' thaI the w lullOO blO'<\! up In finll.Umt (In fact around t I 9!i6), lbrlo<' ,,:m""gs ore: 1101 ! urp.." ng. Thll eumple !JIould CQfI' mct yOli til>l \oOmet;mes Slmuhnk', ,, ;unrng men ages do nOI O<'C (I':mly Ind,cate lhal yOll made I mmal e. bul may be inlonilC 10 m3thoillalical ptopclll e:. or lOlir modd. l'inaJly, If you allached a Scope block 10 your OIl lpU! "Gnal, Ihe rel U\{' of the " ",ulnlloo shou ld be vi,ible when you double d lcl on Seope: hl(ICk. The: Scol'" "rndow Ihal rorl up is not I figure: '" ImSow ;n lhe u\ ual ", n<;(' (;t doc. not a flgo", num!""r, fo. c' ample), bu. 11
t""
"a"e
4.o1v. or od.'S Eu .... l~ 9.1 llul(d. S;muhnlmodc:l ro sludy the dt/Te. enllal equallOn II y J. l' , 'lU'hcd 11\ example 7 A. Vary lb. ""Ilal .alue of y{OJ, and ", udy bow the nun",rrc al ; 0111 . unll ,anC' " IIh Ibr ,mllul cond,tlon. Confi"" "hal " ., . IKI",," III h gu.e 7.4, lh,. slightly
9.2. Outjlur ro rile II'orhl"''''' amillow SIIIIUrJII ~ 11'0I'k.<
I~gu", 98: The
127
Soopt Oulpul
different InlUII cond,t lon. I,,'e ""'ulIOl1S "hich r.ma,n aI~llhe SlIM for evomually d,,'erge from one anOllw:r.
I
"M•. but
9.2
O utput to the Works pace lllld How Simulink Works
To ,.1
rr>OI'e dcl~,iod mform.:llion from Slmuhnk. )'011 may wanl 10 (unMr ~S, tho
outpul of Simuhnk uJml rCluiar MAT LAII ~OO1m""d! In onIor 10 do thIS. yOll shouid mal.e uoe of the Th W(){u~e blod froOllhe SInh LIbrary. for uantp1e. We cou ld modify 11M: modoi of Figure 9 4 by addmg thIS blod. I •• 00." In F,gu", 99
-B
-
~.,
"~
128 in o~r u:omple .... e"·e changed II 10 yout , and set 11 to ~ of elMS doubk :omoy) Run· n;ng Ih. moJel gi,," s the Otce: of num.rical soh •• and u ~ one of MATLAO's (>!hr , SO.... B Instead.) U"ng the Tu Work\p;toCe bioel , .... e Can 00\\ romp.>rC the model> In h gure 94 and Figu", 9.6 . Runnmg Ihe model In I' igu", 96 aglllll g"'es an error me:"""ge on lhe Simu lation l),al OO\IIC' .... ,noJoo,o•• th .. 11 m" sa)'ml: ''Tl'Qtlb]. solvmg algebr.uc loop conlaining '"eeond e x /M4th Function' . t lime: 1.8. Stopptng simulalion n,.,,,, m.y be a sinllulanly In Ih. solulIon," W. alw ge l a .... ;un lng me:ssage In lhe Conunand W' ndow. sa)'mg: " Wllflllng: 'The model • ..,('condcx' does nOI have cCnllnuouS Slates, he nce USlnll the soln. 'V",r i "bleStepDi screte' mltead of t"" soh •• ' o de4 5 • spn:ified In the Configurat ion l'ar.IITJ<'lell dialog." This" ,"di•• m '. of a shlht]y more seriou. problem than ,,'e had .... ith the filll mOde l. a"d e~pl . ;'ts why USi ng the Integrato. blQCk is bcl1er Ihan uSing the Den ... I,,·. blocl ,,'''''n ),ou II>H •• hoio •. 1'01' 0f>C thing, in the model of f igure 9,6. lhe", IS oo coo,'enl.nl way 10 InpUI the In,t,a] conduioos. as t......, is wllh the Inlcgrollo< bloc k Secolld]y. MAll.A B need. 10 sub>mul c " d,ff~n:"1 soh 'cr for oddS Thi rdly. Iher~ il 1M problem tl\llt the \.lI uue root funct ion is not dlffell' nhable: al 0 and not defined 10 lhe left of 0 And finally. the", i5 Ille ISSue ..... e alll';wJy dl>cuued. name:ly that nume:ncal diffe ", nt;nuon i\ more un.table Ihan numerical integrnllon. In spite of nit these problcm~. the simulation "ill ''worh''_t.... pi~ture Ofle 1«1 In lhe Scope IS shown in h 8urc 9. 10. and ... 1}' ",uc.'b "' semble~ I~gure 9. 8.
od.,
od.'
1~8~Jl:
9, 10 Scope Oulput for Ille Second Model
If " f outpul t"" Tt'~ult of t"" >«ood modd 10 the Worhpace as .... e did with Ihe filll rnodel. .... can now eomp.;!re the ffSUIt!.. T"" final "'lulu are IDown In Figure 9. ] I TIle computed poi nt5 from Ihe rlfst mode l are shown ... nh dIamonds; those from the second model are 500"" Wllh Circ les, NOie thai tho: 1..... 0 solutions are a lll105t idenlical ou l to I
9.2. Om/,ull<.> ~ \\'orbpacc "lid /lOll' Sunu/ml llI:.>fh
'"
I .t>. 001 II",,, L . I~y nart 10 di,.., ... The ~u,el IoOluIJon, "u>< tqUlllon. as rompllial with (I.olv 'h ,•• model. • . I ~ shown ",Ih . wild I",.,• and b.blCall, tOl~'-'.~""" w, u", muh. of tile
fi""
"
,
"I
• " "
• • " •
I
.. "
,
" " ".,
..
!
, ,, '
.,/ "
"
,
fill'''' 9. II Cllffi/'Wlhg T"ll Modtls
~;I( . .... I!il) 9.2 Build a Sim ull nk IllOdtI 10 11udy ll1e IlOII hncar /l ulonomou~) initial , ,.I\If
problem IJ(O)
O.
('I. I)
and 10 oulplll the "ppro~im31C rotUl ion 10 1M Worhpac~ Compare your ,..,ullS with tlle UKI roluliQll as cumrulcd by (leol ve Conflnn lhal ll approad"'J i as l i:ol5 1~.
Chapter 10 Solving and Analyzing Second Order Linear Equations Nt" l"n'~ \.CCond law of dynarmes-force Ii ''1 ...1'0 DUll 'Hnes ItteIcr1lKln_II, p/I).itim thaI. In ordrr to undenlalld how 1M ", .... ld 1f11Jfb, llley mu
110M. The mo.1 basic second order differen t,al rqu8110ni ar. liMO< .~u.tiO!lS ""ilh ( <>Mlanl e'''ff,dNlIJ: a';' + by' + ry g(I).
Th...e «I"ahons mod.! a ,,-ide ''lIntly of ph)'.;".' ~11U'bOOI. ,,,,,loolng OSCIllations of ~pon8§. ~Imp'" dectll<: CIrC'\l't~. and tho ",bQlIonS of tumng roru 10 rrod~ '(Mmd and of eltttronslo produce light. In oliltl '''WOhorn. ~\lCh IS tho mOIlOll of .. pt'ndulum. lOt may be able 1(1 arl'fO~"n:lle lilt rc:sull1Dl d,ff...,null equa'"", Ita.ooably ..-ell by' hn. U r d,ffercnllal "'1".1100 .. lib ronstanl rocffic~nll l'ooIuo.1d)'. y,e kno hnrar d,fJcrnllll C1lu3UOII' .... ,lh con~lant codfictenll. 1 seoor.d Unforlun.ldy. we often canOOl find wlulIOII rorm~I~1 for nlor" ~".... ,R;, "'I"atIOns. """" for equatIons ""th ,.... ,,/1h'" ,lH'" .. , '
'.'$'
h,,,,.,
Y"
Such c'luat,onl
h.,-. mlpor1ant
oro.
+ P{1}1I' + q(l)U _.
.pphcatton~
g(I).
II ilCS For ...mp1c. Airy's equal ion.
to P y
" ,,,"
r
0,
.
(10,1)
""
ChapleT /0 So",",, .,' ,'~
(10.2)
OCcurs in the study of v'~rnllQnl of . e,rcular
Ill,",br~nc
and or Waler Wa"oS wnll clr.:: uln.
Iymmclry When stud)"ng 111'lo1 ortkr d'rr.... nl1al cqO~hOM fOl .. h,,,11 n""" r.o!ullOO fOfm"I,,! ""ere una"ul.bk. ". could mm 10 ~,cral othrr "",thods lIy ~JX<:lf)'ng an Inlllal "al .... "' c could compute an appnJ"OUIC no"","".l soIUilOIl lJy leUlng''''' ,ntllal .-:illll's 'uy. ....., could plO! a ~.puamrIC' famIly of al'prmlln;t~ WIUIIOO curo..,s and gel a f~1 for the btb~"ior of a genual solution. AI'cm:u,,·cly. " c could plOl the: d,m:I,OIl field of the: dlff.,_ tnll.] equalion and" ~ '110 dniw <01>1.'10.;00' about the quahln!>,·c btha~ior of solutIons. I'or ~colld ~r .quauons. Ihl'5e methods life mo •• cumhe"",,,,,,; you muM .lp"cify twO condllions to pic~ ou t • ""'que wl ution. We u~uallr specify initial "aluci for the (uflCHon and II, fi"1 de, i';lU\O a, lOme point. ",." We ca" u>C mllllcricallcclmiquel IU compole WI appro1"Ilrlt. r.oluuoo In order to;> graph e!loogh soIul;o;>nS 10 gtlln id • • of lhe:n gmeral brII3vlOr, .... mu~t conM!UCI a tWramrler family of lOlulioru. Sin« thl' mittll does not del.nn' .... thl' mlllal ,lope, " C cannot dnt .. a d''''''''!ion fidd fOf a.l«Ond or«,fy ''lIlucs I/{tg) !.tt and 11(11) 1/1 at 1"' 0 distl""l pomts. The n:,u ltmK problem i5 called a boo,,,I,,')' mil" "robl~m. We shall u· I' h3"e an ...hamnge {We, the mOfe 01"'0105 nontencal and graphICal mel hods. They mo;>re cffecll" .'Y y,eld ""onn.tion about flIOJIC,llc, shared by all solullons of ~ dlf. fc .... nlinl equltlOn, nbout lIle Oscillatory nalu,. of &Olut;onl of an equation, and aboot tIK: rrccl~ ...Ite of decay Of glov. th of &o1""on5. Conlpan5Oll methodl 1"0"00 InfOflilallon on the Wlutlon1 of a ''lIriable cor:fficlCnt rqualoon by companng the: «tu,"on "lIh an appropnate con$tanl cor:fflCicnt equation. Tlul possIbIlity ;1 liuggr~d by lhe: fo;>lkMmg narnple I~, large I. the: coeffici~nt 1/ t on JI~Sstn equallo;>n (I 0 2) 15 ~IOM<: 10 O. Su, 1k»(l's equatIOn ,~clost 10 tho: equation II' + 1/ 0, .. h~ gclltral&olullon Clll br .. nlten ... 1/ /I OOt!I{t 6). Thus one: m'ghl u!,<,cl&olunons 10 Iks!.Cl's eqU3110n to o;cilille and lool roughly h~c ~me .. a~e. for large /. w~ rre~nt. '~s "lt IIlal ''lIIid:tlclslICh co"'pamons.
"'lI(
11t<: geometric melhod is ba!.Cd on dnc'{:liOtI fidds, which a,,, 1101 dIrectly arP'icablcuJ a .ccond order equation. We ~how,ltov.c\'C" how 10 conmuct I n:latcd fi,~t orde, ~Iunllon
/0, I, S«ond Clnkr EqUaI",..< "-111\ MATLAB
'" 10.1
Second Order Etluutions with MATLAB
MATUl.II·, u~uallOOl. for fioom, Ilmbo/w; soIullOl'l' of d,ff~~nlJai "'I"a,,0111 ""Ork perfeclly wcll for Jl:cood onkr Inlt;\)" dlfferenllal equ~tmnl ... nl! COIIsunl eot/lk,.nl', lbc synlaJ< of the colllm~nd I. farm liar. ncepllUI",c mUll 'p«,fy an ad,1I1;on.1 in>ll.1rond " tlOn,
EXIl Ill I,le 10. 1 Consider lhr dlffer.nlJal equ.tion
fl Ol) We can gol a JOlu""" by lypmg: » o d e l · ' Dly .. Dy - 6 'y . lO ' .xplt)'I » 4eo l"e (04. 1)
'"' exp{2' t) ' C2 'exp 1-3' t I ' Cl- S' elCPl
t)
As e~pt<'led. the goncr.>l soIullOn drptoos on 1,.'0 """tmy con~, To sol". 1hr dIffer. ential tqUIIJon "lIh mltlll rorKhllOn1 v(O) 0 and VIOl I . • -e " 'OIIki l)',," u
4 .01"e l 04e1 ,
'yIO ) . 0',
'DyIO) . 1 ' ,
'M 4t5'exp(2'tl'21/S ' exp(-3 ' tl 'S ' oxp(t) MATLAII Can abo JOI,'e ~lIe bQuooary ,.. lue probleml 1... ,'. rominu" . sma; lhe !.arne dlffen:nllal equation. and con»der the boonlWy rondlUOII! vIOl 0 and ~(ln(2)1 10_ w. gt't !be JOI.lIon of lhe boundaty ,-alue problem b) IYJllIII u 4 .01"e(04. 1, ' y I O) .0', ' y {109' Il)) • 10',
'"' S ' exp(2 ' t )-S 'exp (tl EKlIllI l'lc Hl.2 Now consIder too I",ear S('C'QIItl onkr dlffcrr nt>al equation lin
+ ell' + II
0_
( IO 4)
MATl A illS unab'" 10 find an ._'plW;,1 formula for lhe ,e ... ral soIU'IOII_ [nWad . ... (In u!oO ode . 5 10 floo a nu",erical SQIDnon lIu, 04eU IS duigllC'd only for filll onJer ~","on,. . ,at,_ ...,'_, The..,f"",. ",e ron.'ert the '·ortunalely II can deal WIth ,'.... tor. 3, "e " .~- ~. ~cond ord~r >col .. "quatlon 10 a fint ortlcr ,..Clor .qu.llon for y . ,,"h yl 1) rrp~!oO n"nll lhe ",.value .'Id yl l ) ",p..,,..nl,,,g tho , al".. of it' deo"""" , Thus 10 loh •• 1«. ond order dt{(crennal ~uanoo nunlrr1('~Il) In MATI.Af! . .. 'e mu, t "''''nte II In the . : :
.,.c.
fi,,,,
[.'/'" .')[ andl"'n>lale''''formDlafor[V:J('.~·VlJ 'nlOananony ~ . . . ~. IIJl lJ lin) TO llJuliralchow func,ion (Of an mil ... func'lOn If )'011" ulml MATLA Of car . '[, .'[
To'~
lh,. works we rttal,(IOA)mI01l ,lV ,.onddefi .... >~ rh • • IJlt , y) (y (2) / -t" 2 ·y (l) - yll))1
ClL1pr..r 10. NINo' lhc: numental solution defined on lhc: m\e",,1 10,31 ",ch inilial oondiciOn1 )1(0) I, vIOl I t~ bo:: found by c)'r,ng » lxa, yal • odan(rba, 1031, 11 -1]1. and 1\ tan bo:: rloued uJing IhI: Iec:hnlllue, de, .Joped In Char'''' 7 A good "-1) 10 unden.land Ihr bo::lIa, 101 of. gt'ne~ 5OIUli00 or a ~ond order eqUaIJon .s co 1'101 numeneal solulions correspond,ng 10 • wIde range of mlw':l ,.. Iuei and IIUUo] slopes 11.", lJC lhc: MATLAB commands co do '" for Ihls nll",,,I .. , » figur., h old on » tor yO • -2.2 tor ypO • - 1.0.5.1 [ttor, ytor] • od •• 5(rb_, [0 ll. [yO ypO]) I [tRk. ybaltl _ od.U(rb_, [0 -21. [yO ypO]) I plo t (ttor , yfor (:.l» plot (tbalt, yRk ( •• l»
.... ....
n.. n:suh ,S!Jloy,,,,.n Fig"'" 10.1. We ~ iliallhe ",1UIKIIU lottm 10 k,d off for po5,ci"e I and S«IIlIO blow up for
•.,
""gam.., t,
..• .. ... •
.
".". .
.
"'gu", 10.] Numol1Cll SolutionS of (]O 4)
Coold we hive oobuned thiS ""Ihudlvc .nformalion dln:clly from lbe differenual equ. uon Inslead of from lhe ,"'pb of a few ",luIIOl1l7 One IP\II()8th IS 10 look allhe 5OlUlion. of dlffcn:nual equM.ons Ihal are sim,lar 10 thll one. For cnmple. when t IS small. t" is ti<»c 10 l.em. So lhe iml"lio". to (1.4) should 00 dMe co lhe wlulions of clle dlffc.cnlial equalion
V'+)I
0
We krIow WI IhI: ",Iuuons to !he lall ... fqUallon are tllle CUrve •. !'or uamplc. lhe wluclon JaudYln, !he 1000Ial cond,hons )I{O) O. V(O) ] IS jll'>!. sinft). So. one mlJliC e~pcct !he: soluuons of IIIlS ,M,al value problem 10 look like 8ill(l) for ~mall vallM's of t ]n
/0.1 Second O"kr l!quaJo OIIJ 'Hrh MATl.All
(
,
.,
/ ,
,
,
,
+U
h gu", 102: Coml>
0
The gmph of lhe nu merical solunon on h gll'" 10.2 appea.. 10 1."01 off:u / il\Crruu,
Lookmjt a~3in al Ihe: di ff.r.tI!I~1 equallon, " 'f 5« !hI! "hen I i, llliC, lhe / I v' I
or
u" + /1';
O.
Thi, equalion can almo»l be M»,'fl/ .'l'iJ,:l1ly: liS VIICnl M»ullon i,g;,." hy
U(t )
, I + JJ
,
1',
-'/3 du
The .'pootnu.al fUllChon bema mlegnled I. M...-Iy uro for l;t!Jc JIIKIIM ,-,JOIN of.. So. " hen I " large alld ]lOS11,,'., lhe solull ons \bou ld h<-rom. allllOl l ronnant Whtn I i.l.,. and nega",'c, li1e e>pone"lla l lenn is huge, and W. ",peCl!he solunon. 10 b]1lW up. Fm~lIy, let's lool al!he boundMy ,aillC rrobkm ( IO.! )
I.
lien: ". ""~e lhe ... me d,ffe.. "l,. l cq"""on, bul m,'. ~i fiC'd ,-.JutS of 1bt .\OIullOII al IWO dlfferenl PO,"II, , 10) 0 and ,( I) I m (';liltti"1I ""'I/m,1 Consider the i"IIIal ." Ioe probkm
0/' + 'V +,
0,
vlO)
0,
,'(0)
A,
(106)
,,'he 'e Ihe ti r~1 in" ,"1 cOlld"ion is the firM bOll ndary condnion. lI1d in tho II«ODd imiW COIllil hon, the slop<' aI / O il ~J"Xifit
s,_
·..
CJ..pIcr 10 SoIII"g
_",".rn,,,,,,,;;
I for "llIch I1M' SQiutlOll SIII'SM$ the J«OOd boundary condition in ( 10.5), IIKn tile 5OIuhOfl of (10.6) ~ I w,l1 be tht soIUIIOIl of OU. boundary vatue problem.
fioo the ... lue 6
",l/I
15 u\u.11y found by S('lecllng ,'lIriou' nur"""nl ""I ..., for •• al>d wl"ng tile inthal ,.1"" problem (IO,~) ",,,ncricilly "'lIh ode. 5, ",nh an eye til fin'hns m.alue for S fUI .. !Itch llIe
I
S('cond l>oundM)' condlllon In (II},') OJ 5allSfi~, TO"anlllll, end, the following fUllellOn
M ·flkisusdul. fUDc~loD r • trial(l) % find. ylll giv«D that y CO) .0, y'IO) • • "be2 • _It,y) [yUt/ _t " 2·Y(2) - y ( l J)1 Ite yel • o.sd5(rhl2 , (011. (0 _]1, r • y.(en4, I), We Il0l. lila. fOi certam sImple naml'ks (10.6) can be 5Oh~ symbohcally ,."h dlcl v. If tile resuilm, 5OIuhon ,5 deOOled by 1/(1. 5), Ihon , can be
found hy wlv, ng 11(1,.)
I ""II hero or IOlv.
A .. llhe con'lanl ,""elm (1 0 I ). The rode Ihal folio",! shows how '0 lllot np4c 10 find and plO! ,iI< ("umcneal) !.01. IIon, whIch is ~how'n In rl/lure 10.3. NOlO w .1ht boundary conllniool II. m~ned In 1ht fonn of a fUnf;.iQn beond of ya and yb. 1ht '-aluc, of y I~; vI ., 1ht "'0 boundary pomls. TIle !.Oh.r ~"IIS ""Ill the mlllal glleU and lools roo- a nearby wl.tion of lhe dlfferenlial C~ yy - daval(bva ol, ttl, » plot(tt,Yr(l,,))
" "
."
...." ." "''"
....
"" .
,,,
..
10.2
Sccnnd Order
1<: (ILIlitiOIlS
'"
with Simulink
One can uj,C Slinubnk. I.!i w~n a~ od.45 !o _ L. _ n"me"".1 ",luuOl'l' r • elIU~!IOl'S _ \\e WIll IIlu'lrn!e Ilow!o do Illli ,. . . ,_ 0 r.«OI'Iu ortltr ... , lnear Imtllli '-II"" problem
,_
. ""'"in
,po
,
u.
dill + ( 1 + ' )11
yeO) ' 0,
V(O)
r
(10.7)
"'hcr~ Rr( basIcally Iwo ways !O COOltroc! a model ~ ,. or II «1"'1101'1 The fiBt which ". L_ t't uS I"", mod", st.o..n '" Figu", 10 4. tl lilt more ob.iou. _.L _~ ~; .. ~u ...... ,"n!tly... t UK 110-0 '
,
l'lgu,," lOA A M(Xk1 for Equ..UOll (10.7)
I.,
In!ei'""!or bloch coupltd u.>gtthn The 6rsl InkgJalo. has JI' II> '"pul ODd V ... II< Outpul_ The Kcond has V as it< Inpul and V aiM output Si~ the rquatiOft wriln V' IS a funcliOfl o f V and, (or in mml Castl •• 1'>0 11'. !hough!1It 11' krm is Illl,ung in (107)), lhal !ell. us how to hnk up the blocks lie .. \I e U!oN' Polynomial hndian block 10 cOfl'pule I + , l from / in, .ingle , Iep; Ihrn we muillply Ih" b)' V In tilt Product bl(IC l and Ihc"n l11uluply by I m I"" Gam block 101<'! JI' Tilt mitia! rondlUOO in LIlt 6.... Inl<'pl!o. SCts lhe v"luc 11'(0) I; Ihe ina", cond'!lon In !1It ~ond Intrgnllor (called Inlcgt"ltorl litre by lIefault --Slnluhnl alway, ,",illS IN! III block ":unoS model lit diuinUOR. \lluch rOf 110.7) ,,-outd lit
h"..,
In '
(11' . -( I + 1')).
(/I,v')'
If We subs!iIU!e y for (V. V'). " 'c ran ", ... n!e thIS U'IIIJ: m>-tri~ oprr:ltlonl al
y'
0
Yt (
'(1 + 1')) 0
allli we C:tll further rewnte!he roefliclcnt m>-tnAII
.
(IU)
.,
1'llIure IO_S, s",opc Output wIth lhe SQlulion 10 (10.7)
1
-(1: t ») (?' u
'(' ') (' ')
(10.9)
(1+1)00+\0
Thi l5uuem conllrocling lht mood !bown in Hg .. '" 10.6. This nlOlk\ use; "e<:IOf signals
".,..,.....
lJId only a , inllk [nl.gralo .. I" Ih" dlal",m. lhe Com!"", block ha) 'u block pammcl.r ~I tOJ IO. 0, 1, OJ.lheGalnblockh"- .... 10 1IllI1f1~ mulhphcauon. on acoordallCC VI nil e<JU.1hon ( 10.8). NOie lhal In the modtl of fig,,", 10.6, lhr Scope blorl i. corul«'cd 10. ,-«fOr Jign"r The output one gtl~" \OOwn '" FIgure 10.7. NOIe Ihat om: .lCcs plols of bulb .nmponc"u of y. in OIher word, ..,f 1>0111 11 and 11'. a' funeh,,", of "Ill •. Smce II i. Ihe fir" compo''''''!' i, IS pl""ffi on )"110'0" ((he bnglne, Cut·... In Figure 10.7); ,he ",cood com""""m J! app:a,'! '" purple.
'"
!'igon: 10 .7, Scope: Oul",,1 "lIh lhe V~1Or SQluuoo to 110.71
10.3
Comparison Methods
In lh" K(lIOf1. ,,~ d, \.Cuss the Siunn Compan\Oll Tlleo .. m. In acldJlion, .. ~ "'Ill d'>CUIllht n:Jalloo bel ..'«n IhI: a ros of 1.... 0 linu d y ,"d~ptndtnl w luhOm of I StCOOd onl
'/lid ~"I'I!tII' u(a ) u(b) 11<'/"'''''' 1" 1/" 'q!tll/ron
0,
0 fo r mm, (I < 6 (ho" " is not "I,ml~olly ,rro), Lr, t'(z) IH a
1/' .. h,rr r(r)
+ q(z)y
',,""""n
+ r(z)1/
O. 0 for ""'" .r '" !Q. b] .
q(z) fnr all 0 :0:; :.r :0:; b, TMn I'(Z)
Wc wIll defci IhI: proof of thIS Ihton:m bmHy. I'fl'(mil\l in
.,
1/+1/
•
OO.IOJ
0.
1.."1 ,.- he II [lQjll""e n"mhe, If II < .r ::; I,. Ihe n I/z ::> II"', W. n(Wo' . pply 1M ~bo>'e Iheorelll w,lh q(z) 1/ It' "ad r (.1 ) 1/ .1. Thu ' rorrlJWT ",yahoo (10. 10) to the eon, !atll eOl' me;"n! equation y" + ( I, 1\ )1/ O. .. holt grr\ of the Inl.,.,,1 Then lht Snum C~ 1'hro.-"nl rrllplle, lh.1 e""y ~lu!lOn of ( 10.10) Ira, I ,,,'" '"' 1111> inlc,,·al} bil ll1JlllllClll rrnphes Ihn! 00 (0 , "" j. 1M ,.eros of 110 10) I,'2llnot t>r farthrr l pM1 l1w1
.i.
n'"
CII.apr~r JO "',."••, 'OM' A••"""., J"'~
.. s
In ",,!ticu1a., 501ullOnf of (10.10) musl OKinnie D' .r • 'Xl. though the osdllalioo~ may b(comc Ius fre'l"""1 a.;r incrra ... ~. I" ......:d. by tumms 11K aoo-'c comp:1f"on ","",lnd. one conc:lu~. lh:ot tho lA:fI.K of solulloo~ of (10 ]0) in nlU~1 be at lea,( ~ V J,. apan
1"-, --..)
F'l ure 10.8.00..) '''0 rcP"'scnWI"~ .... UIlOO. of (10.101 comp.Md "nil ".01... A. yOll can the graph coofinm Ihr prrru,llOl1' of tbr I"'~~'ou, parlIlrap/l
=.
~
\
Thr Siunn ('o'"P''''''''' 111N11cm can al", be u...-d 10 ~l"d)' the: 1.1:10. of 501ul 10111 10 "nY'1 ~ua!ion 31101 10 Il. s~r, cq"'lIIon. In th. 1.lIer ~a"". a ., ubsl\!UI"}II IS made 10 ellmi. n~l~ lhe V' lentl
10,3, 1 Th ... Illll'rhu:1nl: or Zl'ros \'011 "'")' ha,'. '1OIIc<'
+ "(.I)y' + q(.rlv
O.
(10,11 )
If p( .. ) 0,"" lilt else ,n C'qu.ahon (10 I).'''''' proof of,be 1R••,I":,"g re!ul. follO'l'oJ fro,,, ;u, """Iica"oo of lilt S(Uml Co"'pamon 'fh«ntn hy rotnpanng lbe rquauon ... ,(h ,( ... If. 1'« lilt noorr ,cOlOral "'lua" oo (10. I I). ,be proof'~ b.>scd on .be Wronsl o.an I.e. ~,(r) and Vllr) II<' any 'wo Imt3Jly ,ndcl"'nlknl .oIUHOO. of (10 II). Recall !ha( .he Wron,l,.n of~o\r) ~nd VI("} i.
S"loCC 1/1 and Ih are soI~l;ons of{JO I I ). " (01""'1 tlul ll' loll
11"
" .""",
,..,~rnl.olullon
I" ...••
I
•
uo:
'"
.",oua'
p(z)\\'
IS 11'(x)
("
/..-
1>1)1 chlnge "gIl It IIIU,5 ha, the wne ~_ • h h Ill" lOS In" ( COI111ant ." I~ must be nonzero (or else ill and ill woold be Ion"all y depcndtol). . A""m
Th,. lonnuJa '>how, that \I' (z) doo:s
11'(6)
1I;(b)Ul{b).
S;IICC' 'I, ! be >:ero SOfhe\Oo tit", OO1O'«n 0 and b. Sinula'h. V, mu\! bo,., .. «m beh. ff:n ~ny '" 0 7.trOS of 1/<1.
~;u rdlr 10. 1 The sotallons to Airy's equauon:and S.U.. I'I cq~I'on IR rom.xml "" lmponallt thai [hey ha •• tJe<,n givu n~mcJ, Tbr Airy fullCUOII' Ai{r) and Bi(z) .,. a fun . damc"t,,1 ~t of
