Eyler usullari

Oddiy va takomillashgan Eyler usullarini analitik yechim bilan taqqoslab, har bir qadamdagi farqni kuzating.

Input / Parametrlar

Masalani sozlang

Plot / f(x, y)

Method / Formulalar

Oddiy Eyler

y_{i+1}=y_i+h\,f(x_i,y_i)

Hosila joriy nuqtada baholanadi.

Takomillashgan · Heun

\begin{aligned}\bar{y}_{i+1}&=y_i+h\,f(x_i,y_i)\\y_{i+1}&=y_i+\frac{h}{2}\Bigl[f(x_i,y_i)+f(x_{i+1},\bar{y}_{i+1})\Bigr]\end{aligned}

Pretsenzor bashorati va korrektor aniqlashtirishi.

Aniq yechim

\begin{aligned}y(x)&=\frac{x}{2}-\frac{1}{4}+C e^{-2(x-x_0)}\\C&=y_0-\frac{x_0}{2}+\frac{1}{4}\end{aligned}

C = y₀ − x₀/2 + 1/4; standart boshlang'ich qiymatlarda C = 7/4.

Qadamlar soni5

$n=\left\lceil\frac{x_{\mathrm{end}}-x_0}{h}\right\rceil$

3.000000 oraliq, oxirgi qadam chegaraga moslanadi.

Data / Nuqtalar

ȳ — pretsenzor bashorati $\Delta y=\left|y_{\mathrm{takomil}}-y_{\mathrm{eyler}}\right|$

6 ta nuqta

$i$	$x_i$	$y_i^{\text{Eyler}}$	$y_i^{\text{Takomil}}$	$y(x_i)^{\text{Aniq}}$	$\Delta y$
0	-1.000000	1.000000	1.000000	1.000000	0.000000
1	-0.400000	-0.800000	0.460000	0.077090	1.260000
2	0.200000	-0.080000	0.323200	0.008756	0.403200
3	0.800000	0.136000	0.396064	0.197817	0.260064
4	1.400000	0.452800	0.577953	0.464402	0.125153
5	2.000000	0.749440	0.816536	0.754338	0.067096