numerical solution for initial value problem numerical analysis
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Numerical Solution for Initial Value Problem
Numerical Analysis
Introduction àžï¿œàžàž«àž²àžï¿œàž²à¹àž£ï¿œàž¡àžï¿œàž àžï¿œàž àžï¿œàžàž«àž²àž§ï¿œàž²àžï¿œàž§àž¢àžàž²àž£à¹àžï¿œàžªàž¡àžàž²àž£à¹àžàž
àžàžï¿œàžï¿œàžàžï¿œàž ï¿œàžªàžàžàžàž¥ï¿œàžàžàžï¿œàžà¹àžï¿œï¿œàžàžà¹àžàžï¿œàž²à¹àž£ï¿œàž¡àžï¿œàžà¹àžï¿œï¿œàžàžà¹àž àž§àž à¹àžàž àžï¿œàž§à¹àž¥àžàž ï¿œà¹àž¡ï¿œà¹àžï¿œà¹àž«ï¿œàžï¿œàž²àžàž£àž°àž¡àž²àžàž ï¿œàžï¿œàžà¹àžï¿œï¿œàžàž à¹àžï¿œà¹àž«ï¿œàžï¿œàž²àžàž£àž°àž¡àž²àž
àž àžï¿œàžàž ï¿œàž)àž²àž«àžàžàžï¿œàž²àžàž£àž°àž¡àž²àžàžï¿œï¿œàžà¹àž£àž°àž«àž§ï¿œàž²àžàžï¿œàžàž ï¿œàž)àž²àž«àžàžàžªàž²àž¡àž²àž£àžà¹àžï¿œàžàž²àž£àžàž£àž°àž¡àž²àžàžï¿œàž²à¹àžàžï¿œàž§àž
x0 x1 x2 x3 xn
ð ðð ð= ðáºð,ðá», ðð †ð†ðð, ðáºððá»= ð¶
Well-posed Problemúð à ªµ€Ãn€ ÂŽï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ õ®ŠŽ {®µnµÃŠ·Ã€ oï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðŠâ² = ðáºð¥,ðŠá», ð †ð¥â€ ð, ðŠáºðá»= ðŒ €€»·ᅵ ðó ððŠ n°ÃºÃ° õ®ŠŽ »ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ð¥Ãï¿œ áŸð,ðá¿ ÂŽ à {®µnµÃŠ·Ã€ o€ à ¥®¹ÃÃ¥ªᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œðŠáºð¥á» õ®Š ï¿œ ð¥âáŸð,ðá¿Ã³ {®µÃÊ¥ªnµ {®µÃn€ ÂŽï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ (well-posed problem)
Well-posed Formula: Example¡·µŠ µŽ®µnµÃŠ·Ã€ oï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ xyxy sin1 , 20 x , 00 y
à ÃÃï¿œ ï¿œ ï¿œ xyxyxf sin1,
xyxyxf y cos, 2
ï¿œ Ãï¿œ ï¿œ ÂŒnï¿œ n°Ãï¿œ ºÃ°ᅵ õ®Šᅵ 2,0x ÂŽ à Ãï¿œ ¥ᅵ³€Ã¡¥ᅵᅵ Ãï¿œ Â¥Ãᅵ¥ªÃ³Ãï¿œÃᅵᅵᅵ®µᅵnµÃŠ·Ã€ᅵoï¿œÃï¿œn€ᅵᅵᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ
Classes of MethodsŠ³€µᅵ ï¿œ nµᅵ xy ÃÂ¥ õ®ᅵ ï¿œ ï¿œ ï¿œ x ï¿œ Ãï¿œ »ᅵ ï¿œ nµᅵ ÃÃï¿œ Ãï¿œ ,,, 210 xxx ³à oï¿œ ï¿œ iy Ãà nµŠ³€µ °ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ixy
1. Š³Ã¥ª·žÃÃ¥ªᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ (one step method)
Œᅵ Šᅵ °ᅵ ï¿œ µŠᅵ Š³€µᅵ ï¿œ nµᅵ ³°¥ŒnÃï¿œ ŠŒᅵ hyxhyy iiii ,,1 ,2,1,0i
ÃÂ¥Ãï¿œ ï¿œ hyx ii ,, Ãâ r nª Ãà Ã¥Ú Ãۼðᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ xÚÃÂ¥ nµµᅵ ï¿œ ï¿œ ï¿œ ï¿œ ix ÃÃï¿œ ï¿œ hxi
2. Š³Ã¥ª·®µ¥Ãï¿œ ï¿œ ï¿œ ï¿œ ï¿œ (multi-step method)
Ãà µŠ®µᅵ ï¿œ ï¿œ 1iy Ã¥ᅵ Ãï¿œ oï¿œ šÃï¿œ ¥ᅵ ÊŒoï¿œ nµÃoª€µᅵ ï¿œ ªnµ1 ï¿œ »ᅵ Š³Ãï¿œ ¥ᅵ ª·ᅵ ï¿œ ÀŠŒᅵ Ãï¿œ ï¿œ ï¿œ °ᅵ Œᅵ ŠÃï¿œ Ãï¿œ
m
kkikik
m
kkiki hyxbhyay
101 ,, ,2,1,0i
ÃÂ¥Ãï¿œ ï¿œ maaa ,,, 10 , mbbb ,,, 10 Ãà nµ ªᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ
Classes of Methods õ®Šᅵ Š³Ãï¿œ ¥ᅵ ª·ᅵ ®µ¥ᅵ Ãï¿œ ï¿œ ¹Ãï¿œ €ŠŒᅵ Œᅵ ŠÃï¿œ Ãï¿œ
m
kkikik
m
kkiki hyxbhyay
101 ,, ,2,1,0i
oµᅵ 0ma ®Šº° 0mb Š³Ã¥ª·ž³€ᅵ ï¿œ ï¿œ ï¿œ 1m ï¿œ Ãï¿œ ኵ³ᅵ o°ᅵ Ãï¿œ oï¿œ o°€Œ 1m ï¿œ »ᅵ oµᅵ ðâð = ð ³ µ€µŠ ®µᅵ ï¿œ ðð+ð ï¿œ µᅵ ï¿œ nµᅵ µᅵ ï¿œ oµᅵ ï¿œ ªµᅵ °ᅵ Œᅵ Šà ᅵ oï¿œ ÂŽï¿œ ï¿œ Š³Ãï¿œ ¥ᅵ ª·ᅵ ï¿œ Ãï¿œ ³ÃŠ¥ᅵ ªnµ Š³Ã¥ᅵ ï¿œ
ª·Ã¥ŽÃoï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ (explicit) oµᅵ ðâð â ð ³€ ªà €nŠµ nµᅵ ï¿œ ï¿œ ï¿œ ï¿œ ðð+ð ï¿œ Šµᅵ ï¿œ ï¿œ Ãï¿œ °ᅵ ï¿œ oµᅵ ï¿œ °ᅵ Œᅵ Š Š³Ãï¿œ ¥ᅵ ª·ᅵ ï¿œ Ãï¿œ ³ÃŠ¥ᅵ ªnµ Š³Ã¥ª·žᅵ ï¿œ ï¿œ
Ã¥Š·¥µ¥ᅵ ï¿œ (implicit)
Euler Method õ®ŠŽ {®µnµÃŠ·Ã€ oï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðâ² = ðáºð,ðá», ð †ð†ð, ðáºðá»= 𶠳 õ® îonµᅵ ï¿œ ï¿œ ï¿œ ï¿œ ð ÃnµÃŽ³à ᅵoªnµᅵ ï¿œ ï¿œ ï¿œ ï¿œ ðð = ð+ ðð ð = ð,ð,ð,âŠ,ðµ
š nµŠ³¥³ µŠ³®ªnµ »ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ð= ðâððµÃŠ¥ ªnµᅵ µ ° Ãï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ
Euler MethodÃ¥ᅵ §¬ž ° ÃÂ¥rðŠrï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ (Taylorâs Theorem)
oï¿œ µðŠáºð¥á» Ãï¿œ Ãï¿œ ï¿œ Ãï¿œ ¥ᅵ °ᅵ ï¿œ ï¿œ ®µᅵ nµÃŠ·Ã€ᅵ oï¿œ ó ðŠáºð¥á»âð¶2áŸð,ðá¿ Ãoª
ðŠáºð¥ð+1á»= ðŠáºð¥ðá»+ ðŠâ²áºð¥ðá»áºð¥ð+1 â ð¥ðá»+ ðŠâ²â²áºððá»2! áºð¥ð+1 â ð¥ðá»2 Ãۼðð = 0,1,2,âŠ,ðâ 1 ó µᅵ ï¿œ ðð âáºð¥ð,ð¥ð+1á» Ãۼðîoâ = ð¥ð+1 â ð¥ð ó µᅵ ï¿œ ðŠâ² = ðáºð¥,ðŠá» ³à oï¿œ ï¿œ
ðŠáºð¥ð+1á»= ðŠáºð¥ðá»+ âðáºð¥ð,ðŠðá»+ â22!ðŠâ²â²áºððá»
Euler Method: Formula
ŒŠŠ³ÃÂ¥ ª·žᅵ ï¿œ ï¿œ ï¿œ Euler
ðŠ0 = ðŒ ðŠð+1 = ðŠð + âðáºð¥ð,ðŠðỠ€ nµªµ€ µà ºÃ° áµ³ ·Ãï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ (Local Error) Ã}ï¿œ ï¿œ â22! ðŠâ²â²áºððỠõ®ŠŽ µᅵ ï¿œ ï¿œ ðð âáºð¥ð,ð¥ð+1á»
Euler Method: Geometry Meaningµ µŠ Ãîoï¿œ ï¿œ ï¿œ ï¿œ ðŠð Ãï¿œ Ãï¿œ ï¿œ nµᅵ Š³€µᅵ ï¿œ °ᅵ ðŠáºð¥ðỠʵà ᅵ oªnµðŠðâ² â ðŠâ²áºð¥ðá»= ðáºð¥ð,ðŠðá»
1y
yxfy , ay
Slope ,afay
a 1x
yxfy ,
ay
1y
a 1x
2x
2y
Euler Method: ExampleÃoŠ³Ã¥ª·žᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ Euler à µŠ Š³€µ à ¥° Ž®µnµÃŠ·Ã€ oï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ
12 xyy , 20 x , 5.00 y ÃÂ¥Ãoï¿œ ï¿œ 10N , 2.0h ó ixi 2.0
µᅵ ᅵ 1, 2 xyyxf
îo 5.00 y ó 121 iiii xyhyy 9,,2,1,0 i
0i , 120001 xyhyy
8000.0105.02.05.0
1i , 121112 xyhyy
1520.112.08.02.08.0 2 2i 12
2223 xyhyy
5504.114.0152.12.0152.1 2
Euler Method: Exampleï¿œ Ãï¿œ ¥ᅵ ÃÃï¿œ oï¿œ Š·ᅵ õ®Š ï¿œ ï¿œ ï¿œ ®µᅵ nµÃŠ·Ã€ᅵ oï¿œ ï¿œ Ãï¿œ Ãï¿œ º° xexxy 5.01 2
i x y y(x) |y-y(x)|0 0.0000000 0.5000000 0.5000000 0.00000001 0.2000000 0.8000000 0.8292986 0.02929862 0.4000000 1.1520000 1.2140877 0.06208773 0.6000000 1.5504000 1.6489406 0.09854064 0.8000000 1.9884800 2.1272295 0.13874955 1.0000000 2.4581760 2.6408591 0.18268316 1.2000000 2.9498112 3.1799415 0.23013037 1.4000000 3.4517734 3.7324000 0.28062668 1.6000000 3.9501281 4.2834838 0.33335579 1.8000000 4.4281538 4.8151763 0.3870225
10 2.0000000 4.8657845 5.3054720 0.4396875
Taylor order nµ µŠ Š³ µ¥° »Š€ÃÂ¥rðŠrŠ° »ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ð= ðð ³à oï¿œ ï¿œ
ðáºðá»= ðáºððá»+ ðâ²áºððá»áºðâ ðð á»+ ðâ²â²áºððá»ð! áºðâ ððá»ð + â¯+ ðáºðá»áºððá»ð! áºðâ ððá»ð õ®ŠŽ µᅵ ï¿œ ï¿œ ðð âáºðð,ðð+ðỠµᅵ ï¿œ ðâ²áºððá»= ð൫ðð,ðáºððá»àµ¯, ðâ²â²áºððá»= ðâ²àµ«ðð,ðáºððá»àµ¯, ...,,ðáºðá»áºððá»= ðáºðâðá»àµ«ðð,ðáºððá»àµ¯ ³à oï¿œ ï¿œ
ðáºðð+ðá»= ðáºððá»+ ðð൫ðð,ðáºððá»àµ¯+ ððð ðâ²àµ«ðð,ðáºððá»àµ¯+ â¯+ ððð!ðáºðâðá»àµ«ðð,ðáºððá»àµ¯+ ðð+ðáºð+ ðá»!ðáºðá»àµ«ðð,ðáºððá»àµ¯
Taylor order n: FormulaŠ³Ã¥ª·žᅵ ï¿œ ï¿œ Taylor ° ÂŽï¿œ ï¿œ ï¿œ n ðð = ð¶ ðð+ð = ðð + ðð»áºðá»áºðð,ððá» ð = ð,ð,ð,âŠ,ðµâ ð Ãۼð
ð»áºðá»áºðð,ððá»= ðáºðð,ððá»+ ðððâ²áºðð,ððá»+ â¯+ ððâðð! ðáºðâðá»áºðð,ððá» nµ·¡ µÃ¡µ³ ·à º°ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðð+ð
áºð+ðá»!ðáºð+ðá»áºððỠõ®ŠŽ µᅵ ï¿œ ï¿œ ðð âáºðð,ðð+ðá»
Taylor order n: Exampleï¿œ ï¿œ ÃoŠ³Ã¥ª·žᅵ ï¿œ ï¿œ ï¿œ Taylor ° ÂŽï¿œ ï¿œ ï¿œ 2 ó° ÂŽï¿œ ï¿œ ï¿œ 4 ÂŽ {®µnµÃŠ·Ã€ oï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðâ² = ðâ ðð + ð,ð†ð†ð,ðáºðá»= ð.ð úð µÃŠµÃoÃÂ¥rðŠr° ÂŽï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ 4 ¹ o° Ão° »¡Ž r°ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ð ¹° ÂŽï¿œ ï¿œ ï¿œ ï¿œ ï¿œ 3 ¹Ãï¿œ ï¿œ ðáºð,ðá»= ðâðð + ð ðâ²áºð,ðá»= ð ð ð൫ðâ ðð + ð൯= ðâ² â ðð= ðâ ðð + ð â ðð ðâ²â²áºð,ðá»= ð ð ð൫ðâ ðð + ð â ðð൯= ðâ² â ððâ ð = ðâ ðð â ððâ ð
ðâ²â²â² áºð,ðá»= ð ð ð൫ðâ ðð â ððâ ð൯= ðâ² â ððâ ð = ðâ ðð â ððâ ð
Taylor order n: Exampleà oï¿œ ð»áºðá»áºðð,ððá»= ðáºðð,ððá»+ ðððâ²áºðð,ððá» = ðð â ððð + ð + ðð൫ðð â ððð â ððð + ð൯= áð + ððá൫ðð â ððð + ð൯â ððð ð»áºðá»áºðð,ððá»= ðáºðð,ððá»+ ðððâ²áºðð,ððá»+ ððð ðâ²â²áºðð,ððá»+ ðððððâ²â²â² áºðð,ððá» = áð + ðð + ððð + ððððá൫ðð â ððð൯âáð + ðð + ððððáððð + ð + ðð â ððð â ðððð
Taylor order n: Exampleà o ŒŠÃÂ¥rðŠr° °ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðð = ð.ð ðð+ð = ðð + ð൬ð + ðð൰൫ðð â ððð + ð൯â ððð൚ ó ŒŠÃÂ¥rðŠr° Ãï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðð = ð.ð ðð+ð = ðð + ðááð + ðð+ ððð + ððððá൫ðð â ððð൯âáð + ðð+ ððððáððð + ð
+ ððâ ððð â ðððð õ®Šᅵ ð = ð,ð,ð,âŠ,ðµâ ð
Taylor order n: Exampleîo ð= ð.ð, ðµ= ðð ÂŽ Ãï¿œ ï¿œ ï¿œ ï¿œ ðð = ðð + ðð= ð+ ð.ðð áºð = ð,âŠ,ððỠµŠ³Ã¥ª·ÃÂ¥rðŠr° õ®ŠŽᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ð = ð à oï¿œ
ðð = ðð + ð൬ð + ðð൰൫ððâ ððð + ð൯â ððð൚= ð.ð+ ð.ðáŸð + ð.ð)áºð.ðâ ð+ ðá»â ðá¿= ð.ðð õ®Šᅵ ð = ð, ðð = ð.ð
ðð = ðð + ð൬ð + ðð൰൫ðð â ððð + ð൯â ððð൚= ð.ðð+ ð.ðï¿œð + ð.ð)൫ð.ððâ ð.ðð + ð൯â ð.ðáºð.ðá»àµ§= ð.ðððð
Taylor order n: ExampleŠ³Ã¥ª·ÃÂ¥rðŠr° ÂŽï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ 4 îo
õ®Šᅵ ð = ð ðð = ðð + ðááð + ðð+ ððð + ððððá൫ðð â ððð൯âáð + ðð+ ððððáððð+ ð + ððâ ðððâ ðððð
= ð.ð + ð.ðááð + ð.ðð +áºð.ðá»ðð +áºð.ðá»ððð á൫ð.ðâ ðð൯âáð + ð.ðð + ð.ðððð áð.ðáºðá»+ ð + ð.ðð âáºð.ðá»ðð âáºð.ðá»ððð = ð.ðððð
Taylor order n: Example
õ®ŠŽᅵ ð = ð, ðð = ð.ð
ðð = ðð + ðááð + ðð+ ððð + ððððá൫ðð â ððð൯âáð + ðð+ ððððáððð + ð + ððâ ððð â ðððð = ð.ðððð+ ð.ðááð + ð.ðð +áºð.ðá»ðð +áºð.ðá»ððð á൫ð.ððððâ ð.ðð൯
âáð + ð.ðð + ð.ðððð áð.ðáºð.ðá»+ ð + ð.ðð âáºð.ðá»ðð âáºð.ðá»ððð = ð.ðððððððð
Taylor order n: Example
Exact Taylor order 2 Error
Taylor order 4 Error ðð ðáºððá» ðð È7ðáºððá»â ððÈ7 ðð È7ðáºððá»â ððÈ7
0.00 0.5000000 0.5000000 0.0000000 0.5000000 0.0000000 0.20 0.8292986 0.8300000 0.0007014 0.8293000 0.0000014 0.40 1.2140877 1.2158000 0.0017123 1.2140910 0.0000033 0.60 1.6489406 1.6520760 0.0031354 1.6489468 0.0000062 0.80 2.1272295 2.1323327 0.0051032 2.1272396 0.0000101 1.00 2.6408591 2.6486459 0.0077868 2.6408744 0.0000153 1.20 3.1799415 3.1913480 0.0114065 3.1799640 0.0000225 1.40 3.7324000 3.7486446 0.0162446 3.7324321 0.0000321 1.60 4.2834838 4.3061464 0.0226626 4.2835285 0.0000447 1.80 4.8151763 4.8462986 0.0311223 4.8152377 0.0000614 2.00 5.3054720 5.3476843 0.0422123 5.3055554 0.0000834
Interpolate other valuesÃۼð o° µŠ Šµ nµᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ð(ð.ðð) ó µ€µŠ Ãoª· µŠ Š³€µ nµà nªÃnÃoï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ¡®»µ€ᅵ µŠ° rï¿œ ï¿œ ï¿œ 2 »ᅵ ï¿œ ª°¥nµÃnï¿œ ï¿œ ï¿œ ï¿œ îoðð = ð.ðó ðð = ð.ðó µ o°€ŒÃŠ³€µà oµŠ³Ã¥ª·ÃÂ¥rï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ
ðŠr° ÂŽï¿œ ï¿œ ï¿œ 4 à oï¿œ ðáºð.ððá»â൬
ð.ððâ ð.ðð.ðâ ð.ð൰áºð.ðððððððá»+൬ð.ððâ ð.ðð.ðâ ð.ð൰áºð.ðððððððá»= ð.ðððððð
¹ÃnµŠ· °ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðáºð.ððá»= ð.ððððððð ÂŽ à nµŠ³€µ Ãà o¹€ ªµ€ µà ºÃ° á¥ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ 0.0007525
Interpolate other values
ÃۼðÃo¡®»µ€ᅵ ï¿œ Hermite ³ o° ®µnµŠ³€µ °ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðâ²(ð.ð)ó ðâ²(ð.ð) ¹Ãõà oÃ¥°µ«¥ªµ€ €¡Ž r°ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðâ²áºðá»= ðáºð,ðá» ÂŽ Ãï¿œ ï¿œ ï¿œ ðâ²áºð.ðá»= ðáºð.ðá»âáºð.ðá»ð + ð = ð.ðððððððâáºð.ðá»ð + ð= ð.ððððððð ðâ²áºð.ðá»= ðáºð.ðá»âáºð.ðá»ð + ð = ð.ððððððð âáºð.ðá»ð + ð= ð.ððððððð
Interpolate other valuesÃÂ¥Ão nµ ºÃºÃ° ®µ € Š³ · ·Ã° ¡®»µ€ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ Hermite ³à oo°€Œ µŠµᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ
1.2 3.1799640 2.7399640
1.2 3.1799640 0.1118825 2.7623405 -0.3071225
1.4 3.7324321 0.0504580 2.7724321
1.4 3.7324321 ðáºð.ððá»â ð.ððððððð+áºð.ððâ ð.ðá»áºð.ðððððððá»+áºð.ððâ ð.ðá»ðáºð.ðððððððá»+áºð.ððâ ð.ðá»ðáºð.ððâ ð.ðá»áºâð.ðððððððá»= ð.ððððððð
25
Runge-Kutta Method (RK)
àž£àž°à¹àž àž¢àžàž§àž Runge-Kutta à¹àž-àžàžàž²àž£àžàž£ï¿œàžàžàž£ï¿œàžàž£àž°à¹àž àž¢àžàž§àž à¹àžàž¢ï¿œà¹àž¥àžàž£ï¿œ à¹àžï¿œï¿œàžà¹àž«ï¿œàžàžàžï¿œàžàžàžà¹àžàžàžàžàžàžï¿œàž²àžàžàžàž¥àž²àžàžï¿œàžàžï¿œàžàžª/àžàž¢ï¿œàžàžàžàž£ï¿œàžàž©àž²à¹àž§ï¿œ à¹àžàžàžàž°àž ï¿œà¹àž£àž²à¹àž¡ï¿œàž)àž²à¹àž-àžàžï¿œàžàžàž«àž²àžàžï¿œàžï¿œàžàžï¿œàž¢ï¿œàžàž¢àžï¿œàžàžï¿œàžàžª/àž
26
Taylor Theory in 2 Variables
oµᅵ ðó »° »¡Ž rÂ¥n°¥° ÂŽ o°¥ªnµ®Šº°ÃnµŽᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ð n°ÃºÃ° ÃÀᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ð«=áŒáºð,ðá»|ð †ð†ð,ð†ð†ð áœÃ³ áºð,ðá» ÂŽï¿œ ï¿œ áºð+ ð,ð+ ðá»Â°Â¥ÂŒnÃï¿œ ð« ÃÂŒnÃoªᅵ ï¿œ ï¿œ ðáºð+ ð,ð+ ðá»= ðáºð,ðá»+ ð ððððáºð,ðá»+ ð ððððáºð,ðá»àµš
+áððð ððððððáºð,ðá»+ ðð ðððððððáºð,ðá»+ ððð ððððððáºð,ðá» + â¯
+ ðð! áððáððâðððð
ð=ððððððâðððððáºð,ðá»
27
Runge-Kutta order 2Š³Ã¥ª·ÃÂ¥rðŠr° ° à o€µµᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðáºðð+ðá»= ðáºððá»+ ððâ²áºððá»+ ððð ðâ²â²áºððá»+ ððð! ðâ²â²â² áºððá» ðáºðð+ðá»= ðáºððá»+ ððáºðð,ððá»+ ððð ðâ²áºðð,ððá»+ ððð! ðâ²â²â² áºððỠúð µᅵ ï¿œ ï¿œ ï¿œ ðâ²áºðð,ððá»= ðððð൫ðð,ðáºððá»àµ¯+ ðâ²áºððá» ðððð൫ðð,ðáºððá»àµ¯Ã³ ðâ²áºððá»=ð൫ðð,ðáºððá»àµ¯ à oï¿œ ðáºðð+ðá»= ðáºððá»+ ðàµðáºðð,ððá»+ ðð ðððð൫ðð,ðáºððá»àµ¯+ ð൫ðð,ðáºððá»àµ¯ðð ðððð൫ðð,ðáºððá»àµ¯àµ
+ ððð! ðâ²â²â² áºððá»
28
Runge-Kutta order 2ʥåᅵ ï¿œ ï¿œ ï¿œðáºðð+ðá»= ðáºððá»+ ðàµðáºðð,ððá»+ ðð ðððð൫ðð,ðáºððá»àµ¯+ ð൫ðð,ðáºððá»àµ¯ðð ðððð൫ðð,ðáºððá»àµ¯àµ
+ ððð! ðâ²â²â² áºððá» ÂŽ µŠ Š³ 杭 rï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ððáºðð + ð¶,ðáºððá»+ ð·á»ÃÂ¥Ão§¬ž ° ÃÂ¥rðŠrï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ð = ð ³à oï¿œ ï¿œ
ððáºðð + ð¶,ðáºððá»+ ð·á»= ððáºðð,ð ðá»+ ð¶ ðððð൫ðð,ðáºððá»àµ¯+ ð· ðððð൫ðð,ðáºððá»àµ¯àµšàµ© ððáºðð,ð ðá»+ ðð¶ ðððð൫ðð,ðáºððá»àµ¯+ ðð· ðððð൫ðð,ðáºððá»àµ¯ Ã¥¡ r³à oªnµᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ
ð = ð, ð¶= ððó ð·= ððð൫ðð,ðáºððá»àµ¯
29
Runge-Kutta order 2ʥåᅵ ï¿œ ï¿œ ï¿œðáºðð+ðá»= ðáºððá»+ ðàµðáºðð,ððá»+ ðð ðððð൫ðð,ðáºððá»àµ¯+ ð൫ðð,ðáºððá»àµ¯ðð ðððð൫ðð,ðáºððá»àµ¯àµ
+ ððð! ðâ²â²â² áºððá» ÂŽ µŠ Š³ 杭 rï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ððáºðð + ð¶,ðáºððá»+ ð·á»ÃÂ¥Ão§¬ž ° ÃÂ¥rðŠrï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ð = ð ³à oï¿œ ï¿œ
ððáºðð + ð¶,ðáºððá»+ ð·á»= ððáºðð,ð ðá»+ ð¶ ðððð൫ðð,ðáºððá»àµ¯+ ð· ðððð൫ðð,ðáºððá»àµ¯àµšàµ© ððáºðð,ð ðá»+ ðð¶ ðððð൫ðð,ðáºððá»àµ¯+ ðð· ðððð൫ðð,ðáºððá»àµ¯ Ã¥¡ r³à oªnµᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ
ð = ð, ð¶= ððó ð·= ððð൫ðð,ðáºððá»àµ¯
30
Runge-Kutta order 2
µŠà oª¥ᅵ ï¿œ ï¿œ ï¿œ ððáºðð + ð¶,ðáºððá»+ ð·á»ÃŠ³Ã¥ª·ÃÂ¥rðŠrà €nÃnµÃº° Ã¥ª oµÃoï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ᅵ¡ rÊŒᅵ ï¿œ ï¿œ ï¿œ ððð൫ðð,ðáºððá»àµ¯+ ðððáðð + ð¶,ðáºððá»+ ð·ð൫ðð,ðáºððá»àµ¯á ªᅵÃŠà Š·€Ã®o ŒŠ° ° õª ° ÂŽ r¹Ãº°ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ŠŒà ° ŒŠᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ Runge-Kutta ° °ᅵ ï¿œ ï¿œ ï¿œ(RK2) ðð+ð = ðð + ð ï¿œðððáºðð,ððá»+ ððð൫ðð + ð¶,ðð + ð·ðáºðð,ððá»àµ¯àµ§
31
RK2: Midpoint Formula
ðð = ð, ðð = ð, ð¶= ððó ð·= ðð ðð+ð = ðð + ððàµðð + ðð,ðð + ðððáºðð,ððá»àµ±
ð = ð,ð,ð,âŠ,ðµâ ð
32
RK2: Modified Euler Formula
ðð = ðð, ðð = ðð, ð¶= ðó ð·= ð
ðð+ð = ðð + ðð ï¿œðáºðð,ððá»+ ð൫ðð+ð,ðð + ððáºðð,ððá»àµ¯àµ§ ð = ð,ð,ð,âŠ,ðµâ ð
33
RK2: Heunâs Formulaðð = ðð, ðð = ðð, ð¶= ðððó ð·= ððð
ðð+ð = ðð + ððáðáºðð,ððá»+ ððáðð + ððð,ðð + ððððáºðð,ððá»á ð = ð,ð,ð,âŠ,ðµâ ð
¹Ãï¿œ ï¿œ nµ·¡ µÃ¡µ³ ·à ° à µ€ ŒŠ º°ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ð¶àµ«ðð൯
34
RK2: Example
ÃoŠ³Ã¥ª·žᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ Runge-Kutta ÂŽ {®µnµÃŠ·Ã€ oï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðâ² = ðâ ðð + ð, ð†ð†ð, ðáºðá»= ð.ð ¡Šo°€ oª¥ᅵ ðµ= ðð,ð= ð.ð, ðð = ð.ðð óðð = ð.𠌊 µŠ Š³€µ µŠ³Ã¥ª·žnµÃà oÃnï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ŒŠ »¹à µᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðð+ð = ð.ðððð â ð.ðððððð â ð.ðððð+ ð.ðð𠌊ᅵ Euler ŽÚᅵ ï¿œ ï¿œ ï¿œ ðð+ð = ð.ðððð â ð.ðððððð â ð.ðððð+ ð.ðð𠌊ᅵ Heun ðð+ð = ð.ðððð â ð.ðððððð â ð.ðððð+ ð.ððððð ð = ð,ð,ð,âŠ,ð
35
RK2: Exampleµᅵ ï¿œ ðâ² = ðâ ðð + ð, ð†ð†ð, ðáºðá»= ð.ðó ðµ= ðð,ð= ð.ð, ðð = ð.ðð, ðð = ð.ð ¹Ãà oªnµᅵ ï¿œ ï¿œ ðáºð,ðá»= ðâ ðð + ð
ŒŠ »¹à µ º°ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðð+ð = ðð + ðð൬ðð + ðð ,ðð + ðððáºðð,ððá»àµ° ÃÂ¥ ·Ã¡µ³ᅵ ï¿œ ï¿œ ï¿œ ðð + ðððáºðð,ððá»= ðð + ðð൫ðð â ðð¢ð + ð൯ = ðð + ð.ðð ൫ðð âáºð.ððá»ð + ð൯= ðð + ð.ððð â ð.ððððð + ð.ð
= ð.ððð â ð.ðððð+ ð.ð
36
RK2: Exampleµᅵ ï¿œ ðð + ðð = ð.ðð+ ð.ð à nµ Ãï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðáºð,ðỠ³à oï¿œ ï¿œ
ððàµðð + ðð,ðð + ðððáºðð,ððá»àµ±= ð.ðàµáºð.ððð â ð.ðððð+ ð.ðá»âáºð.ðð+ ð.ðá»ð + ðàµ
= ð.ðàµð.ððð â ð.ððððð + ð.ð â൫ð.ðððð + ð.ððð+ ð.ðð൯+ ðൠ= ð.ðàµð.ððð â ð.ððððð + ð.ð â ð.ðððð â ð.ðððâ ð.ðð + ðൠ= ð.ðððð â ð.ðððððð + ð.ððâ ð.ððððð â ð.ððððâ ð.ððð+ ð.ð
= ð.ðððð â ð.ðððððð â ð.ðððð+ ð.ððð
37
RK2: ExampleÂŽ à ŒŠ »¹à µ õ®ŠŽ {®µÃº°ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ
ðð+ð = ðð + ððàµðð + ðð,ðð + ðððáºðð,ððá»àµ± = ðð + ð.ðððð â ð.ðððððð â ð.ðððð+ ð.ððð
= ð.ðððð â ð.ðððððð â ð.ðððð+ ð.ððð
38
RK2: Example
ðð ðáºððá» Midpoint Method
Error Modified Euler
Error Heun's Method
Error
0.00 0.5000000 0.5000000 0.0000000 0.5000000 0.0000000 0.5000000 0.0000000 0.20 0.8292986 0.8280000 0.0012986 0.8260000 0.0032986 0.8273333 0.0019653 0.40 1.2140877 1.2113600 0.0027277 1.2069200 0.0071677 1.2098800 0.0042077 0.60 1.6489406 1.6446592 0.0042814 1.6372424 0.0116982 1.6421869 0.0067537 0.80 2.1272295 2.1212842 0.0059453 2.1102357 0.0169938 2.1176014 0.0096281 1.00 2.6408591 2.6331668 0.0076923 2.6176876 0.0231715 2.6280070 0.0128521 1.20 3.1799415 3.1704634 0.0094781 3.1495789 0.0303627 3.1635019 0.0164396 1.40 3.7324000 3.7211654 0.0112346 3.6936862 0.0387138 3.7120057 0.0203944 1.60 4.2834838 4.2706218 0.0128620 4.2350972 0.0483866 4.2587802 0.0247035 1.80 4.8151763 4.8009586 0.0142177 4.7556185 0.0595577 4.7858452 0.0293310 2.00 5.3054720 5.2903695 0.0151025 5.2330546 0.0724173 5.2712645 0.0342074
39
Runge-Kutta order 4
ðð+ð = ðð + ððáºðð + ððð + ððð + ððá» ðð = ððáºðð,ððá» ðð = ðð൬ðð + ðð,ðð + ððð൰
ðð = ðð൬ðð + ðð,ðð + ððð൰ ðð = ððáºðð+ð,ðð + ððá» ð = ð,ð,ð,âŠ,ðµâ ð nµ·¡ µÃ¡µ³ ·à º°ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ð¶àµ«ðð൯
40
RK4: ExampleÃoŠ³Ã¥ª·žᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ Runge-Kutta ° ÂŽï¿œ ï¿œ ï¿œ 4 ÂŽ {®µnµÃŠ·Ã€ oï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðâ² = ðâ ðð + ð, ð†ð†ð, ðáºðá»= ð.ð
¡Šo°€ oª¥ᅵ ðµ= ðð, ð= ð.ð, ðð = ð.ðð óðð = ð.ð
µᅵ ï¿œ ðáºð¥,ðŠá»= ðŠâ ð¥2 + 1, ðµ= ðð, ð= ð.ð, ðð = ð.ðð ó ðð = ð.ð ð = 0 ðŠ1 = ðŠ0 + 16áºð1 + 2ð2 + 2ð3 + ð4á» ðŸ1 = ððáºðð,ððá»= 0.2áºðŠ0 â ð¥02 + 1á» = 0.2áº0.5â 02 + 1á»= 0.3000000
41
RK4: Example
ð2 = âð൬ð¥0 + â2,ðŠ0 + ð12൰= ð.ðð൬ð+ ð.ðð ,ð.ð+ ð.ðð ൰
= ð.ð൫ð.ððâ ð.ðð + ð൯= ð.ððððððð
ð3 = âð൬ð¥0 + â2,ðŠ0 + ð22൰= ð.ðð൬ð+ ð.ðð ,ð.ð+ ð.ðððð ൰
= ð.ð൫ð.ðððâ ð.ðð + ð൯= ð.ððððððð ð4 = âðáºð¥1,ðŠ0 + ð3á»= ð.ððáºð.ðáºðá»,ð.ð+ ð.ððððá» = ð.ð൫ð.ððððâ ð.ðð + ð൯= ð.ððððððð
42
RK4: ExampleÂŽ Ãà oï¿œ ï¿œ ï¿œ ï¿œ ï¿œ
ðð = ð.ð+ ððáºð.ððððððð+ ðáºð.ðððððððá»+ ðáºð.ðððððððá»+ ð.ðððððððá»= ð.ððððððð
ð = ð ðð = ðð + ððáºðð +ððð +ððð +ððá» ð²ð = ððáºðð,ððá»= ð.ð൫ðð â ððð + ð൯
= ð.ð൫ð.ðððððððâ ð.ðð + ð൯= ð.ððððððð
43
RK4: Exampleð2 = âð൬ð¥1 + â2,ðŠ1 + ð12൰= ð.ðð൬ð.ð+ ð.ðð ,ð.ððððððð+ ð.ðððððððð ൰
= ð.ð൫ð.ððððððððâ ð.ðð + ð൯= ð.ððððððð
ð3 = âð൬ð¥0 + â2,ðŠ0 + ð22൰= ð.ðð൬ð.ð+ ð.ðð ,ð.ððððððð+ ð.ðððððððð ൰
= ð.ð൫ð.ððððððððâ ð.ðð + ð൯= ð.ððððððð ð4 = âðáºð¥1,ðŠ0 + ð3á»= ð.ððáºð.ðáºðá»,ð.ððððððð+ ð.ðððððððá» = ð.ð൫ð.ððððððâ ð.ðð + ð൯= ð.ððððððð
à oï¿œ ðð = ð.ððððððð+ ððáºð.ððððððð+ ðáºð.ðððððððá»+ ðáºð.ðððððððá»+ð.ðððððððá»= ð.ððððððð
44
RK4: Example
ðð RK4 ðáºððá» Error ðð ðð ðð ðð 0.00 0.5000000 0.5000000 0.0000000 0.3000000 0.3280000 0.3308000 0.3581600 0.20 0.8292933 0.8292986 0.0000053 0.3578587 0.3836445 0.3862231 0.4111033 0.40 1.2140762 1.2140877 0.0000114 0.4108152 0.4338968 0.4362049 0.4580562 0.60 1.6489220 1.6489406 0.0000186 0.4577844 0.4775628 0.4795407 0.4976925 0.80 2.1272027 2.1272295 0.0000269 0.4974405 0.5131846 0.5147590 0.5283923 1.00 2.6408227 2.6408591 0.0000364 0.5281645 0.5389810 0.5400626 0.5481771 1.20 3.1798942 3.1799415 0.0000474 0.5479788 0.5527767 0.5532565 0.5546301 1.40 3.7323401 3.7324000 0.0000599 0.5544680 0.5519148 0.5516595 0.5447999 1.60 4.2834095 4.2834838 0.0000743 0.5446819 0.5331501 0.5319969 0.5150813 1.80 4.8150857 4.8151763 0.0000906 0.5150171 0.4925189 0.4902690 0.4610709 2.00 5.3053630 5.3054720 0.0001089
45
RK4 vs Other MethodsŠ³Ã¥ª·žᅵ ï¿œ ï¿œ Runge-Kutta ° ÂŽï¿œ ï¿œ ï¿œ 4 ÃoµŠ®µnµ¢{rï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ 4 ŠÃn° à °ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ °µÃ¥à oï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ᅵŠ³Ã¥ª·žᅵ ï¿œ ï¿œ Euler À µ ³ÃŠᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ( à °ᅵ ï¿œ ï¿œ ï¿œ )Ã}ï¿œ ï¿œ ð𠊳媷žᅵ ï¿œ ï¿œ Runge-Kutta ° ÂŽï¿œ ï¿œ ï¿œ 4 ªnµŠ³Ã¥ª·žᅵ ï¿œ ï¿œ ï¿œ ï¿œ Runge-Kutta ° ÂŽï¿œ ï¿œ ï¿œ 2 ( À µ °ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ᅵà °ᅵ ï¿œ ï¿œ ï¿œ ðð ) ኵ³Š³Ã¥ª·žᅵ ï¿œ ï¿œ Runge-Kutta ° ÂŽï¿œ ï¿œ ï¿œ 4Ãoõª µŠ®µnµ€µ ªnµÃ}ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ 2 Ãnµᅵ° Ãn³ à °ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ
46
RK4 vs Other Methods¡·µŠ µ{®µᅵ ï¿œ ï¿œ ï¿œ ðâ² = ðâ ðð + ð, ð†ð†ð, ðáºðá»= ð.ð Ã¥ᅵ ÃoŠ³Ã¥ª·žᅵ ï¿œ ï¿œ ï¿œ Euler ¡Šo°€ oª¥ᅵ ð= ð.ðððŠ³Ã¥ª·žᅵ ï¿œ ï¿œ Euler ŽÚᅵ ï¿œ ï¿œ ï¿œ (RK2)¡Šo°€oª¥ᅵ ð= ð.ðð óŠ³Ã¥ª·žᅵ ï¿œ ï¿œ Runge-Kutta 4 (RK4) ÂŽï¿œ ï¿œ ð= ð.ð ʥå»ᅵᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ð= ð.ð,ð.ð,ð.ð,ð.ð,ð.ð
47
RK4 vs Other Methods
ðð Exact Euler ð= ð.ððð Modified Euler ð= ð.ðð Runge-Kutta order 4 ð= ð.ð
0.0 0.5000000 0.5000000 0.5000000 0.5000000 0.1 0.6574145 0.6554982 0.6573085 0.6574144 0.2 0.8292986 0.8253385 0.8290778 0.8292983 0.3 1.0150706 1.0089334 1.0147254 1.0150701 0.4 1.2140877 1.2056345 1.2136079 1.2140869 0.5 1.4256394 1.4147264 1.4250141 1.4256384
48
Multi-Step Methodoµ°· ·ÃŠ € µŠÃ·° »¡Ž rï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ð ðð ð = ðáºð,ðá»Ã®º° nªᅵ ï¿œ ï¿œ áŸðð,ðð+ðῠ³à oï¿œ ï¿œ
ðáºðð+ðá»â ðáºððá»= ðâ²áºðá»ð ððð+ððð = ðáºð,ðá»ð ððð+ððð
ÂŽ Ãï¿œ ï¿œ ï¿œ ï¿œ ðáºðð+ðá»= ðáºððá»+ ðáºð,ðá»ð ððð+ððð
Ãï¿œ ï¿œ ðáºðá» oª ¡®»µ€ ª Š³€µ nµà nªᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ð·áºðá»ÃŠµ³ µ€µŠ °· ·ÃŠ à o¹ÃÃ}¡®»ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ᅵµ€ Šžᅵ ï¿œ ï¿œ ð®µà oµ µ » o°€Œᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ áºððâð,ððâðá», áºððâð+ð,ððâð+ðá», ... , áºðð,ððá»Ã³ðáºððá»â ðð ³à oï¿œ ï¿œ
ðáºðð+ðá»â ðð + ð൫ð,ð·áºðá»àµ¯ð ððð+ððð
49
Multi-Step Method
ŠŒà ° ¡®»µ€ ª Š³€µÃà µ€µŠ ÃoÃï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ð·áºðá»Ã oÃnï¿œ ï¿œ à ³ ª à » º° µŠÃoï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ᅵš nµ ºÃºÃ° Â¥o° ® ° ·ª ÂŽï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ
Š³Ã¥ª·®µ¥à µ€µŠ Ãnà oÃ}ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ 2 »n€ º°ᅵ ï¿œ
1. Š³Ã¥ª·Ã¥ŽÃoï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ Ã}Š³Ã¥ª·žÃï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðð+ð à €n¹ÃÂŽ µŠ®µnµᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðáºðð+ð,ðð+ðá» 2. Š³Ã¥ª·Ã¥Š·¥µ¥ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ¹Ãï¿œ ï¿œ € µ nª ¹ÃÂŽï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðáºðð+ð,ðð+ðá»
50
Multi-Step Method: Explicit caseŠ³Ã¥ª·®µ¥ÃÃ¥ŽÃoï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ Š žÃï¿œ ï¿œ ï¿œ ð= 𠌊 °ᅵ ï¿œ ï¿œ Adams-Bashforth ° ÂŽ nµÃÊŒÃï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ
ðð+ð = ðð + ðÏ ððððáºððâð,ððâðá»ðð=ð
° ÂŽï¿œ ï¿œ ï¿œ ð ððð ððð ððð ððð ð¬ð
1 0 1 ððð ðâ²â²áºðŒðá» 2 1 3/2 -1/2 ððððð ðâ²â²â² áºðŒðá» 3 2 23/12 -16/12 5/12 ðððð ðáºðá»áºðŒðá» 4 3 55/24 -59/24 37/24 -9/24 ðððððððð ðáºðá»áºðŒðá»
51
Adams-Bashforth FormulaAdams-Bashforth ° ÂŽï¿œ ï¿œ ï¿œ 3 (AB3) ðð = ð¶ð,ðð = ð¶ð,ðð = ð¶ð ðð+ð = ðð + ðáðððððáºðð,ððá»â ðððððáºððâð,ððâðá»+ ððððáºððâð,ððâðá»á
áºð = ð,ð,âŠ,ðµâ ðá» nµ·¡ µÃ¡µ³ ·Ãº°ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðððð ðáºðá»áºðŒðỠõ®ŠŽ µᅵ ï¿œ ï¿œ ðŒð âáºððâð,ðð+ðá»
Adams-Bashforth ° ÂŽï¿œ ï¿œ ï¿œ 4 (AB4) ðð = ð¶ð,ðð = ð¶ð,ðð = ð¶ð,ðð = ð¶ð ðð+ð = ðð + ðáðððððáºðð,ððá»â ðððððáºððâð,ððâðá»+ ðððððáºððâð,ððâðá»âððððáºððâð,ððâðá»á áºð = ð,ð,âŠ,ðµâ ðá»
nµ·¡ µÃ¡µ³ ·Ãº°ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðððððððð ðáºðá»áºðŒðỠõ®ŠŽ µᅵ ï¿œ ï¿œ ðŒð âáºððâð,ðð+ðá»
52
Multi-Step Method: Explicit case
Š³Ã¥ª·®µ¥ÃÃ¥ŽÃoï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ Š žÃï¿œ ï¿œ ï¿œ ðâ ð (ð†ð†ð) µ µŠ Š³ µ¥š nµ ºÃºÃ° ³à ᅵoŠŒᅵÃᅵᅵᅵ°ᅵ ŒᅵŠÃï¿œ}ᅵᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ
ðð+ð = ððâð+ ðÏ ððððáºððâð,ððâðá»ðð=ð
ð ð ððð ððð ððð ððð ð¬ð
1 1 2 ððð ðâ²â²â² áºðŒðá» 3 3 8/3 -4/3 8/3 ðððððð ðáºðá»áºðŒðá»
53
Multi-Step Method: Explicit case ŒŠ °ᅵ ï¿œ ï¿œ Milne ðð+ð = ððâð + ðáðððáºðð,ððá»â ðððáºððâð,ððâðá»+ ðððáºððâð,ððâðá»á áºð = ð,ð,âŠ,ðµâ ðá» nµ·¡ µÃ¡µ³ ·à º°ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðððððð ðáºðá»áºðŒðỠõ®ŠŽ µᅵ ï¿œ ï¿œ ðŒð âáºððâð,ðð+ðá»
ÃۼððÃ}õª Ãóú°ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ð= 𠌊 õ®ŠŽ Š žᅵ ï¿œ ï¿œ ï¿œ ðâ ð ³Ãoõª °ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ð o°¥ᅵªnµ ŒŠ õ®ŠŽ Š žᅵ ï¿œ ï¿œ ï¿œ ï¿œ ð= ð À° ÂŽÃ¥ª ÂŽï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ
54
Multi-Step Method: Implicit caseŠ³Ã¥ª·Ã¥Š·¥µ¥ᅵ ï¿œ ï¿œ ï¿œ ï¿œ
Š žÃï¿œ ï¿œ ï¿œ ð= ð : ðð+ð = ðð + ðÏ ððððáºððâð,ððâðá»ðâðð=âð
ŒŠ ÃÊ¥ªnµᅵ ï¿œ ï¿œ ŒŠ °ᅵ ï¿œ ï¿œ Adams-Moulton ° ÂŽ nµÃï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ° ÂŽï¿œ ï¿œ ï¿œ ð ðð,âð ððð ððð ððð ð¬ð
1 0 1 âððð ðâ²â²áºðŒðá» 2 1 1/2 1/2 â ðððððâ²â²â² áºðŒðá» 3 2 5/12 8/12 -1/12 â ðððððáºðá»áºðŒðá» 4 3 9/24 19/24 -5/24 1/24 âððððððð ðáºðá»áºðŒðá»
Ãۼðððâð+ð < ðŒð < ðð+ð
55
Multi-Step Method: Implicit caseŠ³Ã¥ª·Ã¥Š·¥µ¥ᅵ ï¿œ ï¿œ ï¿œ ï¿œ
Š žÃï¿œ ï¿œ ï¿œ ðâ 𠌊 Ãnµ à º° ŒŠᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ Simpson (ð= ð, ð= ð) ðð+ð = ððâð + ððáŸðáºðð+ð,ðð+ðá»â ððáºðð,ððá»+ ðáºððâð,ððâðá»á¿ áºð = ð,ð,âŠ,ðµâ ðá» nµ·¡ µÃ¡µ³ ·à º°ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ â ðððððáºðá»áºðŒðỠõ®ŠŽ µᅵ ï¿œ ï¿œ ðŒð âáºððâð,ðð+ðá»
áºÃ° óÃo ŒŠÃ¥ŽÃoó劷¥µ¥ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ õÃ}o° ®µnµᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðâð,ðâð+ð,âŠ,ðð à ¥n° oª¥Š³Ã¥ª·žÃÃ¥ªᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ (single step method) à ¥n° ÃnÃoŠ³Ã¥ª· °ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œRunge-Kutta order 4
56
Predictor-Corrector Method
àž£àž°à¹àž àž¢àžàž§àž àž 1 à¹àž-àžàžàž²àž£à¹àžï¿œàžª/àžàž£ 2 àžª/àžàž£àž£ï¿œàž§àž¡àžï¿œàž à¹àžàž¢àžª/àžàž£àž«àž3ï¿œàžàžàž°à¹àž-àžàžª/àžàž£à¹àžàžàžï¿œàžà¹àžï¿œàž à¹àž£ àž¢àžàž§ï¿œàž² àžª/àžàž£àžï¿œàž§àž)àž²àžàž²àž¢ (Predictor Formula) à¹àž¥àž°àž àžàžª/àžàž£àž«àž3ï¿œàžà¹àž-àžàžª/àžàž£à¹àžàž¢àžàž£àž¢àž²àž¢ àž ï¿œàž¡ àžàž§àž²àž¡à¹àž¡ï¿œàžàž¢)àž²àžª/àžàžàž§ï¿œàž² à¹àž£ àž¢àžàž§ï¿œàž² àžª/àžàž£àžï¿œàž§à¹àžï¿œ (Corrector Formula)
àžàž²àž£à¹àžï¿œàžª/àžàž£àžï¿œàž§à¹àžï¿œ à¹àž£ àž¢àžàž§ï¿œàž² àžàž²àž£àž)àž²àžàž³)1àž²àž àž²àž¢à¹àž (inner iteration)
àžï¿œàžàžà¹àžï¿œàž/ï¿œàžï¿œàž§àž)àž²àžàž²àž¢à¹àž¥àž°àžï¿œàž§à¹àžï¿œ àž ï¿œàž¡ àžï¿œàž²àžàž¥àž²àžà¹àžàž¥ï¿œï¿œàžàžàžàž¢/ï¿œà¹àžàžï¿œàžàžï¿œàžà¹àžàž¥ï¿œà¹àž àž¢àžàžï¿œàž àž¡àžàž°àžï¿œ1àžà¹àž¥ï¿œàž§àžàž²àž£àž)àž²àžàž³)1àž²àž àž²àž¢à¹àžàžàž²àžàžàž°à¹àž¡ï¿œàž¥/ï¿œà¹àžï¿œàž²
57
Predictor-Corrector Method ŒŠ °ᅵ ᅵ ᅵ Adams-Bashforth-Moulton (ABM4)
Predictor ðð+ðáºðá» = ðð + ðððáŸðððð â ððððâð + ððððâð â ðððâðá¿ Corrector ðð+ðáºð+ðá»= ðð + ðððáððð+ðáºðá» + ðððð â ðððâð + ððâðá ÃoŠ³Ã¥ª·à ÃÃ¥ªÃnï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ RK áºÃ°®µnµᅵ ðð,ðð,ððó ðð,ðð,ðð,ðð 1. õªᅵ ï¿œ ï¿œ ðð+ð oª¥ ŒŠ ªᅵ ï¿œ ï¿œ õµ¥ᅵ ï¿œ îonµÃµª Ã}ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðð+ðáºðá» 2. Ãonµᅵ ï¿œ ðð+ðáºðỠáºÃ° õªᅵ ï¿œ ï¿œ ðð+ðáºðá» 3. õªᅵ ï¿œ ï¿œ ðð+ðáºð+ðá» oª¥ ŒŠ ªᅵ ï¿œ ï¿œ Ãoï¿œ 4. õª nµᅵ ï¿œ ï¿œ ï¿œ ðð+ðáºðá»Ã³ õÃõà o°ᅵ ï¿œ ï¿œ ï¿œ 3 Š³ Ãï¿œ ï¿œ ï¿œ ï¿œ ï¿œ áðð+ðáºð+ðá»â ðð+ðáºðá»á< ð
58
Predictor-Corrector MethodÂŒn ŒŠ ª õµ¥ᅵ ï¿œ ï¿œ ï¿œ ï¿œ - ªÃo°ºÃÃï¿œ ï¿œ ï¿œ
ŒŠ ª õµ¥Ão ŒŠ °ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ Milne
ðð+ðáºðá» = ððâð + ððð áŸððð â ððâð + ðððâðῠŒŠ ªÃoÃo ŒŠᅵ ï¿œ ï¿œ ï¿œ ï¿œ Simpson
ðð+ðáºð+ðá»= ððâð + ððáðð+ðáºðá» + ððð + ððâðá ®Šº° ŒŠ ªÃoÃo ŒŠᅵ ï¿œ ï¿œ ï¿œ ï¿œ Hamming
ðð+ðáºð+ðá»= ððáððð â ððâð + ððáðð+ðáºðá» + ððð â ððâðáá
59
Predictor-Corrector Method: Example
ÃoŠ³Ã¥ª·žª õµ¥ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ - ªÃo®µnµ°ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ð ûᅵ ï¿œ ï¿œ ð= ð.ð ° {®µnµÃŠ·Ã€ oï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œðâ² = ð + ðð Àᅵ ð= ðÃۼð ð= ðÃÂ¥Ãoï¿œ ï¿œ ð= ð.ð
õ® îoï¿œ ï¿œ ï¿œ ÂŒn ŒŠ ÃÃoº°ᅵ ï¿œ ï¿œ ï¿œ ï¿œ
ŒŠ ª õµ¥ᅵ ï¿œ ï¿œ ï¿œ ðð+ðáºðá» = ðð + ððáºðð,ððỠŒŠ ªÃoï¿œ ï¿œ ï¿œ ðð+ðáºð+ðá»= ðð + ððáðáðð+ð,ðð+ðáºðá»á+ ðáºðð,ððá»á
60
Predictor-Corrector Method: Example
ðð = ð,ðð = ð,ðáºðð,ððá»= ð
ððáºðá»= ð+áºð.ðá»áð + ððá= ð.ð
ðáðð,ððáºðá»á= ð + ð.ðð.ð = ð.ðððð
ð= ð ððáºðá»= ð+ ð.ðð áŸð.ðððð+ ðá¿= ð.ðððð
61
Predictor-Corrector Method: Example
ðáðð,ððáºðá»á= ð + ð.ððððð.ð = ð.ððð
ð= ð ððáºðá»= ð+ ð.ðð áŸð.ðððð+ ðá¿= ð.ðððð ð= ð ððáºðá»= ð.ðððð ðð = ð.ð, ðð = ð.ððððâ ðáºð.ðá»
62
Predictor-Corrector Method: Example
óà oï¿œ ððáºðá»= ð.ðððð
ððáºðá»= ð.ðððð
ððáºðá»= ð.ðððð
ððáºðá»= ð.ðððð â ðáºð.ðỠªµ€ µà ºÃ° ° ŒŠ ªÃoº°ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ â ðððððâ²â²â² áºðŒðá» Ãۼððð < ðŒð < ðð+ð
63
Predictor-Corrector Method: Example
Š³€µᅵ ï¿œ ðâ²â²â² áºðŒðá» oª¥ᅵ ðâ²â²â² áºðá»= âðó ð= ð.ðà oªµ€ µà ºÃ°ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ à €n÷ᅵ ï¿œ áâð.ðððð áºðá»áâ ð.ððà ððâð à ªnµ à ¥€ ªµ€Ã€nï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ 3 D.P.
ðâ² = ðáºð,ðá»= ð + ðð
ðâ²â² = ð ðâ²ð ð = ð+ ðð ðð ð+ ððð = ðáð + ððáâ ððð = ðð ðâ²â²â² = ð ðâ²â²ð ð = â ððð
High ODE and System of ODE
{®µnµÃŠ·Ã€ o À € µŠÃ·° »¡Ž r° ÂŽï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðÃÂ¥Ãªà €ŠŒà Ã}ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðáºðá»= ð൫ð,ð,ðâ²,âŠ,ðáºðâðá»àµ¯ Ã¥€ÃºÃ° à nµÃŠ·Ã€ oï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðáºððá»= ð¶ð, ðâ²áºððá»= ð¶ð, ..., ðáºðâðá»áºððá»= ð¶ð
High ODE and System of ODE
Ãۼðîo ð= ðð, ðâ² = ðð, ..., ðẠðâðá»= ðð š Ãà oº°Š³ € µŠᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ
ððâ² = ðð ððâ² = ðð
ððâ² = ðð
â® ððâðâ² = ðð
ððâ² = ðáºð;ðð,ðð,âŠ,ððỠóúð à nµÃŠ·Ã€ oï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ððáºððá»= ð¶ð, ððáºððá»= ð¶ð, ..., ððáºððá»= ð¶ðÃۼð ð= ðð
High ODE and System of ODE
Ã¥à oÊŒà € µŠÃª ðŠr÷° »¡Ž rï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ à oÃ}ï¿œ ï¿œ ï¿œ ðâ²áºðá»= ð൫ð,ð൯ Ã¥€ᅵ
ð= ðð Ãۼð ð= ðð
ÃÂ¥Ãï¿œ ï¿œ
ðáºðá»= ൊ
ððáºðá»ððáºðá»â®ððáºðá»àµª ; ð൫ð,ð൯=ÛÛÛÛðð൫ð;ð൯ðð൫ð;ð൯â®ðð൫ð;ðàµ¯Û Û
ÛÛÛ=Û àµŠ
ððâ² áºðá»ððâ² áºðá»â®ðáºð;ðð,ðð,âŠ,ððá»àµª
ðáºððá»= ð¶ðð¶ðâ®ð¶ð
High ODE and System of ODE
Š³Ã¥ª·žÃÃo®µ à ¥{®µnµÃŠ·Ã€ oï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðâ² = ðáºð,ðỠÀᅵ ð= ððÃۼð ð= ðð o°ᅵ ᅵà ÃÂ¥ Š·€µ õŠrï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ð,ðâ²,ðáºð,ðá»,ðð óðð,ðð,ðð,ððà ŒŠÃï¿œ ï¿œ ï¿œ ï¿œ Runge-Kutta
®Šº° ŒŠ°ºÃÃï¿œ ï¿œ Ã} Š·€µÃª ðŠrï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ð,ðâ²,ð൫ð;ð൯,ððó ð²ð,ð²ð,ð²ð,ð²ð
RK4 in vector form ŒŠÃï¿œ ï¿œ ï¿œ RK4 º°ᅵ
ðð+ð = ðð + ðð൫ð²ð + ðð²ð + ðð²ð + ð²ð൯ ð²ð = ðð൫ðð,ðð൯ ð²ð = ðð൬ðð + ðð,ðð + ð²ðð൰
ð²ð = ðð൬ðð + ðð,ðð + ð²ðð൰ ð²ð = ðð൫ðð+ð,ðð + ð²ð൯ áºð = ð,ð,âŠ,ðµâ ðá»
ABM4 in vector form
ŒŠ ª õµ¥ᅵ ï¿œ ï¿œ ï¿œ ðð+ðáºðá» = ðð + ððð ï¿œðððð â ððððâð + ððððâð â ðððâð൧ ŒŠ ªÃoï¿œ ï¿œ ï¿œ ðð+ðáºðá» = ðð + ððð ï¿œððð+ð + ðððð â ðððâð + ððâð൧ ÃÂ¥Ãï¿œ ï¿œ ðð = ðáðð;ðáºððá»á
High ODE and System of ODE: Example
µ {®µnµÃŠ·Ã€ oï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðâ²â² = ððð,ðáºðá»= ð,ðâ²áºðá»= âð ®µnµŠ³€µ °ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œðáºð.ðá» oª¥Š³Ã¥ª·žᅵ ï¿œ ï¿œ ï¿œ RK4 õ® îoï¿œ ï¿œ ï¿œ ðâ² = ð
ÂŽ Ãï¿œ ï¿œ ï¿œ ï¿œ ðâ²â² = ðâ² = ððð ðáºðá»= ðáºðá»ðáºðá»àµš, ð൫ð,ð൯= ðáºðá»ððð൚, ððáºðá»= ðáºðá»ðáºðá»àµš= á
ðâðá
High ODE and System of ODE: Example
à oŠ³ € µŠᅵ ï¿œ ï¿œ ï¿œ ðâ² = ð൫ð,ð൯ ¡Šo°€ÃºÃ° à ᅵ ï¿œ ï¿œ ððáºðá»= á
ðâðá Ã¥Š³Ã¥ª·žᅵ ï¿œ ï¿œ ï¿œ Runge-Kutta ° ÂŽï¿œ ï¿œ ï¿œ 4
ðð+ð = ðð + ðð൫ð²ð + ðð²ð + ðð²ð + ð²ð൯ ð²ð = ðð൫ðð,ðð൯= áºð.ðá» âð
áºðá»áºðá»ð൚= áâð.ðð á
ðð + ð²ðð = áðâðá+ ððáâð.ðð á= á
ð.ððâð á ðáºðá»ðáºðá»àµš
High ODE and System of ODE: Example
ð²ð = ðð൬ðð+ ðð,ðð + ð²ðð൰= áºð.ðá» âðáºð.ððá»áºð.ððá»ð൚= á
âð.ðð.ððððá ðð + ð²ðð = á
ð.ððâð á ðáºðá»ðáºðá»àµš
ðð + ð²ðð = áðâðá+ ððá âð.ðð.ððððá= á
ð.ððâð.ððððá ðáºðá»ðáºðá»àµš
ð²ð = ðð൬ðð + ðð,ðð + ð²ðð൰= áºð.ðá» âð.ððððáºð.ððá»áºð.ððá»ð൚= á
âð.ððððð.ðððð á
High ODE and System of ODE: Example
ð²ð = ðð൫ðð,ðð + ð²ð൯= áºð.ðá» âð.ððððáºð.ðá»áºð.ððððá»ð൚= á
âð.ððððð.ðððð á
ðð + ð²ð = áðâðá+á
âð.ððððð.ðððð á= áð.ððððâð.ððððá ðáºðá»ðáºðá»àµš
µᅵ ï¿œ ðð = ðð + ðð൫ð²ð + ðð²ð + ðð²ð + ð²ð൯ = á
ðâðá+ ððááâð.ðð á+ ðá âð.ðð.ððððá+ ðáâð.ððððð.ðððð á+áâð.ððððð.ðððð áá = á
ð.ðððððâð.ðððððá ÂŽ Ãï¿œ ï¿œ ï¿œ ï¿œ ðáºð.ðá»= ð.ðððððó ðâ²áºð.ðá»= âð.ððððð
High ODE and System of ODE: Example
ÃoŠ³Ã¥ª·žª õµ¥ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ - ªÃoáºÃ°®µnµᅵ ï¿œ ï¿œ ðÃۼð ð= ð.ð õ® îoï¿œ ï¿œ ï¿œ ðâ²â² + ð= ð ó ð ð ðâ² = ð 0.0 1 -1
0.1 0.895171 -1.094838
0.2 0.781397 -1.1787736
0.3 0.659816 -1.250857
High ODE and System of ODE: Example
õ® îoï¿œ ï¿œ ï¿œ ðð = ð,ðð = ðð,ð= ð.ð
ðâ² = ð ðâ²â² = ðâ² = âð
µᅵ ï¿œ ð= áððá ðð = á
ðâðá, ðð = áð.ððððððâð.ððððððá, ðð = áð.ððððððâð.ððððððá, ðð = áð.ððððððâð.ððððððá
µᅵ ï¿œ ð= áðâðá ðð = á
âðâðá, ðð = áâð.ððððððâð.ððððððá, ðð = áâð.ððððððâð.ððððððá, ðð = áâð.ððððððâð.ððððððá
High ODE and System of ODE: Example
Ão ŒŠ ª õµ¥ᅵ ï¿œ ï¿œ ï¿œ ï¿œ - ªÃoï¿œ ï¿œ ABM4
ŒŠ ª õµ¥ᅵ ï¿œ ï¿œ ï¿œ ðð+ðáºðá» = ðð + ððð ï¿œðððð â ððððâð + ððððâð â ðððâð൧
ððáºðá»= ðð + ððð ï¿œðððð â ðððð + ðððð â ððð൧= áð.ððððððâð.ððððððá+ ð.ðððáððáâð.ððððððâð.ððððððáâ ððáâð.ððððððâð.ððððððá+ ððáâð.ððððððâð.ððððððáâ ðáâðâðáá = á
ð.ððððððâð.ððððððá
High ODE and System of ODE: Example
ððáºðá»= áâð.ððððððâð.ððððððá
µᅵ ï¿œ ððáºðá»= áð.ððððððâð.ððððððá ó ð= á
ðâðá
ŒŠ ªÃoï¿œ ï¿œ ï¿œ ðð+ðáºðá» = ðð + ððð ï¿œððð+ð + ðððð â ðððâð + ððâð൧ ððáºðá»= ðð + ð.ðððáðððáºðá»+ ðððð â ððð + ððá= á
ð.ððððððâð.ððððððá ððáºðá»= áâð.ððððððâð.ððððððá
ððáºðá»= ðð + ð.ðððáðððáºðá»+ ðððð â ððð + ððá= áð.ððððððâð.ððððððá
High ODE and System of ODE: Example
ððáºðá»= áð.ððððððâð.ððððððá ððáºðá»= áð.ððððððâð.ððððððá
áððáºðá»â ððáºðá»áâ = ð.ðà ððâð
à ªnµ à ¥ᅵ ï¿œ ï¿œ ï¿œ € ªµ€Ã€n¥õ¹ « ·¥€ õînÃï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ 5 (5 D.P.) ðáºð.ðá»= ð.ðððððð ó ðâ²áºð.ðá»= âð.ððððð
Divided Different Method for BVP
{®µnµ°ᅵ ï¿œ ï¿œ ï¿œ ï¿œ (BVP) ° € µŠÃ·° »¡Ž r÷à oï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðâ²â²áºðá»+ ðáºðá»ðâ²áºðá»+ ðáºðá»ðáºðá»= ðáºðá» ð < ð¥< ð ðáºðá»= ð¶, ðáºðá»= ð·
µŠ®µ à ¥Ã· ªà ³ o° à ÃÂ¥ {®µÃŠ³ n°ÃºÃ° îoÃ} {®µÃŠ³ à €nï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œn°ÃºÃ° Ã¥ᅵ ï¿œ ï¿œ ï¿œ
1. à ÃÂ¥ÃÀ n°ÃºÃ° °ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðà nªᅵ ï¿œ ï¿œ áŸð,ðá¿Ã}à ° »Ãnnªº°ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðð,ðð,âŠ,ðð 2. à ÃÂ¥ € µŠÃ·° »¡Ž rÃ} € µŠ š nµ ºÃºÃ°ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ 3. à Ã¥úð à nµ° Ãõ® îoÃ}úð à ÃÃ¥ª ÂŽ nµ°ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ð ûÃnnª nµÃï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ
Differential Equation to Divided Different Form
ðâ²áºððá»â ððâ² = ðððáºðð+ð â ððâðá» ðâ²â²áºððá»â ððâ²â² = ðððáºðð+ð â ððð + ððâðá» ðáºðá»áºððá»â ððáºðá»= ðððáºðð+ð â ððð+ð + ððð â ðððâð + ððâðá»
µᅵ ï¿œ ðâ²â²áºðá»+ ðáºðá»ðâ²áºðá»+ ðáºðá»ðáºðá»= ðáºðá» ÃۼðÚ Ã}š nµ ºÃºÃ° ³à oï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðððáºðð+ð â ððð + ððâðá»+ ðáºððá» ðððáºðð+ð â ððâðá»+ ðáºððá»ðð = ðáºððá»
áºð = ð,ð,âŠ,ðâ ðá»
Divided Different Formula for Linear BVP
îo ðð = ðáºððá», ðð = ðáºððỠó ðð = ðáºððỠŠŒÃ®€nà oï¿œ ï¿œ ï¿œ ï¿œ ൬ð â ðððð൰ððâð +൫âð+ ðððð൯ðð +൬ð + ðððð൰ðð+ð = ðððð ð = ð,ð,âŠ,ðâ ð
Ãۼðà nµŽ ž³à oŠ³ € µŠÃ·à o€Ã€ Š· r° € Š³ · ·ÃÃ}À Š· r µ€Ãªᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œÃÂ¥ ʵ nµ» ªŠ³ € µŠ Àᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðâ ð € µŠÃ³€ ªà €nŠµ nµÃµªᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðâ ð ªᅵº°ᅵ ðð,ðð,âŠ,ððâðà ³ Ãï¿œ ï¿œ ï¿œ ï¿œ ðð = ð¶Ã³ ðð = ð·
Existing of Solution
oµÃº° µ ° nªᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ð Ãõîoï¿œ ï¿œ ð< ððŽÃÂ¥Ãï¿œ ï¿œ ðŽ= ðŠðð±ðâ€ðâ€ðÈ7ðáºðá»È7À Š· r°ᅵ ï¿œ ï¿œ ï¿œ ï¿œ € Š³ · ·Ãʳ € µŠᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ áð â ððððáððâð +൫âð+ ðððð൯ðð +áð +ððððáðð+ð = ðððð, áºð = ð,ð,âŠ,ðâ ðỠ³€Ãª Ã¥€»€ n€Ão¹€ à ¥ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œÃn° ó€Ã¡¥ à ¥Ã¥ªᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ
Divided Different for BVP: Example
®µᅵ ï¿œ ð Ãï¿œ ð= ð.ð µ {®µnµ°ᅵ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ðâ²â² + ððâ² + ð= ð, ðáºðá»= ð, ðáºðá»= ðÃۼðÃoï¿œ ð= ðð
ðð = ð ðð = ðð ðð = ðð ðð = ð ðð = ðð
Ãï¿œ ï¿œ ðð,ðð,ððÃï¿œ áð â ððððáððâð +൫âð+ ðððð൯ðð +áð + ððððáðð+ð = ðððð ³à oï¿œ ï¿œ
áð â ðððð áððâð +൫âð+ ðð൯ðð +áð + ðððð áðð+ð = ðððð
Divided Different for BVP: Example
õ®ŠŽÃn³ nµᅵ ï¿œ ï¿œ ð ³à oï¿œ ï¿œ ð = ð áð â ðððð áðð +൫ðð â ð൯ðð +áð + ðððð áðð = ðððð ð = ð áð â ðððð áðð +൫ðð â ð൯ðð +áð + ðððð áðð = ðððð ð = ð áð â ðððð áðð +൫ðð â ð൯ðð +áð + ðððð áðð = ðððð
à nµᅵ ï¿œ ï¿œ ð= ðð, ðð = ðð, ðð = ðð, ðð = ðð, ðáºðá»= ð, ðáºðá»= ð
³à oŠ³ € µŠᅵ ᅵ ᅵ ᅵ ᅵ
âð.ðððð ð.ððððð ðð.ðððð âð.ðððð ð.ððððð ð.ððððð âð.ðððð൩ðððððð൩= ð.ððððððð.ððððððð.ðððððð൩
Divided Different for BVP: Example
®µ à ¥° Š³ € µŠÃ·à oï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ à oà ¥ᅵ ï¿œ ï¿œ ðð = âð.ððððððð, ðð = âð.ððððððð, ðð = ð.ðððððð