常微分方程数值解法PPT教案

1、会计学1常微分方程数值解法常微分方程数值解法 )2 . 1()()1 . 1(,),(0yaybaxyxfy第1页/共73页 )2 . 1()()1 . 1(,),(0yaybaxyxfy.)(,()(0 xadttytfyxy第2页/共73页 )2 . 1()()1 . 1(,),(0yaybaxyxfy第3页/共73页 xadttytf)(,(. ),.,2 , 1 , 0(.ninabhihaxi 第4页/共73页).(2)()(1)().(2)()(1)(111iiiiiiiiyhxyxyhxyyhxyxyhxy ,)(2)()(2)()()(1111 iiiiiiiiiixxyhxy

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

3、hxyxyxyh 第8页/共73页).(2)(2),()(,()()(1111111iiiiiiiiiyhyhyxfxyxfhyxyhe 第9页/共73页第10页/共73页 1)0(1 , 02yxyxyyyxyyxf2),( . 9,.,2 , 1 , 0)2(1 iyxyhyyiiiii第11页/共73页 1)0(1 , 02yxyxyyyxyyxf2),( )2(1111 iiiiiyxyhyy. 9,.,2 , 1 , 0)1(2)1(8121 ihhxhyyyiiii第12页/共73页校校正正值值隐隐式式预预测测值值显显式式)()(),(),(1111 iiiiiiiiyxhfyyy

4、xhfyy.)()()(22110nnyyyyyyy第13页/共73页校校正正值值隐隐式式预预测测值值显显式式)()(),(),(1111 iiiiiiiiyxhfyyyxhfyy 1)0(1 , 02yxyxyy. 9,.,2 , 1 , 0)2()2(11111 iyxyhyyyxyhyyiiiiiiiiii第14页/共73页校校正正值值隐隐式式预预测测值值显显式式)()(),(),(1111 iiiiiiiiyxhfyyyxhfyy第15页/共73页)(2)(21kkyhhe ).(2)(21iiyhhe 第16页/共73页 kiikhehE1)()(第17页/共73页第18页/共73页

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

6、11111 kkkkkkkkkkxxxyxfxyxfhyxyxyxfxyxfhdxxyxfkk),(),(2111 kkkkkkyxfyxfhyy第22页/共73页),(),(2111 kkkkkkyxfyxfhyy,)(12)()(1331 iiiiixxhOhyhe 3)(12)(abfRT 第23页/共73页),(),(2111 kkkkkkyxfyxfhyy. 1,.,2 , 1 , 0)()(),(),(1111 niyxhfyyyxhfyyiiiiiiii梯梯形形校校正正欧欧拉拉预预测测),(,(),(211iiiiiiiiyxhfyxfyxfhyy 第24页/共73页),(),(

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

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

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

10、(,()2,2(),(641211311121113211KKhyhxfKKhyhxfKyxfKKKKhyyiiiiiiii第35页/共73页 hKxyxyii )()(1 ),(11111iiiiyxfKhKyy ),(),()2(112111211hKyxfKyxfKKKhyyiiiiii )2(,()2,2(),(641211311121113211KKhyhxfKKhyhxfKyxfKKKKhyyiiiiiiii一阶一阶二阶二阶三阶三阶第36页/共73页 ),(),(),()(112121211122111hyhxfKhyhxfKyxfKKKKhyymimimiiiimmii 第37页

11、/共73页 ),()(,(),(121212121211122111hKyhxfhxyhxfKyxfKKKhyyiiiiiiii 原则原则: 使使ei(h) = y(xi) yi的阶数尽可能高的阶数尽可能高第38页/共73页 ),()(,(),(121212121211122111hKyhxfhxyhxfKyxfKKKhyyiiiiiiii 原则原则: 使使ei(h) = y(xi) yi的阶数尽可能高的阶数尽可能高).()(2)()()()(312111hOxyhxyhxyhxyxyiiiii )(),(),(211122112hOyxfyhKxhyxfKiiii )(),(),(),(21

12、1111211hOyxfyKyxfxhyxfiiiiii 第39页/共73页 ),()(,(),(121212121211122111hKyhxfhxyhxfKyxfKKKhyyiiiiiiii 原则原则: 使使ei(h) = y(xi) yi的阶数尽可能高的阶数尽可能高).()(2)()()()(312111hOxyhxyhxyhxyxyiiiii )(),(),(),(2111112112hOyxfyKyxfxhyxfKiiiiii . )()()()()(),(),(),(),(3122212113111112221121111hOxyhxyhxyhOyxfyKyxfxhyxhfyxhf

13、yyiiiiiiiiiiiii 第40页/共73页 ),()(,(),(121212121211122111hKyhxfhxyhxfKyxfKKKhyyiiiiiiii 无穷多解，从而有许多无穷多解，从而有许多2阶阶R-K公式公式).()(2)()()()(312111hOxyhxyhxyhxyxyiiiii ).()()()(312221211hOxyhxyhxyyiiii 2112221 第41页/共73页 ),()(,(),(121212121211122111hKyhxfhxyhxfKyxfKKKhyyiiiiiiii 2112221 ),(),()2(112111211hKyxfKy

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

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

16、0 , 02yxyy 2313114221211321111122111143211)(),()2()2,2()2()2,2()(),(226hKyhKyhxfKKhyKhyhxfKKhyKhyhxfKyyxfKKKKKhyyiiiiiiiiiiiiii第49页/共73页 1)0(5 . 0 , 02yxyy 23142213211221143211)()2()2()(226hKyKKhyKKhyKyKKKKKhyyiiiiii第50页/共73页第51页/共73页431212kkkkkfffhyy 线性多步线性多步.)(,()(11 kkxxkkdxxyxfyxy第52页/共73页.)(,()

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

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

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

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

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