NUMERICAL INTEGRATION OF DIFFERENTIA．
k
EQuATIoNs 一 oN o≦x＜oo BYTATsuJIRo SHIMIZU
Let us consider the numerical integration of the ordi皿ary differential equation塞イ（x，・）・n・≦・≦姻…nce…h…1・…nb・ing・assum・旬，・h・…i・k・・w・
that the numerical integration on O≦x≦b can apProximate the true solution pre− cisely as desired， if the step size h may be chosen properly sma11． In this paper we do not consider rounding errors． If the numerical integration． is to be required fbr a very long interva1，0r the solu− tign is not known how血r it exists， the above theorem can not be applied． Mathe− ＝ノてx，ア）on O≦x≦oo （assuming or matically it is to integrate the solution of dx not the existence、 of the solution on O≦x＜oo）． ’ If we asSume the existence of the solution on O≦x＜◎o， the accumulated・trunca− ti・n’err・・s can n・t be e・tim・t・d・wh・n W・i・t・gr・t・With・t・p・セ・乃（h・w・v…sm・11 乃may bQ）the solution on O≦x＜o◎． Ordinary the皿merical integratioh is perfbrmed by choosing h，，乃2，…，偏… properly in order to diminish the a㏄umulated errors． But such a process is not succSessf㌔1． ． ・ As we have shown the accumulated errors can not be made smaller than a given small quantity by such a process ’in genera1． 『． 』 「 F…he eq・・…n呈一・，・（旬一1・he・㏄一1・t’・d・…in・・ea・e・血・・価・・ly・・ we integrate on O≦x＜◎o however small we may choose the seque皿㏄of the step sizes h1，t． h2， … ， hn， … ．1） Since the accumulated errors， the fbrmula of which is known， can not be esti− mated precisely on O≦x＜oo even fbr a simple differential equation， we shall show without using the accumulated e宜ors−fbrmula， that some皿umerical methods give numerical integrations which behave similarly to the tme solutions on O≦x＜◎o（f‘）r some simple diffヒrential equations）． Definition：Let the differential equatign ． il’i−・f（・， ・）〕 ，μ） 1）T．ShimiZu：Contribution to the Theory of Numerical Integration of Non・linear Differential Equations Oり． TRU Mathematics， Vol．7，1971， pp．50−57． ［63］64
T．SHIMIZU
be・9i・en；wh・・ρ・ f（x，’V）i・an・1yti岬th・eSpe・t・t・x・ndアi・0≦x＜。。・lyl＜。。・ and let− the solution −y（x）． of（1）withッ⑩e加o皿0≦x＜。◎． Further we assume㌫rρlthr興゜W興ρ゜噸゜；s’S−lsati←x≧x°：αi be』ce蘭n
（1） ア（X）→O for− x→oo。 Where．〆’＞0，・〆＜O and y’→O whenγ（xo）＞0 、 γ”＜0，ア’＞Oand〆→O whenγ（κo）＜0， q・）“アω→αキ・㊤・・→…，α≧Owhere〆’＞0・γ’＜O and −y→O wh・n・7（x・）＞0 α＜、O where〆’＜0，〆．＞O and〆→O whenγ（x・）＜0ダ （IID γ（x）→aキO fbr．x→oo，α＞O where），”・ぐ0，〆＞O and〆→O whenγ（xo）＜0 〆Owh・・e〆’＞0，〆＜0・nd．〆→O・whenアぴ・）＞0， ⑲γω→・・f・・．・→・・wh・・eツ”＞0・・〆’〈0・・d〆〉。、 ．yω→一。。 ．．．whe・eγ’＜0・rγ”＞O andγ’＜0・ Then the solution qf（1）iS said． 狽潤@satisfy the‘co耳dition（A）． Theorem：ff／−tlie÷soletieff．。具9−s噸《rs．−thecoh《lition．（A），−the・−numerical−integra・．． tiopッヵof（1）hy Euleピs血ethod Sptisfies gne of l the fb皿oWing fbur conditions re− sp㏄tively， Whep the、 sequen㏄qf step sizes乃1， h2，… ， h・・，… is properly chosen．（1） 、 γ。→0．fot x。→。。． 、
（II） ．・．み→α、fbば。→。。 ”、 『
．．ｿID ． 万→d fbrXn→。。
（IV）． γ・→°°（or −O◎）fgr XE→°°
⑰・T・！。．a・e、th？・alUes・t・＝・・一乃・＋乃・＋…輪（〔・2・・・う・ ． Proof 3 First we put戊ゾ（κ，7）≡g（x，ア）mthe fbllowi皿g． Letッ（x節）be the lvalues of the tmC solution of（1）at xのw与ere l−
・x・’−h・＋乃・＋…＋乃・＠−1・2・…），then−．．
A、）二胸＋、9幅。ω）†与〆（㊧．’1、、
On the other hand， Euler「s method ， ．．． 』、．． ・「e 『 ． み＋・＝．γ・＋hg（Xi，γ・）・．・ Putting en＝．7（Xs）一・みWe have 1．． ・、 − 1 ・∵ 二、『εカ＋・＝＝・en（1＋・・‘3i．（・・’））＋♀ア〃（・・L』』w・…↓〃（・・）・霧…（・…e｛・e曲・・・・・・…輌・…g・h・mea・二・・1・・血…em
respectively． T…c ⑩・C・・ ・・s・・h・・．・・（」c・）〉輌（・・）〉・…dん岨’・h・・〃・；…≦η＜11． ・．’・・．−、→
・．晦晦・〆＞0．w・、 haΨ．・ω≧坤τah≧”・・．°n・the°ihe嘩騨W夕h°豆se
恥uch that∴．，．ご．・t．．一．：二 三．、、．． ＿． ．
NUMERICAL INTEGRATION OF DIFFERENTIAL EQUATIONS ON O皐くoo 65
9（κ。，万） −1乃。∫（xπ，ル）1≦η＜’1 y祁 fbrル＋、＝ぬ＋iち9（x。，み）， we haveぬ＞O fbr an n． By assumptionア（x。）→0， we hqve〃→O for xn→o◎． Now we shaU shOw th聖ye ti can choose仏， which satisfies the above two conditions alldΣhゴ→。◎fbrη→◎。． ∫＝1 Consider the sequence of i亘tervals〃1−1≦x≦〃1（m＝1，2，3，…）， and put ・（M）一。畷。｛max；多・−1∫（…）1｝・ w・・re ・’浮煤@．an・胸・・垣・・…輌・・みく・ω・nd・（xn）＜Kw・hav・レ・1＜K・
Hence we can put K forレ1， and consider the magnitudes of the fonctionψ（｝n）． Now・1f we choose h，，， such that］h．q（m）＝η＜1 and we take the step size he・ throughout the inter’va1．m−1≦x≦m（〃1＝1，2， …）r−when we integrate o110≦x＜◎o・ Thus we can choose乃％satisfシing all conditions． The other case Can be treated Perfectly similarly． ・ 。 The cases（II），（IID． Since the solutionア（x）satisfies y（x）→αfbr y”＞0，−putting γ一α＝｝∼．we have 芸一・（・，r＋at）…書・（x，・）・ By y’＝γ’，γ”＝Y”and Y→O fbr x→◎◎． Euler’s method fbr the equation Y’＝g（κ，y＋α）gives 、 ’
Yn．、一・Yn＋hg（Xn， Yn＋α）． ． −Now
γtx。）一α一γ（x。）＞O when〆’＞0，〆＜O if y（x。）＞0 ＜0 〆’＜0，γ’＞0 ＞0 ＞0 ア”＞0，」〆＜0 ＜0 ＜0 γ”＜0，y’＞0 ・＜0． Each case belollgs to one of the two cases，of（1）， and】陥→O fbr xn→oo． Euler’s methodみ＋1＝み十んg（xヵ， y。）is satis五ed byみ一α＝Yn， and Euler’s method gives unique solution， and hen㏄the solution by Euler，s method】r”＋1 ＝】㌦十乃確（xヵ， 】∼十α）is nothing other than｝㌦『yヵ一α． Hence 7抄→αfbr x”→◎◎． The above relation holds Perfectly shnilarly fbr the other casesα＞O andα＜0・ The eaぷθ（IV）． Let us consider the caseγ”＜O and y（X）→oo． By choosing ん ∂9 ≦η〈1we haveみ〉γ（xn）fbr all n（puttingγo≧」ノ（xo））s㎞皿arly assuch that hヵ ∂γ in the case（D． Of collrse rg（x，ア）’contains x andγ， but byγ’＞0，ッ”＜O we have， fbr large values of x，γ（x）＜Lx． Hence we can take Lx fbrレ（副o田n−1≦x≦m， （〃1＝〃lo，〃le十1，… ）． Thus we can 6btah1（ρ（吻as sirnilarly as in the case（1）． By y（x）→◎◎we have γヵ→◎o… tt ・ The caseア”＞0，夕→−c◎can be treated．similarly as’ above、 Byγ（xn）＞y．「andy（Xn）→一。。 wc haveル→一。。・… 』 ・ 一… 一一’三∠
v．．
66
1T． SHIMIZU The caseγ”＞0， y→◎◎． In this case， by夕’＞0，9《三；；層γ（壌＞0． Hence・f・r・a’su伍dently small乃we have 8（x。，み）＞0．Hen㏄夕。．、＞y。．． ・ ：了：・ 一 ・… ∬・’・ご㌔ Byγ”＞O wC’have 9（hr。＋、，1・（ぬ。、））＞g（xn， y（xh）），・and g（xh＋Cγ亮」≧g（x＃，1，蕗）． Now y。ぞiL万r㎏』（ぬ，』匁）、giyes A＋i」γo＝〃Σ9（Xj， y5）． Hen㏄y。→◎o・fbT n→◎。． Th。 c嬬ヅ’＜0，♪→一。・da。も。 t，浸。d，imi1。，1y。、。b。ve−・，一 REMARK I． 111versely if We haveみ→O orα負）r xn→6◎， when we、integrate nu孟erically the solution． of（1）．by』Euler’s method， ChOosing． the seque皿ce of step sizes h，， h2，…1prope｝1y， and satisfyingアノ≦0， Yn”f≦O for all n respectively as fbr the cases（1）and（II）， then the true．． solut！on wrU．satisfy y（κ）→010rα． fbr x→o◎， Sin6e the true solution will satisfy”ア’（xn）≦0， y”（xn）≦O resp㏄tively fbr』some simple 《lifferential lequatiol1． We can not assert thatγ（x）→αf（）r the case（III）even whenン潟→αsatisfying み’＞0，．ytt’＞Ol．In中is case there may㏄cur thatγ（x）→αor◎o． REMARK II． Defi〃∫励〃．’If for a numerical integration of some differeritial equa占 tion the accumulated trqncation −errors e。（A）by a numerical method A are smaUer than those eヵ（8｝of a methodβin abso1Ute valueS fbr all n≧ηo fbr’asequgttce of step sizes乃1，乃2，…properly chosen，輌t is said that the method∠4 is nloオe a㏄Urate tban the methodβfbr such Sequence of step−sizes乃1，乃2，…． Suppose a solution of a di1笛erential equation exists on O≦x≦ゐ， and the nulnericalintegration is perfbrmed．by some numerical n鴫thod on O≦x≦b（convergent in
Dahlquist’s sense）．， ， 一 ！ It is evident thqt、amethod，．thQ lopal， truncation error of which is hρNp， is more a㏄urate than a n輿hod， that．ρf W匝ch is hgNe， fbr a su伍ciently small step s鋭ze乃， when p＞4 andεo r O（θo being the initial eπor）． Prooゾ： The accumulated ttuncation error can be exptessed by ・・ i1＋・募ω）…（1＋h＆t∼・・））＋眺（1＋h＆f（・・））…（1＋弓チ（・・一・））＋…〉 ＋砿一・（1＋・募（・・））＋・・N・，・i． 1是t’ ・・一 ｩ（1＋み筈（・・））…（踊31（・・一・））＋…職・一・（1＋」；チ（・・））＋聴・ bO’the．error by the method、4 and ． ・ ！・ …． ㌧． ・・⊇（1＋h＆f（・・’））…（1＋・募（・’・一・））＋…＋・・凡・一・（1＋ん砦（・・う）滅・’ be that by the methodβ． ． 、ここ．．、・一 ・．ハら，‘・．｛しnd凡，《are analytic jn O≦x≦b，1γ1＜◎◎， and limited in absQlute values「on o≦x≦ゐ・Henoe we can take le”1＜leヵl fbr su伍cient1弊3ma皿一々カーeボ： ＿ご：）一・・…● ● NUMERICAL】NTEGRATK）N OF DIF F［EiRENTIAL EQUATIONS ON O；〈1；．x＜∞ 67 REMARK III． The result in Remark II does not hold fbr the numerical integra・ tion by some numerical method o110≦x≦◎o泊1 genera1． For面stan㏄： So】［ve numerica皿y the equation〆＝−y with．γ（q）＝1 by Euler，s methodア昂＋1＝γ外一h7h and by．Taylor’s expansio皿method of the s㏄ond order ・。＋・一・ ・・。＋芸・。（・・一の・ Put eカ＝γ（Xヵ）−Yn add εお＝y（Xn）−Zn， then ．． ノ ．． ・・ ． ・。＋・一・。（1一の＋♀・〃（・。），（・〃（・・．）−e…θり 書w（1−h）n＋・・（1−h）・一・＋…＋e・”・h｝ 一♀・畔伽＋謀譜1｝・ On the other hand， ・。＋F」ξ・・’…h …＋… i1−・＋♀）カ＋1−1 1 一
