Differensial tenglamalarni taqribiy yechimlarini aniqlash. Runge –kutta va adams usullari

9.2-Maple 7 dasturi
Рунге-Кутта usulida hisoblash(9.1-масала):
> DE:=diff(y(t),t)=t+cos(y(t)/sqrt(5));

> ysol:=dsolve({DE, y(1.8)=2.6}, numeric, method=rkf45);

> ysol(1.9);

> ysol(2.0);

> with(DEtools):
DEplot(diff(y(x),x$1)=x+cos(y(x)/sqrt(5)),y(x),x=-2..3, [[y(1.8)=2.6]],y=-4..5, linecolour=sin(x*Pi/5),stepsize=.05);

Runge-Kutta usulida hisoblash dasturini quyidagicha tuzamiz.


9.4- DASTUR
5 REM RUNGE – KUTTA USULI
10 DEF FNE (X;Y)=X+COS(Y/SQR (5))
20 REM “boshlang’ich qiymat ,qadam, berilgan kesma yuqori chegarasi:”
22 READ X, Y, H, B
24 REM “boshlang’ich qiymat ,qadam, berilgan kesma yuqori chegarasi qiymatlari:”
26 DATA 1.8, 2.6, 0.1, 2.8
30 PRINT
32 PRINT “Birinchi tartibli differentsial tenglama ”
33 PRINT “ Y¹=F(X,Y) uchun”
34 PRINT “Koshi masalani RUNGE – KUTTA usulida”
36 PRINT “ taqribiy yechimini topish”
38 PRINT
40 N=(B-X)/H
42 FOR I=1 TO N
43 Q1=H*FNE(X,Y)
44 Q2=H*FNE(X+H/2,Y+Q1/2)
45 Q3=H*FNE(X+H/2,Y+Q2/2)
46 Q4=H*FNE(X+H,Y+Q3)
47 Y=Y+(Q1+2Q2+2Q3+Q4)/6
49 X=X+H
50 PRINT “X (“;USING “###.###”;I:
52 PRINT “)=”;USING “###.###”;X;
54 PRINT “ F(“;USING “###.###”;I;
56 PRINT “)=”;USING “###.###”;Y
59 NEXT I
90 END
Birinchi tartibli differentsial tenglama
Y¹=F(X,Y) uchun
Koshi masalasini RUNGE – KUTTA usulida
taqribiy yechimini topish.
X(1)=1.900 Y(1)=2.8201
X(2)= 2.000 Y(2)= 3.0408
X(3)= 2.100 Y(3)= 3.2619
X(4)= 2.200 Y(4)= 3.4831
X(5)= 2.300 Y(5)= 3.7045
X(6)= 2.400 Y(6)= 3.9260
X(7)= 2.500 Y(7)= 4.1478
X(1)= 2.600 Y(1)= 4.3700
X(9)= 2.700 Y(9)= 4.5931
X(90)= 2.800 Y(90)= 4.8172