4. Runge-Kutta Formula For Differential Equations. A. Euler Formula B. Runge-Kutta Formula C. An Example for Fourth-Order Runge-Kutta Formula
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1 NCTU Deprme o Elecrcl d Compuer Egeerg Seor Course <Dymc Sysem Alyss d Smulo> By Pro. Yo-Pg Ce. Ruge-Ku Formul For Derel Equos A. Euler Formul B. Ruge-Ku Formul C. A Emple or Four-Order Ruge-Ku Formul To solve e derel equos umerclly e mos useul ormul s clled Ruge-Ku ormul wc s bee wdely used umercl lyss. A. Euler Formul For dymc sysem wou pu s geerlly epressed s e ollowg rs-order derel equo: ( & ( α (- were s e l me d ( α s e l codo. Te problem o solve ( (- or > s clled e l vlue problem or IVP bre. For emple e ollowg equo & ( (- s IVP d s soluo c be obed closed orm s below: e However (- s gve more complced suc s s (- & ( (- e s oe dcul or mpossble o d ( closed orm sce mos o e derel equos do o ve closed orm soluo. To deermme e soluo umercl meods re requred. Te smples umercl meod s clled e Euler ormul wc ws propsed by Euler 768. W e use o ed grd sze e grd pos log re orderly deoed s were (-5 d e vlue ( s deed s (-6 -
2 NCTU Deprme o Elecrcl d Compuer Egeerg Seor Course <Dymc Sysem Alyss d Smulo> By Pro. Yo-Pg Ce - were ( α s gve. Accordg o e deo o dervve we ve lm & (-7 wc mples e dervve o ( c be ppromely epressed s & (-8 Subsug (-8 o (- leds o α (-9 d e α (- wc s e well ow Euler ormul. Clerly order o ceve soluo more precsely e grd sze sould be s smll s possble. However reducg e grd sze s ece sce e cos o clculo me my crese remedously. Now le s roduce Tylor s epso o epl e error cused by Euler ormul (-. Accordg o Tylor s epso uco y( couous c be epressed s (- wose ger order dervves re & (- && (- &&& 5 5 (- Vewg rom e bove equos we ve d or d us (- c be rewre s && & (-5 e we ve
3 NCTU Deprme o Elecrcl d Compuer Egeerg Seor Course <Dymc Sysem Alyss d Smulo> By Pro. Yo-Pg Ce - && & (-6.e. O (-7 were O (-8 Comprg (-7 w (- we ow e erm O s e error o Euler ormul wc s proporol o. B. Ruge-Ku Formul I order o reduce e error o urer mody Euler orm (- s e ollowg orm: δ γ β β (-9 were β β γ d δ re vrbles o be deermed d ew erm δ γ s cluded. Sce depeds o ( d s Tylor s epso ( d s gve s e ollowg orm y (- were ll e coeces re cos. Is prl dervves re (-
4 ( ( y ( y ( NCTU Deprme o Elecrcl d Compuer Egeerg Seor Course <Dymc Sysem Alyss d Smulo> By Pro. Yo-Pg Ce ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (- (- (- (-5 Clerly ll e coeces c be derved rom e bove prl dervves d epressed s ( ( ( ( ( (. Subsug em o (- yelds ( ( ( ( ( ( ( ( e γ d ( δ e ( ( ( ( ( ( γ δ ( ( ( ( ( γ ( δ ( γ ( γδ Clerly (-9 c be rewre s ( δ ( O( (-6 (-7 O [( β β ( ] [ β γ ( β δ ( ] [ β γ ( 6β γδ ( β δ ( ] - (-8
5 NCTU Deprme o Elecrcl d Compuer Egeerg Seor Course <Dymc Sysem Alyss d Smulo> By Pro. Yo-Pg Ce I we w o e e precso o rom (-7 d (-8 we ve ( [( β ( ] ( O β (-9 [ β γ ( β δ ( ] O( Sce &( ( e erm ( ( ( ( ( ( ( ( & As resul rom (-9 d (- we ob c be cged o e ollowg orm: ( ( ( ( ( wc leds o [( β β ( ] [ β γ ( β δ ( ] (- (- β β (- ( β δ ( ( ( ( β γ (-.e. β γ (- ( β δ (-5 Sce ere re our vrbles β β γ d δ ree equos (- (- d (-5 e coce o ese vrbles s o uque. Te mos commo oe s β β γ δ ( (-6.e. rom (-8 we ve e pprome ssoluo s below: ( ( ( (-7 wc s clled e secod-order Ruge-Ku ormul. For e coveece o progrmmg e ormul s oe rerrged s below: were ( (-8 ( ( (-9-5
6 NCTU Deprme o Elecrcl d Compuer Egeerg Seor Course <Dymc Sysem Alyss d Smulo> By Pro. Yo-Pg Ce -6 I cse ger precso s requred we oem employ e ger order Ruge-Ku ormul. For emple precso o s eeded e e our-order Ruge-Ku ormul mus be used wc s gve s 6 (- were (- Ts ormul s bee wdely ppled o lo o pplcos egeerg due o s smplcy d ccepble ccurcy. C. A Emple or Four-Order Ruge-Ku Formul Ne le s use emple o sow e progrmmg o e our-order Ruge-Ku ormul MATAB. Cosder e ollowg equo: & (.5 (- d d e soluo ( or. Te closed orm soluo c be obed s 5 e. (- Now le s pply e our-order Ruge-Ku ormul o solve (- umerclly. I e sep sze s se s. e e precso wll be e order o 8. Accordg o (- we ve 6 (- were
7 Te progrmmg MATAB s gve s below: NCTU Deprme o Elecrcl d Compuer Egeerg Seor Course <Dymc Sysem Alyss d Smulo> By Pro. Yo-Pg Ce >> % Four order Ruge-Ku meod >>.5; % l codo (.5 >>.; % sep sze.sec >> -*(-; -*(/-; >> -*(/-; -*(-; >> ((**/6; >> e((-(-.5*ep(-; % umercl error >> (; >> or :599 % ol smulo me 6 sec >> -*((-; -*((/-; >> -*((/-; -*((-; >> (((**/6; >> ((*; % -s >> e((-(-.5*ep(-(; % umercl error >> ed >> plo(; lbel( ; ylbel( ( ( >> plo(e; lbel( ; ylbel( e( e(
8 NCTU Deprme o Elecrcl d Compuer Egeerg Seor Course <Dymc Sysem Alyss d Smulo> By Pro. Yo-Pg Ce From e bove umercl resuls o e( s ru e precso s less wc s deed roud e order o ( 8. I MATAB some srucos re provded o solve e derel equos suc s ode d ode5. I c ese srucos re lso mplemeed by e cocep o Ruge-Ku meod. Now le s use e sruco ode o solve e sysem (- d use e sruco ode5 o solve e sysem descrbed below: (.5s( s & (.5 (-5 wose soluo c o be epressed by closed orm. Te progrmmg MATAB s gve s below: Cree m-le: rs.m uco drs( d-; Cree m-le: rs.m uco drs( d-.5*s(s(*; >> % ey e ollowg srucos >> []ode(@rs[:.:6].5 >> plo(; lbel( ; ylbel( ( ( >> % ey e ollowg srucos >> []ode5(@rs[:.:6].5 >> plo(; lbel( ; ylbel( ( -8
9 NCTU Deprme o Elecrcl d Compuer Egeerg Seor Course <Dymc Sysem Alyss d Smulo> By Pro. Yo-Pg Ce ( Problems P.- Derve e ollowg rd-order Ruge-Ku ormul: were 6 ( ( ( P.- Solve e ollowg sysem:. ( e s( 5 & ( -9
4. Runge-Kutta Formula For Differential Equations
NCTU Deprme o Elecrcl d Compuer Egeerg 5 Sprg Course by Pro. Yo-Pg Ce. Ruge-Ku Formul For Derel Equos To solve e derel equos umerclly e mos useul ormul s clled Ruge-Ku ormul
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