Runge–Kutta RK4 usuli

Har qadamda hosilaning to‘rtta qiymati bilan Koshi masalasini hisoblang; barcha oraliq koeffitsientlar va yechim grafigini ko‘ring.

y' = f(x, y), y(x₀) = y₀

Klassik oshkor to‘rtinchi tartibli Runge–Kutta sxemasi.

INPUT / PARAMETERS

Koshi masalasi parametrlari

RK4 bosqich formulalari

\begin{aligned}k_0&=h f(x_i,y_i)\\k_1&=h f\left(x_i+\frac{h}{2},y_i+\frac{k_0}{2}\right)\\k_2&=h f\left(x_i+\frac{h}{2},y_i+\frac{k_1}{2}\right)\\k_3&=h f(x_i+h,y_i+k_2)\\y_{i+1}&=y_i+\frac{k_0+2k_1+2k_2+k_3}{6},\quad x_{i+1}=x_i+h\end{aligned}

ANALYTIC / EXACT

y(x) = -x^2 - 2x - 2 + 3e^{(x - x_0)}

Kiritilgan birinchi tartibli differensial tenglama va boshlang‘ich shartdan analitik chiqarildi.

VISUALIZATION / RK4

RK4 yaqinlashuvini ushbu misolning aniq yechimi bilan solishtiring.

RESULTS / STEP TABLE

i	xᵢ	yᵢ (RK4)	k₀	k₁	k₂	k₃	y aniq
0	0.0000	1.000000	0.100000	0.105250	0.105512	0.111551	1.000000
1	0.1000	1.105513	0.111551	0.118379	0.118720	0.126423	1.105513
2	0.2000	1.224208	0.126421	0.134992	0.135420	0.144963	1.224208
3	0.3000	1.359576	0.000000	0.000000	0.000000	0.000000	1.359576

Yakuniy yaqinlashuv: y(0.3000) ≈ 1.359576· aniq: 1.359576