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

4.8.22 · D1 · Maths › Numerical Methods › ODE solvers — Euler's method (derivation, global error)

Is page par har woh symbol build kiya gaya hai jo parent note tumhare samne phaink deta hai — $y^{'}$ , $f (x, y)$ , $h$ , $x_{n}$ , $y_{n}$ , $O (h)$ , $L$ , $ξ$ , $e^{L (x - x_{0})}$ — ek aise samajhdaar 12-saal ke bachche ke liye jo inhe pehle kabhi nahi dekha. Upar se neeche padho; koi cheez tab tak nahi aayegi jab tak woh earn nahi ho jaati.

1. Ek function aur uska graph — sabse pehli cheez

Sabse pehle: ek function ek machine hai. Tum use ek number $x$ dete ho, woh ek number return karta hai, jise hum $y$ kehte hain. Hum likhte hain $y = g (x)$ : " $y$ equals $g$ of $x$ ."

Simple words: $x$ input hai (kitna daayein), $y$ output hai (kitna upar).
Picture: har input ke liye point $(x, y)$ plot karo. Dots mil ke ek curve banate hain — wahi graph hai.
Kyun zaroori hai: poora topic ek unknown curve $y (x)$ dhundhne ke baare mein hai. Yeh bilkul clear hona chahiye ki $y$ ek height hai jo $x$ ke badhne ke saath change hoti hai.

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

2. Slope — kitna steep, aur kis direction mein

Kisi bhi seedhi line par do points lo. Slope jawaab deta hai: har 1 step daayein ke liye, kitne steps upar?

$slope = \frac{rise}{run} = \frac{change in y}{change in x}$

Simple words: rise over run. Positive = uphill, negative = downhill, zero = flat.
Picture: line ke saath laga ek right triangle — horizontal leg run hai, vertical leg rise hai.
Kyun zaroori hai: Euler ka poora trick hai "slope ki direction mein chalo." Agar tum triangle se slope nahi padh sakte, toh baad mein kuch bhi samajh nahi aayega.

3. Derivative $y^{'}$ — ek curve ka slope

Ek curve bend karti hai, isliye uski steepness har point par alag hoti hai. Derivative curve ka slope hai ek single point par — wahan curve ko bas thoda sa chhune wali tangent line ka slope.

Hum ise $y^{'}$ likhte hain (padho "y prime") ya $\frac{d y}{d x}$ (padho "dee-y by dee-x").

Simple words: $y^{'}$ = ek point par curve ki instantaneous steepness.
Picture: curve mein zoom karo jab tak woh seedhi na lage; us seedhi line ka slope $y^{'}$ hai.
Kyun zaroori hai: differential equation $y^{'}$ se bani hai. Woh is poore show ka star hai.

4. Tangent line — woh object jis par Euler actually chalta hai

Simple words: tangent woh seedhi line hai jo curve ko ek point par usi slope ke saath chhuye jitni wahan curve ki hai.
Picture (upar figure): violet mein curve, ek point par orange mein uski tangent. Dhyaan do tangent curve ko door jaane par chhod deti hai — woh gap Euler ki error ka seed hai.
Kyun zaroori hai: Euler ek chhote step par curve ki jagah uski tangent use karta hai. Tangent aur curve ke beech ka gap hi woh local truncation error hai jiska naam parent note mein hai.

5. Function $f (x, y)$ — har point par ek slope

Yahan ek badi baat hai. Normally slope ke liye pehle curve chahiye. Lekin ek differential equation ise ulta kar deti hai: woh tumhe ek rule $f (x, y)$ deti hai jo plane ke kisi bhi point $(x, y)$ par ek slope deta hai — curve jaane bina.

$y^{'} = f (x, y)$

Simple words: "meri unknown curve ki steepness, jab bhi woh point $(x, y)$ se guzre, $f (x, y)$ ke barabar hai."
Picture: points ki ek grid par ek tiny arrow banao slope $f (x, y)$ ke saath. Yahi direction field hai — chhote arrows ka poora meadow.
Kyun zaroori hai: $f$ akela information hai jo humein milta hai. Curve chhupa hua hai; arrows diye gaye hain. Euler ka kaam hai ek aisa path trace karna jo inhi arrows ke tangent rahe.

6. Initial Value Problem — ek starting pin

Arrows poore plane mein bhare hain, isliye infinitely many curves unhe follow karti hain (har height se ek). Woh curve chunne ke liye hum ek point fix karte hain:

$y (x_{0}) = y_{0}$

Simple words: "input $x_{0}$ par, height $y_{0}$ hai" — meadow mein ek pin thoka hua.
Picture: ek dot jis se curve guzarni chahiye; sirf ek arrow-following path us dot se thread hoti hai.
Kyun zaroori hai: pin ke bina koi unique answer compute nahi hota. $x_{0}, y_{0}$ bas pin ke coordinates hain — subscript $0$ ka matlab "zeroth / starting wala."

7. Subscripts $x_{n}, y_{n}$ aur step $h$ — walker ke footprints

Euler barabar horizontal strides mein chalta hai. Inhe naam do:

$h$ = step size, ek stride ki fixed horizontal distance.
$x_{n}$ = $n$ strides ke baad $x$ : $x_{n} = x_{0} + nh$ .
$y_{n}$ = wahan height ka humara estimate (sahi height nahi — ek computed guess).
Simple words: $n$ ek counter hai (0, 1, 2, …). $x_{n}, y_{n}$ $n$ -th footprint hai.
Picture: dots ki ek staircase daayein ki taraf march karti hai, horizontally har ek $h$ door.
Kyun zaroori hai: update formula $y_{n + 1} = y_{n} + h f (x_{n}, y_{n})$ poori tarah inhi labels mein likhi hai. $y_{n + 1}$ = "agla height," $y_{n}$ = "current height."

8. Big-O notation $O (h)$ — smallness measure karna

Jab hum kehte hain error $O (h)$ hai ("order h") toh matlab: $h$ aadha karo, aur error roughly aadhi ho jaaye. $O (h^{2})$ matlab: $h$ aadha karo, error quarter ho jaaye.

Simple words: $O (h^{k})$ = "step ki $k$ -th power ki tarah shrink karta hai."
Picture: log-log graph par error vs $h$ plot karo — $O (h)$ slope 1 ki seedhi line hai, $O (h^{2})$ steeper slope-2 line.
Kyun zaroori hai: poora punchline — "Euler is first order" — statement hai ki global error $O (h)$ hai.

9. Baaki bache symbols: $ξ$ , $L$ , $M$ , $e^{L (x - x_{0})}$

Yeh sirf error bound mein aate hain. Inse milo taaki yeh scary na lagein.

$ξ$ (Greek "ksi") — koi unknown point $x_{n}$ aur $x_{n} + h$ ke beech. Taylor's theorem promise karta hai discarded term exactly $\frac{h ^{2}}{2} y^{''} (ξ)$ hai ek aise point ke liye; hume kabhi uski value nahi chahiye, bas yeh ki woh exist karta hai. Sochho "ek mystery point jo step ke andar rehta hai."
$y^{''}$ — derivative ka derivative: kitni tezi se slope khud change hota hai, matlab curve kitna bend karta hai (curvature). Zyada bend = tangent curve ko jaldi chhod deta hai = badi error.
$M$ — bending ki ek ceiling: $∣ y^{''} ∣ \leq M$ interval par har jagah. Ek single number jo worst curvature bound karta hai.
$L$ — Lipschitz constant (dekho Lipschitz Continuity): ek number jo kehta hai "agar do starting heights $d$ se differ karti hain, unke slopes zyada se zyada $L d$ se differ karte hain." Yeh measure karta hai ki nearby solution curves kitni tezi se alag ho sakti hain.
$e^{L (x - x_{0})}$ — exponential growth factor. Pehle ki gayi errors baad ke har step par re-amplify hoti hain; worst-case pile-up is exponential ki tarah badhta hai. Bada $L$ (curves tezi se fan out hote hain) = zyada harsh amplification.

Prerequisite map

Yeh map seedha parent topic mein jaata hai. Taylor branch Taylor Series Expansion mein develop hoti hai; forward-difference view Finite Difference Approximations mein rehta hai; amplification factor Lipschitz Continuity se bound hota hai; aur jahan chhota $h$ bhi fail ho jaata hai wahan tum Numerical Stability, Backward Euler & Implicit Methods, aur higher-order Runge-Kutta Methods se milte ho.

Equipment checklist

Graph se curve ki height padhna

Haan —

(x, y)

horizontal

x

par, height

y

wala point hai.

Do points se slope compute karna

rise over run =

\frac{Δ y}{Δ x}

y^{'}

ka matlab words mein bolna

ek point par curve ki tangent ka slope.

\frac{d y}{d x}

ko same cheez ki tarah padhna

y

mein tiny change divided by

x

mein tiny change —

y^{'}

ke identical.

f (x, y)

kya deta hai explain karna

woh slope jo solution ko

(x, y)

se guzarte waqt hona chahiye.

Kyun

f (x, y)

ko sirf integrate nahi kar sakte

f

y

chahiye, jo abhi nahi pata — isliye stepwise method.

y (x_{0}) = y_{0}

kya karta hai batana

poore direction field mein se ek curve select karta hai.

y_{n}

aur

y (x_{n})

mein farq karna

y_{n}

= humara estimate;

y (x_{n})

= sahi height.

O (h)

O (h^{2})

interpret karna

h

aadha karo → error aadhi (

O (h)

) vs error quarter (

O (h^{2})

y^{''}

kya measure karta hai batana

kitni tezi se slope change hota hai — curve ka bending/curvature.

Lipschitz constant

L

kya measure karta hai

nearby solution curves kitni tezi se alag ho sakti hain.

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

kahan se aata hai explain karna

per-step amplification

(1 + L h)^{N} \to e^{L (x - x_{0})}

ka baar baar hona.