Implementing ODE solvers from scratch — Euler, RK4: Foundations

5.4.23 · D1 · Coding › Scientific Computing (Python) › Implementing ODE solvers from scratch — Euler, RK4

Euler ya RK4 ki ek bhi line padhne se pehle, tumhe un sabhi symbols ka pura control hona chahiye jo woh use karte hain. Yeh page har ek ko bilkul scratch se banata hai — pehle plain words, phir ek picture, phir kyun is topic ko iske zaroorat hai. Upar se neeche padho; har block upar wale par lean karta hai.

1. Do variables ka function: $f (t, y)$

Picture. Socho tum ek pahadi par khade ho. Tumhari position describe hoti hai ki tum zameen par kahan ho ( $t$ ) aur kitne upar ho ( $y$ ). Rule $f$ ek chota sa arrow hai jo us spot par laga hua hai aur batata hai ki aage kitni steeply chadhna hai.

Figure — Implementing ODE solvers from scratch — Euler, RK4

Topic ko iske zaroorat kyun hai. Hum usually algebra se $y (t)$ nahi dhundh sakte, lekin $f$ hamesha kuch aisa hota hai jisme hum numbers plug kar ke evaluate kar sakte hain. Poora method yeh hai: $f$ padho, ek step lo, $f$ phir se padho. Agar computable slope nahi hai toh walk karne ke liye kuch nahi hai.

Agar $f$ sirf $y$ par depend karta hai (arrow horizontally tumhari position ko ignore karta hai), toh hum ODE ko autonomous kehte hain.
Agar $f$ $t$ par bhi depend karta hai (arrows time ke saath change hote hain), toh yeh non-autonomous hai — aur tumhe shifted time plug karna yaad rakhna chahiye. Yahi parent page par teesra [!mistake] hai.

2. Derivative aur prime: $y^{'}$ aur $\frac{d y}{d t}$

Picture. Solution curve draw karo. Ek point par, ek ruler sirf curve ko touch karte hue rakh do — tangent line. Iska steepness (rise over run) us point par $y^{'}$ hai.

Do notations kyun? $y^{'}$ short aur clean hai; $\frac{d y}{d t}$ visibly yaad dilata hai ki hum kis variable ke against differentiate kar rahe hain ( $t$ ). Parent ODE ko $\frac{d y}{d t} = f (t, y)$ likhta hai — yeh literally kehta hai "curve ki slope wahi hai jo slope machine report karta hai." Yeh ek equation hi poora problem hai.

Dekho Finite Difference Approximation of Derivatives ki kaise hum is idealized slope ko kuch aisa banate hain jo computer compute kar sake.

3. Limit: $h \to 0 lim$

Picture. Curve par do points lo jo horizontally $h$ distance par hain, aur unke beech seedhi line draw karo (ek secant). Jaise tum doosre point ko paas slide karte ho — $h$ chota hota jaata hai — woh secant line pivot karti hai jab tak tangent nahi ban jaati. Limit wahi final tangent slope hai.

Topic ko iske zaroorat kyun hai. Exact derivative limit se define hoti hai $y^{'} (t) = lim_{h \to 0} \frac{y ( t + h ) - y ( t )}{h} .$ Computer $h$ ko bilkul zero tak nahi le ja sakta — woh zero se divide karna hoga. Toh Euler ka trick hai limit ko jaldi rok do: ek small-but-real $h$ chuno aur secant slope ko tangent ke stand-in ke roop mein accept karo. "Jaldi ruke" aur "true limit" ke beech ka gap wahi error hai jiske peechhe hum baaki topic mein lagte hain.

4. Step size $h$ aur index $n$ : $t_{n}$ , $y_{n}$ , $t_{n + 1}$

Picture. $t$ -axis ke saath evenly spaced fence-posts ki ek row, har ek ke beech $h$ gap. Post number $n$ $t_{n}$ par baith hai; uske upar humara height guess $y_{n}$ baith hai.

Topic ko iske zaroorat kyun hai. Poora algorithm ek loop hai: $(t_{n}, y_{n})$ se $(t_{n + 1}, y_{n + 1})$ banao, phir repeat karo. Subscript notation yeh likhne ka tarika hai ki "current post se next post." Yahi $y_{n + 1} = y_{n} + h f (t_{n}, y_{n})$ ka matlab hai.

5. Taylor expansion — jahan update formula janam leta hai

$y (t_{n} + h) = start y (t_{n}) + straight-line part h y^{'} (t_{n}) + bend jo hum ignore karte hain \frac{h ^{2}}{2} y^{''} (ξ) + \dots$

Picture. Seedha tangent curve ka pehla draft hai. $h^{2}$ term pehla correction hai jo draft ko real curve ki taraf modhta hai; jitne zyada terms rakho, utna tight fit.

Topic ko iske zaroorat kyun hai. Series ko slope term ke baad cut karo → tumhe Euler milta hai, aur pehla pheka hua piece ( $\propto h^{2}$ ) Euler ka per-step error hai. Zyada terms match karte raho → tumhe RK4 milta hai, tuned taaki errors $h^{4}$ level tak cancel out ho jaayein. Poori detail Taylor Series Expansion mein hai.

6. Big-O notation: $O (h)$ , $O (h^{2})$ , $O (h^{4})$

Order	$h$ halve karo → error ×	Reader ka takeaway
$O (h)$ (Euler global)	$\frac{1}{2}$	aadha error — slow
$O (h^{4})$ (RK4 global)	$\frac{1}{16}$	solahwan hissa error — fast

Topic ko iske zaroorat kyun hai. Ek hi reason hai kyun RK4 Euler se behtar hai — exponents ki comparison. $O (\cdot)$ ke bina tum nahi keh sakte "har halving par 16× better." Parent ka error table $O$ -notation ko concrete banane ke alawa kuch nahi hai.

7. Weighted average — RK4 ka dil

Picture. Hop ke along chaar compass readings li gayi hain (start, middle, middle again, end). Dono middle readings zyada trusted hain, toh woh scale par do baar zyada bhari hain.

Topic ko iske zaroorat kyun hai. RK4 ka step $y_{n + 1} = y_{n} + \frac{h}{6} (k_{1} + 2 k_{2} + 2 k_{3} + k_{4})$ exactly height + step × (weighted-average slope) hai. Denominator $6 = 1 + 2 + 2 + 1$ isliye hum $6$ se divide karte hain. Yeh Simpson's Rule ko mirror karta hai, jo midpoint ko bhi heavily weight karta hai — analogous, identical nahi (Simpson $1, 4, 1$ use karta hai).

Prerequisite map

Har box is page ka ek symbol ya idea hai; arrows dikhate hain ki kya kya feed karta hai. Notice karo ki dono update rules Taylor expansion par baithe hain, jo khud derivative par baith hai, jo limit par baith hai — poora tower Section 3 par tika hai.

Equipment checklist

Khud test karo: kya tum reveal karne se pehle har ek ka answer de sakte ho?

f (t, y)

physically kya return karta hai?

Ek single number — woh slope jo solution ko point

(t, y)

par honi chahiye.

Autonomous vs non-autonomous

f

Autonomous sirf

y

par depend karta hai; non-autonomous

t

par bhi depend karta hai, isliye tumhe shifted times plug karne chahiye.

y^{'}

\frac{d y}{d t}

geometrically kya matlab hai?

Us point par solution curve ki tangent line ki steepness.

lim_{h \to 0}

kya poochta hai?

Ek expression kis value ki taraf approach karta hai jaise

h

zero ki taraf shrink hota hai (kabhi pahunche bina).

y_{n}

aur

y (t_{n})

mein kya difference hai?

y_{n}

humara marching estimate hai;

y (t_{n})

true (usually unknown) value hai — unka gap error hai.

h

kya hai?

Fixed horizontal step size — ek hop ki width, jisme

t_{n + 1} = t_{n} + h

Taylor expansion humein yahan kya deta hai?

Solution rebuild hota hai start + slope·

h

+ curvature·

h^{2}

+ … ke roop mein; ise truncate karne se Euler aur RK4 milta hai.

O (h^{4})

ka kya matlab hai jab hum

h

half karte hain?

Error

2^{4} = 16

times drop ho jaata hai.

Local vs global error order?

Local ek-step error hai; global accumulated error hai saare steps ke baad — ek power of

h

weaker.

RK4 slopes ko 6 se kyun divide karte hain?

Kyunki weights

1, 2, 2, 1

ka sum

6

hai, jo ise ek true weighted average banata hai.