ON NUMERICAL INTEGRATION OF DIFFERENTIAL EQUATIONS BY USING THE SECOND METHOD OF LIAPUNOV

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ON NUMERICAL INTEGRATION OF DIFFERENTIAL

EQUATIONS BY USING THE SECOND

METHOD OF LIAPUZNOV

BY

HIRosHI MORIMOTO AND HIDEo IMURA

h缶oduction． This paper is concerned with the problem for’numer輌cal integration of ordinary differential equations with initial◎onditions． It is hardly possible to solve non・1inear differential equations analytically． So numerical integration method is the only remaining altemative to solve non・1inear ditferential equatio亘s in general and many available formUlae fbr numerical integration are well known． But it may be㎞possible to avoid di伍culties caused by fixed step s靴zes fbr numerical integra● tion． Step sizes have to vary a㏄ording to the behavior of the solution． For instance， the solution of van der Pol’s equatiQn

（1）鵬÷＋k（・・−1）緩＋炉・

with initial◎onditions 血（2） x＝2 and ＝O at τ＝O is shown in Fig．1． 2 1 一一 y2’一一 7 31ソ2 2τ 5コソ2 −一 0−1−2 一一一一一一−一一一一一一一丁亭2鳶（3／2−lo92）一一一≡≡一一 t Fig．1 For numerical integration of vertical parts of the solution， step sizes have to be small enough． On the contrary， fbr horiZontal parts they n㏄d not to be small． Moreover， for large k such as 100， it is di伍cUlt to calculate horizontal parts by small step sizes． Sin㏄such phellomena arise from non−linearities of van der Pol’s equati皿， the values of’at which step sizes must be varied can not be assumed， previously． In this paper， we sha皿propose some method to Vary step sizes by using the so− called s㏄ond method of Liapunov．［25］

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26 _{H．MORIMOTO AND H． IMUR．A}

1．Second皿etho己of Liapロno▼． A． M． Liapunov dealt with stabiHty of ordinary differential equations by his s㏄ond method． At present， the second method of Liap皿ov is・耳Ot oply used tg esl ablish Stabi晦ΦporemS but、 a1So plays、㎞portant 「°

ﾄ二、鷲11劉麟蒜e三。蒜d’na「y砥「輌’・qnat’

（・） −∵芸一∫（・，・）．

wh・・e・i・a・n−vect°「gnd f（ちx）is an刀lv蜘「・麺｝i°n which is defin°d in 1×R” and is cont口1uous in （t， x）柚ere l denotes the interval O≦t＜oo and R外 does EucHdean n−space． Let V（’， x）be a continuous scalar function defined on an open set S． For con− ・ウenience，1et V（t，．x）’haVe「continuous P’artial．derivativεs df the fヒst．ordet Viith reSpect

to（ちx）．．「．．．．． ⊥ ’『．・』’ 囁 ’

FOr V（ii，”x）and the system（3）we shall．define Ptt（「， x）as fbllows．． tt．．’

’（・）一’∴

@v・（’，・）一砦＋票蝋’，・．

where・二‘‘・〉’． denotes． the scalar product． Let x（t）be a Solution‘・of（3）staying in．5． Then，防〔’，．X（τ））』『is C6htinuous・and differentiablc∋with respect toぬ． l TherefOre， we． shall obtain．the ， follOWing’ exPressiOn （・）鰍，rt’））一警（・・（’））＋｝／（・，・（’M・，埠））・ 2．Nロmerica1血ltegration．．．．、 Now， we．shall consider numerical integration．of a system of differential equations

（・）緩一綱．．’

with initial con砒ions （7） x＝Xo at t＝’o where x is an n−v㏄tor and’f（x）is an〃−v㏄tor function whiCh is de丘ned in R扮and contlnuous ln x． We suppose that the fbrlnula fbr numerical血tegration is co皿vergent and round・ off errors are丑eglected．．．`§sume．that xn at t＝＝tn has been calculated as the・true・vqlUe Qf．the．solution， then

、、．．・．・・x＝x。＋∫（Xh）（t−t。）・．．、

denot撃the． straight line passing thrgugh the point（ん， xn）・ ’・，．フ FOr a．Certain−Step SiZe h，・ −’ ．．・．、、 t ・一．．−tt ．．．． x。．、＝Xn＋∫（Xn）〃・； is the apProximate value at’”＋1＝’％十．h． ’−Let．’xn＋1． be坐e true value． at．・’＝㍍＋ゴand V（x） be a continuous scalar funqtion de丘ned on an open set S． Moreoveらlet V（x）Satisらlocally a Lipsch晦conditi叫 and the Splution x（t）of（6）with（7）stay in S． For sma11乃， there exists a neibor・

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h・・dσ・f・・s・・hth・tσ⊂S，ぷ∈U・q・d・，，・∈q・’1 L・t L b・th・Up・chit・・c・n− …n・・f・V（x）wi・h・邸・ct t・・i・豆Th…t亘・f・11・wご・xp・es・i・・曲・・b陥麺・

w（：1・1川1・、，n。、es，hl窒誌㍑チ竺｜を＋1一鋤．：．・・∵・・

1㎞・・e・h・p・・b1・m i・．・・4eq・n剛・h五㎡・・i皿I W・・i・・，・upP…磁艇） and γ（x）are cont伍ubuSli ’differe血til’ble up tg寧e des益ed brder，伍en we ca五cai− ・㎡…d耽divldt2 and・d3vldt3 as fun・・i・n・Of；吐・m（5）．a・d（6）・・ On the other hand，．， V（x）＝＝ V（Xn）＋牛・。）（’一’。）＋；鵬Z（鞠X’二’1）・＋諜（・（’。＋・（’一’。））X・一’。）・．（’。≦’≦’。＋、，and O＜θ＜1）． Then， ’「’』1 ． … ，・一『： ∴ t・ ’・．だ：、，庭・・）一・V（Xn）＋誓（・・）h＋捲ζ（・・）糾orh・）・・’∴∴

士hei⇒逗・͡i、・Wilfbe吊P6ssib16、6 ca1、。1。、e・「∵∵≡∴・・ハ

_{．．1∴．．㍉屈．’．．㍉∴㌧γ（X。＋・）7γ（X。ぶ・），∵・／一 1・／．＼∴…’、「』恋，∴・} up to the term of乃2・；：；： ’・・㌃1，、；．：・．＝．／，・，・∴ ：．、．．．：．，「．．．、／．t、．；．・，；．：t！

、。㌃霊’。認竃ii）i霊；：麟監震蕊誌C鑑、。隠竃

SO．that’．…1∵！〕・・i．・’・．：・二・・：’／i・：l t／1∴−．・∴』・∴・．・．・．1…．：／一・…．∴呪．・’○・・’1、∴、、1。、’「：∴，，．・・パ；・’・・γ（x。￥、）一レ（x。♀i）！’〆・∵’一” t：一・‘・○は isf§malr enouきhプニ・Accordihgly， we 8han．obtai丘ithe・・criterio11・for．dete血血血g：Step sizes aS fOllows二；・”…．・∴・・「，“＿f・i’t・，・∵∵ t．Ol．；… 』・・∴．：．∵．．．、・，（9）」レて”＋1）一レてX”＋1）1≦εL ．’∫．．：1．：； wh・・e ez・ d…tes・・m・11 valui・d・p・ndi・g・・L・・・…h・…p・i・ea・’一’・S・・i・fyi・・．・h…i…i・・（9）h・・b㏄・d…麺・d・・h・ value of x at t＝＝ tn＋1 wirbe not always calcpt［ated bY ／、 x・＋∫（為）h ・・di醐l be cal皿1…d．・’b・・”・，・W1・血岬・f・−icali…9r・・i・㎏・i・・t・…

the R皿9各Kutta meth

_{Hd・‘・、．／}

∫ ノ i ． ∼

、，㌶・1。鵠霊・ユ・Van；㏄ご1｝i・er・n（頑血唾・・q’・’…（・）

〆 t ！（10）．

↓

手

く

克、・黶D＝ γ

一 X

＝＝

些凌砂万

、’ ーノ〆ぐ t／ with initial conditions

（11）・・ x−2and・一一争・・t−0・

For this system， we shall obtain ；．．．．

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28

_{H．MORIMOTO AND H． IMURA}

’・2） ⊆：蕊（X．3 −Xn 3）乃

For th6 system（10）， we shaU define V（x， y）as ifollows．

｛13）．V（x，γ）琴x＋y．

Then， for（10）and（13）， from（5）we shall obtain as fbllows．砦一ナ・（芸一・＋・裂一一・＋｛k（・・−1）−1｝（・＋k（等一り｝・ From（12），玖扇・㌃）一・・一（y・＋k（誓一x”）｝・＋・・＋・… Therefbre， V（・…，・…）一玖扁・・痛・）一丁［一・・＋｛・（…−1）−1｝ト＋晴3−・・）｝］h・＋鍋 On the other hand， the Lipschitz ◎onstant of アてx，γ） is 1． The皿， considering terms up to乃2， the criterion for determining step sircs is

㈹丁［一・・＋臨・−1）−1｝｛・・＋k（誓3−・g｝］・・＜・

whereεis a suitable constant determined aocording to the desiered a㏄uracy・of the solqtion and does not depend on lthe Lipschitz constant・ −S・art伍9・f・・m・頑・1…d・・i…（11），−1…一…d…＝＝一争k， w・ca・dee・d・ the step size fbr calculati’ng xn＋1 andγヵ＋1 by（14）as soon as we have obtained xn andみ． ’ ←2，6666）夕

，。 ∫（L④一

、、．14 ∫ ‘ 、で←＿＿＿一一一一軌8−5∼2L5−5 _ 60 −『一『°卜在0−4∼2’ 一4プ’’ 50 _’ f育 ω 、、 30 ・、20−1∼1、25−2．、 20 、、 10 、一1．732 、 1，732 F一一一一一一2LO −1．5 −1．0 一α5 0 （瓦5 1．0 、エ5

20

一10 、一20 、_、一30 、、一40 L24−2∼仕25−3_@ 、一50 ㌧、一60 @ −67．14 ‘＿1 へ、、イ一70 （−1，−6688）（2，−6666） Fig．2

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For k＝100，ε・＝O．OOOOl and the Euler method for numerical integration， the result calculated by a digital computor will be shown in Fig．2． In Fig．2． values de皿oted between dotted lines， for．instance 8．0−4∼2．0−4＝8．0×10−4∼2．0×10−4， will show step sizes which are used for calculation．

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ON NUMERICAL INTEGRATION OF DIFFERENTIAL EQUATIONS BY USING THE SECOND METHOD OF LIAPUNOV