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O ’z b e k ist o n r e sp u b L ik a si o L iy va o ’r t a m a X su s t a ’lim V a z ir L ig I o ’zbek jsto n r e sp u b L ik a siBog'liq boshqarish tizimlarini kompyuterli Elektr mash. fan.i
X
У,
f( x „ y ,)
¥ ( x ,.y .)
0
0
1
0
0
1
0.1
1
0,05
0.005
2
0.2
1,005
0,1005
0,0100
3
0.3
1.0150
0.1522
0,0152
4
0.4
1.0303
0.2061
0,0206
5
0,5
1.0509
0.2627
0,0263
6
0,6
1,0772
0,3232
0,0323
7
0.7
1.1095
0,3883
0.0388
8
0.8
1.1483
0.4593
0,0459
9
0.9
1.1942
0,5374
0,0537
10
1,0
1.2479
D ifferensial tenglam a uchun K oshi m asalasini E yler usulida yechishga
Paskal tilid a tuzilgan dastur matni:
program eyler I; uses crt;
var a, b,у 0,у .real; n: integer;
function f(x,y:real);real;
begin
f:=0.5*x*y
end;
procedure eyler (a, b,yl ;real;n; integer ;var у : real);
8 6
var h,x:real; i.integer;
begin
h :=(b-a)/n;
x :=a;
\vriteln('x= \x :6 :2 ,' y = \ y l : 10:6);
f o r i: = l to n do
begin
y : = f ( x ,y l) * h + y l;
x := x + h ;
w riteln ('x= ',x:6:2,' y= ',y:1 0 :6 );
y l : = y ;
end;
end;
begin
clrscr;
w r ite ('a = ); read(a); w r ite ('b = ); read(b);
w rite('n —) ; r e a d ( n); w riteC yO = ); read(yO);
e y le r (a, b,yO, n.y);
end.
M iso l. B erilgan dasturdan foydalanib, ushbu
y ' ( x ) = y ( x ) - 0 , 5 x ' + x - l
ten g la m a n in g y ( 0 ) = \ shartni qanoatlantiruvchi taqribiy yechim ini aniqlang
( a =
0
. b =
1
, n =
1 0
deb oling).
Y e c h ish . T ek sh irib k o ‘rish m um kinki, berilgan K oshi m asalasi aniq
y ( x ) = 0,5x~ +1 y ech im g a ega. Q uyidagi ja d v ald a K oshi m asalasining aniq va Ey
ler u su lid a topilgan taqribiy yechim lari keltirilgan.
X
T a q rib iy
y e c h im
A n iq y e c h im
0 . 0
1 . 0 0 0 0 0 0
1 . 0 0 0 0 0 0
0 . 1
1 . 0 0 0 0 0 0
1 .0 0 5 0 0 0
0 . 2
1 .0 0 9 5 0 0
1 . 0 2 0 0 0 0
0.3
1 .0 2 8 4 5 0
1 .0 4 5 0 0 0
0 .4
1 .0 5 6 7 9 5
1 .0 8 0 0 0 0
0 .5
1 .0 9 4 4 7 4
1.1 2 5 0 0 0
0 . 6
1 .1 4 1 4 2 2
1.1 8 0 0 0 0
0 .7
1 .1 9 7 5 6 4
1.2 4 5 0 0 0
0 . 8
1.262821
1 .320000
0 .9
1.3 3 7 1 0 3
1.4 0 5 0 0 0
1 .0
1.4 2 0 3 1 3
1.5 0 0 0 0 0
87
Runge-K utta usuli.
Ushbu
y [ = f i ( x . y r y 2, - y j
Уг - и Х , У , . У г — У„)
(
6
.
1 2
)
У .
= f . ( x , y , , y 2.....
У . )
oddiy d ifferen sial ten g lam alar tizim i berilgan bo' lib, uning [а,б] oraliqdagi
y , ( x j =
У,
о-
У г (х « ) = У я ’
■■■■У,,(х„) = у„ 0
(6.13)
boshlangM ch shartlarni qanoatlantiruvchi yechim ini topish talab qilinsin ( x „ = a ) .
A gar
У,
' f
Y =
у 2
v a
F —
A
. Уп.
. f n .
belg ilash lar kiritsak, (6.12) va (6.13) ni quyidagi
У = F(x,Y),
(6.14)
Y(X0) = Қ
(6.15)
k o 'rin ish d a y o zish im iz m um kin. Bu yerda
Ую
(6.1 4 ) ten g lam alar tizim ining (6.15) b o sh la n g 'ic h shartlarni qan o at
lantiruvchi yechim ini R unge-K utta usuli y o rd am id a topam iz. B un in g uchun
x = c r + , h, К = F(x[ t Y.), i = \,2 ....,n belgilash I am i kiritib, quyidagi hisoblashlar
ketm a- k etligini bajaram iz:
x l = a + ih;
k, = F (x ,,Y ,)* h ;
k, = F(x, + h / 2 .Y ,+ k l / 2 ) * h :
k, = F {x : + h / 2 . Y >+ k 1 / 2 )* h;
k4 = Ғ (х : + h . )' + k})* h;
yM = Y + ( k l + 2 k , + 2 k , + k J) / 6
(6.16)
Bu y erd a qad am h = (b -a )/n .
H iso b lash lar ketm a-ketligi / = 1 dan « - 1 g a c h a takroriy ravishda hisobla
nadi v a (6 .1 6 ) form uladan differensial ten g lam an in g y , = y ( x , ) taqribiy sonli
y ech im lari topiladi.
E y le r u su lid a y o ‘l q o 'y ila d ig a n xatolik h tartibda, R unge-K utta usulida y o ‘l
q o 'y ilg a n x ato lik e sa h' tartib d a b o 'la d i. A g ar 0 < h < 1 ekanligini h iso b g a olsak,
u holda R u n g e-K u tta usulining aniqligi Eyler u su lin in g an iq lig ig a nisbatan yuqori
ekanligi kelib chiqadi.
D ifferensial ten g lam alar tizim i uchun K oshi m asalasini R unge-K utta usulida
yechishga Paskal tilid a tuzilgan d astu r m atni:
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Bosh sahifa
Aloqalar
Bosh sahifa
O ’z b e k ist o n r e sp u b L ik a si o L iy va o ’r t a m a X su s t a ’lim V a z ir L ig I o ’zbek jsto n r e sp u b L ik a si
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