Maruza mashg’ulotlari

Runge-Kutta usuli. Bir qadamli oshkor usullarning boshqa bir necha xillari ham majud bo‘lib, ularning ichida amalda eng ko‘p ishdlatiladigani Runge-Kutta usuli hisoblanadi. Usul shartiga ko‘ra har bir yangi tugun nuqtadagi echimni topish uchun funksiyani 4 marta har xil argumentlar uchun hisoblash kerak. Bu jihatdan Runge-Kutta usuli hisoblash uchun nisbatan ko‘p vaqt talab qiladi. Lekin Eyler usulidan ko‘ra aniqligi yuqori bo‘lganligi uchun, undan amalda keng foydalaniladi.
Usulning ishchi formulasi quyidagicha yoziladi:

bu erda ;

Demak, formulalardan ko‘rinib turibdiki, Eyler usuli birinchi tartibli Runge-Kutta usuliga mos keladi.
Eyler usuliga mos algoritm blok-sxemasi.

algoritm asosida paskal dasturlah tilidagi dasturi

Program Eyler;

var a,b,x0,y0,x,y,h:real;

Function f(x,y:real):real;

Begin

f:=<функция куриниши>;
end;

Begin

Write(‘a,b=’); readln(a,b);

Write(‘y0=’); readln(y0);

x0:=a;

Write(‘h=’);readln(h);

writeln(‘x0=’,x0,’ y0=’, y0 );
x:=x0;y:=y0;
while x< b do
begin
y:=y+h*f(x,y);
Writeln(‘x=’,x; ‘ y=’,y);
x:=x+h;
end;

Readln;

end.

Runge-Kutta usuliga mos blok-sxema.

algoritm asosida paskal dasturlah tilidagi dasturi:

Program R__kutta;

var a,b,x0,y0,h,x,y,k0,k1,k2,k3:real;
function f(x,y:real):real;
begin
f:=. . .;
end;

Begin

Write(‘a,b=’);readln(a,b);

Write(‘y0,h=’);readln(y0,h);

x:=a;y:=y0;
while x
begin
k0:=f(x,y);
k1:=f(x+h/2,y+h*k0/2);
k2:=f(x+h/2,y+h*k1/2);
k3:=f(x+h,y+h*k2);
y:=y+(k0+2*k1+2*k2+k3)*(h/6);
x:=x+h;
Writeln(‘x=’,x,’ y=’,y );
end;
readln;
end.