1、會計學(xué)1常微分方程數(shù)值解法常微分方程數(shù)值解法 )2 . 1()()1 . 1(,),(0yaybaxyxfy第1頁/共73頁 )2 . 1()()1 . 1(,),(0yaybaxyxfy.)(,()(0 xadttytfyxy第2頁/共73頁 )2 . 1()()1 . 1(,),(0yaybaxyxfy第3頁/共73頁 xadttytf)(,(. ),.,2 , 1 , 0(.ninabhihaxi 第4頁/共73頁).(2)()(1)().(2)()(1)(111iiiiiiiiyhxyxyhxyyhxyxyhxy ,)(2)()(2)()()(1111 iiiiiiiiiixxyhxy

2、yhxyxyxyh 第5頁/共73頁).(2)(,()()(11iiiiiyhxyxfxyxyh ).(2)(,()()(21iiiiiyhxyxhfxyxy ,)(2)()(2)()()(1111 iiiiiiiiiixxyhxyyhxyxyxyh 第6頁/共73頁).(2)(,()()(21iiiiiyhxyxhfxyxy )(2)(21iiyhhe ).(2)(21iixyhhe 第7頁/共73頁).(2)()()(111iiiiyhxyxyxyh ).(2)(,()()(2111iiiiiyhxyxhfxyxy ,)(2)()(2)()()(1111 iiiiiiiiiixxyhxyy

3、hxyxyxyh 第8頁/共73頁).(2)(2),()(,()()(1111111iiiiiiiiiyhyhyxfxyxfhyxyhe 第9頁/共73頁第10頁/共73頁 1)0(1 , 02yxyxyyyxyyxf2),( . 9,.,2 , 1 , 0)2(1 iyxyhyyiiiii第11頁/共73頁 1)0(1 , 02yxyxyyyxyyxf2),( )2(1111 iiiiiyxyhyy. 9,.,2 , 1 , 0)1(2)1(8121 ihhxhyyyiiii第12頁/共73頁校校正正值值隱隱式式預(yù)預(yù)測測值值顯顯式式)()(),(),(1111 iiiiiiiiyxhfyyy

4、xhfyy.)()()(22110nnyyyyyyy第13頁/共73頁校校正正值值隱隱式式預(yù)預(yù)測測值值顯顯式式)()(),(),(1111 iiiiiiiiyxhfyyyxhfyy 1)0(1 , 02yxyxyy. 9,.,2 , 1 , 0)2()2(11111 iyxyhyyyxyhyyiiiiiiiiii第14頁/共73頁校校正正值值隱隱式式預(yù)預(yù)測測值值顯顯式式)()(),(),(1111 iiiiiiiiyxhfyyyxhfyy第15頁/共73頁)(2)(21kkyhhe ).(2)(21iiyhhe 第16頁/共73頁 kiikhehE1)()(第17頁/共73頁第18頁/共73頁

5、第19頁/共73頁).()()()(,(111kkxxxxxyxydxxydxxyxfkkkk .)(,()()(11 kkxxkkdxxyxfxyxy).,()(,(1kkxxxxhfdxxyxfkk 第20頁/共73頁.)(,()()(11 kkxxkkdxxyxfxyxy 11)()(,(kkkkxxxxdxxFdxxyxf).(,()(,(2)()(,()(,(2)(,(111111 kkkkkkkkkkxxxyxfxyxfhyxyxyxfxyxfhdxxyxfkk),(),(2111 kkkkkkyxfyxfhyy第21頁/共73頁).(,()(,(2)()(,()(,(2)(,(1

6、11111 kkkkkkkkkkxxxyxfxyxfhyxyxyxfxyxfhdxxyxfkk),(),(2111 kkkkkkyxfyxfhyy第22頁/共73頁),(),(2111 kkkkkkyxfyxfhyy,)(12)()(1331 iiiiixxhOhyhe 3)(12)(abfRT 第23頁/共73頁),(),(2111 kkkkkkyxfyxfhyy. 1,.,2 , 1 , 0)()(),(),(1111 niyxhfyyyxhfyyiiiiiiii梯梯形形校校正正歐歐拉拉預(yù)預(yù)測測),(,(),(211iiiiiiiiyxhfyxfyxfhyy 第24頁/共73頁),(),(

7、2111 kkkkkkyxfyxfhyy第25頁/共73頁 1)0(1yxyy第26頁/共73頁 1)0(1yxyy1 . 22 . 0)( 1 . 09 . 122)()2(1111 kkkkkkkxxyhhxxhyhy方程方程第27頁/共73頁 1)0(1yxyy2)2(22)1()2)2(1(11hhxhxhhyhhkkk 方程方程第28頁/共73頁)(,()2(,2(4)(,(6)(,(11111kkkkkkkkxxxyxfxxyxxfxyxfhdxxyxfkk )(,()2(,2(4)(,(61111hxyhxfhxyhxfxyxfhkkkkkk )(,()2(,2(4)(,(611

8、111hxyhxfhxyhxfxyxfhyykkkkkkkk 第29頁/共73頁)(,()2(,2(4)(,(611111hxyhxfhxyhxfxyxfhyykkkkkkkk )(,(16180)()4(5kkkyfhhe ,)()(1618015)5(5iiiixxhOyh 第30頁/共73頁)(,()2(,2(4)(,(611111hxyhxfhxyhxfxyxfhyykkkkkkkk .21hh 431212iiiiifffhyy 第31頁/共73頁)(,()2(,2(4)(,(611111hxyhxfhxyhxfxyxfhyykkkkkkkk 431212iiiiifffhyy );

9、()(90)(16180)2()(51)(51)(511hOyhyhhekkkkk 第32頁/共73頁)(,()2(,2(4)(,(611111hxyhxfhxyhxfxyxfhyykkkkkkkk 431212iiiiifffhyy 第33頁/共73頁)(,()()(,()()(111 yhfxydxxyxfxyxyixxiiii hKxyi )(1 ),(11111iiiiyxfKhKyy ),(),()2(112111211hKyxfKyxfKKKhyyiiiiii第34頁/共73頁)(,()()(,()()(111 yhfxydxxyxfxyxyixxiiii hKxyi )(1 )2

10、(,()2,2(),(641211311121113211KKhyhxfKKhyhxfKyxfKKKKhyyiiiiiiii第35頁/共73頁 hKxyxyii )()(1 ),(11111iiiiyxfKhKyy ),(),()2(112111211hKyxfKyxfKKKhyyiiiiii )2(,()2,2(),(641211311121113211KKhyhxfKKhyhxfKyxfKKKKhyyiiiiiiii一階一階二階二階三階三階第36頁/共73頁 ),(),(),()(112121211122111hyhxfKhyhxfKyxfKKKKhyymimimiiiimmii 第37頁

11、/共73頁 ),()(,(),(121212121211122111hKyhxfhxyhxfKyxfKKKhyyiiiiiiii 原則原則: 使使ei(h) = y(xi) yi的階數(shù)盡可能高的階數(shù)盡可能高第38頁/共73頁 ),()(,(),(121212121211122111hKyhxfhxyhxfKyxfKKKhyyiiiiiiii 原則原則: 使使ei(h) = y(xi) yi的階數(shù)盡可能高的階數(shù)盡可能高).()(2)()()()(312111hOxyhxyhxyhxyxyiiiii )(),(),(211122112hOyxfyhKxhyxfKiiii )(),(),(),(21

12、1111211hOyxfyKyxfxhyxfiiiiii 第39頁/共73頁 ),()(,(),(121212121211122111hKyhxfhxyhxfKyxfKKKhyyiiiiiiii 原則原則: 使使ei(h) = y(xi) yi的階數(shù)盡可能高的階數(shù)盡可能高).()(2)()()()(312111hOxyhxyhxyhxyxyiiiii )(),(),(),(2111112112hOyxfyKyxfxhyxfKiiiiii . )()()()()(),(),(),(),(3122212113111112221121111hOxyhxyhxyhOyxfyKyxfxhyxhfyxhf

13、yyiiiiiiiiiiiii 第40頁/共73頁 ),()(,(),(121212121211122111hKyhxfhxyhxfKyxfKKKhyyiiiiiiii 無窮多解，從而有許多無窮多解，從而有許多2階階R-K公式公式).()(2)()()()(312111hOxyhxyhxyhxyxyiiiii ).()()()(312221211hOxyhxyhxyyiiii 2112221 第41頁/共73頁 ),()(,(),(121212121211122111hKyhxfhxyhxfKyxfKKKhyyiiiiiiii 2112221 ),(),()2(112111211hKyxfKy

14、xfKKKhyyiiiiii第42頁/共73頁 ),()(,(),(121212121211122111hKyhxfhxyhxfKyxfKKKhyyiiiiiiii 2112221 )2,2(),(111211121KhyhxfKyxfKhKyyiiiiii第43頁/共73頁 ),()(,(),(121212121211122111hKyhxfhxyhxfKyxfKKKhyyiiiiiiii 2112221 第44頁/共73頁 ),()2,2()2,2(),(22631142113111211143211hKyhxfKKhyhxfKKhyhxfKyxfKKKKKhyyiiiiiiiiii第45

15、頁/共73頁 ),()2,2()2,2(),(22631142113111211143211hKyhxfKKhyhxfKKhyhxfKyxfKKKKKhyyiiiiiiiiii第46頁/共73頁 1)0(5 . 0 , 02yxyy 2121121132111112211113211)2()2(,()2()2,2(),(64KKhyKKhyhxfKKhyKhyhxfKyyxfKKKKhyyiiiiiiiiiii第47頁/共73頁 1)0(5 . 0 , 02yxyy 2121321122113211)2()2(64KKhyKKhyKyKKKKhyyiiiii第48頁/共73頁 1)0(5 .

16、0 , 02yxyy 2313114221211321111122111143211)(),()2()2,2()2()2,2()(),(226hKyhKyhxfKKhyKhyhxfKKhyKhyhxfKyyxfKKKKKhyyiiiiiiiiiiiiii第49頁/共73頁 1)0(5 . 0 , 02yxyy 23142213211221143211)()2()2()(226hKyKKhyKKhyKyKKKKKhyyiiiiii第50頁/共73頁第51頁/共73頁431212kkkkkfffhyy 線性多步線性多步.)(,()(11 kkxxkkdxxyxfyxy第52頁/共73頁.)(,()

17、(11 kkxxkkdxxyxfyxy第53頁/共73頁 3132313212321231131213222)(! 3)()()()()()()()()()()()()(iikkkkkkkkkkkkkkkkkkkkkkxxFxFxxxxxxxxxFxxxxxxxxxFxxxxxxxxxRxLxF 第54頁/共73頁 3132313212321231131213222)(! 3)()()()()()()()()()()()()(iikkkkkkkkkkkkkkkkkkkkkkxxFxFxxxxxxxxxFxxxxxxxxxFxxxxxxxxxRxLxF .2)2)(1()3)(1(2)3)(2(

18、)()(32122 kkkFttFttFtttLxL第55頁/共73頁.2)2)(1()3)(1(2)3)(2()()(32122 kkkFttFttFtttLxL.51623122)2)(1()3)(1(2)3)(2()()()(321130120110110121211 kkkkkkkkkxxkkFFFhyFdtttFdtttFdttthydttLhydxxLyxykk第56頁/共73頁.5162312)(3211 kkkkkFFFhyxy).,(5),(16),(23123322111 kkkkkkkkyxFyxFyxFhyy第57頁/共73頁.)1(! 3)()(! 3)()(331)

19、4(313htyxxFxRiiik .)(83)3)(2)(1()(6)1()(6)(4)4(011)4(40131)4(431hydttttyhdttyhdxxRixxkk 第58頁/共73頁).,(9),(37),(59),(5524443322111 kkkkkkkkkkyxFyxFyxFyxFhyy)()(720251)()(552)5(hOhyyxyhekkk 第59頁/共73頁.)(30303 iijjikjijFxxxxxL),(),(5),(19),(9243322111 kkkkkkkkkkyxfyxfyxfyxfhyy)()(72019)()(55)5(hOhyyxyhek

20、kk 第60頁/共73頁),(),(5),(19),(9243322111 kkkkkkkkkkyxfyxfyxfyxfhyy)()(72019)()(55)5(hOhyyxyhekkk ),(),(5),(19),(924937595524332211143211 kkkkkkkkkkkkkkkkyxfyxfyxfyxfhyyffffhyy第61頁/共73頁).(! 4)3)(2)(1(! 3)2)(1(! 2)1()()()()5(43424433 hyttttFtttFttFtFxRxNxFkkkk kkkkkkxxxxkxxkkdxxRdxxNydxxyxfyy444)()()(,(3

21、344第62頁/共73頁 kkkkxxxxkkdxxRdxxNyy44)()(334 404342443! 3)2)(1(! 2)1()(4hdtFtttFttFtFdxxNkkkkxxkk.2563343832084434244434244 kkkkkkkkFFFFhFFFFh第63頁/共73頁.256334)(43424434 kkkkxxFFFFhdxxNkk,2,43242434 kkkkkkkFFFFFFF.33432143 kkkkkFFFFF).,(2),(),(2342234)(33221132134 kkkkkkkkkxxyxfyxfyxfhFFFhdxxNkk第64頁/共73頁).,(2),(),(234)(33221134 kkkkkkxxyxfyxfyxfhdxxNkk).,(2),(),(2343322114 kkkkkkkkyxfyxfyxfhyy).