ODE solvers — Euler's method (derivation, global error)

4.8.22 · Maths › Numerical Methods

Hum ek Initial Value Problem (IVP) solve karna chahte hain: $\frac{d y}{d x} = f (x, y), y (x_{0}) = y_{0}$

Function $f$ humein kisi bhi point $(x, y)$ par solution curve ka slope batata hai. Dikkat yeh hai: hamare paas $y (x)$ ka formula nahi hai — sirf uska slope hai. Euler's method "mujhe slope pata hai har jagah" ko "main numerically curve sketch kar sakta hoon" mein convert karta hai.

WHY — woh problem jis par hum attack kar rahe hain

WHY hum sirf integrate nahi kar sakte? Kyunki $f$ khud $y$ par depend karta hai, jo humein abhi pata nahi. Humein ek stepwise, self-bootstrapping method chahiye: jo hamare paas abhi hai use karke next point estimate karo.

HOW — method ko first principles se derive karna

Hum ise do tareekon se derive karte hain taaki acchi tarah se yaad rahe.

Derivation 1: Taylor series (rigorous route)

True solution $y (x)$ ko $x_{n}$ ke aas-paas step $h$ ke liye expand karo: $y (x_{n} + h) = y (x_{n}) + h y^{'} (x_{n}) + \frac{h ^{2}}{2} y^{''} (ξ)$

Yeh step kyun? Taylor's theorem exact hai last term ke saath jo kisi $ξ \in (x_{n}, x_{n} + h)$ par evaluate hota hai. Yeh humein precisely batata hai ki hum kya throw away kar rahe hain.

Ab $y^{'} (x_{n}) = f (x_{n}, y_{n})$ substitute karo (ODE yahi deta hai humein) aur $h^{2}$ term drop karo: $y_{n + 1} = y_{n} + h f (x_{n}, y_{n})$

Derivation 2: Tangent line / finite difference (intuitive route)

Derivative ko forward difference se approximate karo: $y^{'} (x_{n}) \approx \frac{y _{n + 1} - y _{n}}{h}$

Yeh step kyun? Yeh derivative ki definition hai limit $h \to 0$ lene se pehle. Hum $h$ ko finite rakhte hain kyunki computer limits nahi le sakta.

Ise $f (x_{n}, y_{n})$ ke barabar set karo aur $y_{n + 1}$ ke liye solve karo — aapko wahi boxed formula milega.

Figure — ODE solvers — Euler's method (derivation, global error)

Local vs Global error — "Euler sirf first order kyun hai" ka core

WHY global error $O (h)$ hai, $O (h^{2})$ nahi

Yeh hai steel-manned reasoning (skip mat karo — yeh woh 20% hai jo 80% marks dilata hai):

Har step $\sim \frac{h ^{2}}{2} ∣ y^{''} ∣$ size ka error add karta hai.
$x = b$ tak pahunchne ke liye hum $N = \frac{b - x _{0}}{h}$ steps lete hain — toh $N$ badhta hai jab $h$ chhota hota hai.
Naive total: $N \times O (h^{2}) = \frac{b - x _{0}}{h} \cdot O (h^{2}) = O (h)$ .

Toh $h$ ka ek power kho jaata hai kyunki chhote $h$ ke liye zyada steps chahiye.

$e^{L (x_{n} - x_{0})}$ factor kyun hai? Errors sirf add nahi hote — pehle ke errors $f$ se propagate hote hue amplify hote hain. Lipschitz constant $L$ measure karta hai ki nearby solutions kitni tez diverge hote hain; exponential worst-case growth hai.

Worked Examples

Common Mistakes (steel-manned)

Recall Ek 12-saal ke bachche ko explain karo (Feynman)

Socho tum fog mein chal rahe ho aur tum sirf dekh sakte ho ki path kis direction mein point kar raha hai bilkul wahan jahan tum khade ho. Tum poora path nahi dekh sakte. Toh tum us direction mein face karo, ek chhota sa step lo, phir se naya direction dekho, ek aur chhota step lo. Har step mein tum thoda off hote ho kyunki path curve hua jab tum seedha chale — lekin chhote steps tumhe close rakhte hain. Chhote steps lene se tum real path ke zyada close rehte ho, lekin finish karne ke liye tumhe bahut zyada steps chahiye.

Flashcards

What IVP does Euler's method solve?

y^{'} = f (x, y), y (x_{0}) = y_{0}

— ek ODE jisme ek known starting value ho.

State the Euler update formula.

y_{n + 1} = y_{n} + h f (x_{n}, y_{n})

, with

x_{n + 1} = x_{n} + h

Euler ko Taylor series se derive karo.

y (x_{n} + h) = y_{n} + h y^{'} (x_{n}) + \frac{h ^{2}}{2} y^{''} (ξ)

;

y^{'} = f

substitute karo,

O (h^{2})

term drop karo.

Euler ka local truncation error kya hai?

\frac{h ^{2}}{2} y^{''} (ξ) = O (h^{2})

— ek single step ka error.

Global error order kya hai aur kyun?

O (h)

—

N = (b - x_{0}) / h

steps mein se har ek

O (h^{2})

hai, toh

N \cdot O (h^{2}) = O (h)

; ek power kho jaata hai.

Agar

h

half karo, toh global error ka kya hoga?

Roughly half ho jaata hai (first-order method).

Error bound mein

e^{L (x - x_{0})}

kyun hai?

Pehle ke errors propagate hote hue amplify hote hain;

L

(Lipschitz constant) worst-case exponential growth bound karta hai.

Forward vs Backward Euler?

Forward current point par slope use karta hai

f (x_{n}, y_{n})

(explicit); Backward

f (x_{n + 1}, y_{n + 1})

use karta hai (implicit,

y_{n + 1}

ke liye solve karo).

Euler ka geometric interpretation kya hai?

Current point par tangent line ko horizontal distance

h

tak follow karo.

Connections

Taylor Series Expansion — derivation aur error analysis ka engine
Runge-Kutta Methods — higher-order successors (RK4 global mein $O (h^{4})$ hai)
Backward Euler & Implicit Methods — stiff equations ke liye
Numerical Stability — kyun Example 3 blow up hua
Lipschitz Continuity — guarantee karta hai ki error bound exist karta hai
Finite Difference Approximations — forward difference = Euler ka derivation route