Differensial tenglamalarni taqribiy yechimlarini aniqlash. Eyler usuli

200 END
Birinchi tartibli differentsial tenglama
Y¹=F(x,y) uchun
Koshi masalasini Eylerning ketma-ket yaqinlashish usulida taqribiy yechimini topish.
X(1)=1.900 Y(1)=2.8197 Z(1)=2.8197
X(2)= 2.000 Y(2)= 3.0402 Z(2)=3.0406
X(3)= 2.100 Y(3)= 3.2611 Z(3)=3.2617
X(4)= 2.200 Y(4)= 3.4823 Z(4)=3.4830
X(5)= 2.300 Y(5)= 3.7037 Z(5)=3.7044
X(6)= 2,400 Y(6)= 3.9251 Z(6)=3.9259
X(7)= 2.500 Y(7)= 4.1468 Z(7)=4.1476
X(1)= 2.600 Y(1)= 4.3688 Z(1)=4.3697
X(9)= 2.700 Y(9)= 4.5914 Z(9)=4.5926
X(90)= 2.800 Y(90)= 4.8150 Z(90)=4.8166
9.2- Paskal tili dasturi
{Birinchi tartibli differentsial tenglama }
{Y¹=F(X,Y) uchun}
{Koshi masalasini Eylerning ketma-ket yaqinlashish usulida taqribiy yechimini topish}


program ailer2(output,input);
label 11;
function fne(x,y:real):real;
begin fne:=x+cos(y/sqrt(5)); end;
var
x,y,x1,y1,h,b,z,eps:real;
i,n:integer;
begin
eps:=0.001;
writeln(' x=','y=',' h=',' b=');readln(x,y,h,b);
n:=trunc((b-x)/h);
for i:=1 to n do
begin
y1:=y;
y:=y+h*fne(x,y);
11: z:=y;
y:=y1+(fne(x,y1)+fne(x+h,z))*h/2;
x:=x+h;
if abs(z-y)>eps then goto 11;
writeln;
write(' x(',i:2,')=',x:8:4);
write(' y(',i:2,')=',y:8:4);
write(' z(',i:2,')=',x:8:4) ;
end;
end.


3. Eylerning takomillashgan usuli.
Quyidagi
y´=f (x,y)
birinchi tartibli differentsial tenglamani
y(x₀)=y₀
boshlang’ich shartni qanoatlantiruvchi yechimni quyidagi takomillashtirish usullari bilan topamiz.
Birinchi takomillashtirish usuli.
Yuqoridagi masalani yechishda oraliq qiymatlarini aniqlaymiz:
x_i+1/2=x_i+h/2 , y_i+1/2=y_i+hf (x_i,y_i)/2
f _i+1/2=f (x _i+1/2,y _i+1/2), i=0, 1, 2, …..n. (9.8)
bu takomillashtirish bo‘yicha yechimning qiymatlarini quyidagicha topamiz.
y _i+1=y_i+hf (x _i+1/2,y _i+1/2), i=0, 1, 2, …..n. (9.9)
Ikkinchi takomillashtirish yoki Eyler – Koshi usuli. Birinchi navbatda aniqligi yaxshi bo‘lmagan yaqinlashishni
=y_i+hf (x _i,y _i), i=0, 1, 2, …..n.
ko‘rinishda topamiz va = ni hisoblaymiz.
Ikkinchi takomillashtirilgan yaqinlashishni quyidagicha topamiz.
y _i+1=y_i+ [f (x _i,y _i)+ ] (9.10)
Eylerning birinchi va ikkinchi takomillashgan usullarida qoldiq hadnig tartibi, har qadamda O(h³) bo‘ladi.
Har bir nuqtadagi xatolikni baholash takroriy hisob yordamida bo‘ladi, ya’ni hisob qadam bilan takrorlanadi va ning qiymatini aniqligi ( qadamda) quyidagicha baholanadi:
bu yerda y(x) – differentsial tenglamaning aniq yechimi.