Stiff equations — implicit methods, backward Euler: Foundations

4.8.26 · D1 · Maths › Numerical Methods › Stiff equations — implicit methods, backward Euler

Yeh the stiff-equations topic ka Foundations child hai. Hum maan ke chalte hain ki aapne is notation mein se kuch nahi dekha. Hum har symbol ko use karne se pehle earn karte hain.

1. Time ki function $y (t)$ kya hoti hai?

Ek aisa number imagine karo jo time ke saath badle: ek thande hote coffee cup ka temperature, ek bouncing-phir-settle hoti swing ki height, ya battery mein bacha charge.

Figure — Stiff equations — implicit methods, backward Euler

Upar ki blue curve dekho: har moment $t$ par (bottom axis par ek jagah) curve kisi height par hai. Woh height hi single number $y (t)$ hai. Poori blue curve function hai; uspar ek dot ek value $y (t)$ hai. Curve ko left-to-right padhna system ko evolve hote dekhna hai.

Is topic ko yeh kyun chahiye: stiff equations mein sab kuch is baare mein hai ki aisi curve zero ki taraf kaise decay karti hai aur computer use kitni achhi tarah trace kar sakta hai.

2. Derivative $y^{'}$ — curve ki slope

Computer poori curve store nahi kar sakta; woh sirf abhi ki value aur kitni tezi se badal raha hai ka rule jaanta hai. Yahi rate of change derivative hai.

Green line tangent hai — woh straight line jo curve ko touch karti hai aur us ek jagah par uski slope se match karti hai. Uski steepness hi number $y^{'}$ hai.

"Derivative" kyun, koi aur tool kyun nahi? Kyunki ek differential equation ek aisi machine hai jo aapko har point par slope deta hai aur curve dubara banane ko kehta hai. Slope bilkul wahi language hai jo woh bolta hai.

3. Differential equation $y^{'} = f (t, y)$ aur uska starting point $y_{0}$

Ise zor se padho: "y ki rate of change current time aur current $y$ se involve hone wale kisi formula ke barabar hai."

Lekin sirf slope-rule kaafi nahi hai. Har jagah slope jaanna curve ki shape batata hai, kahan se shuru hoti hai nahi. Infinite saari parallel curves ka slope-rule same hota hai; hume ek pin down karni hogi.

Is topic ko yeh kyun chahiye: poora subject ek master IVP, $y^{'} = λ y$ with $y (0) = y_{0}$ , study karta hai, kyunki yeh sabse simple decay-rule hai aur har method ka behaviour isme clearly dikhta hai.

4. Exponential $e^{λ t}$ — pure decay/growth ki shape

Lekin $e^{λ t}$ $y^{'} = λ y$ ka sirf ek hi solution nahi hai — koi bhi constant multiple bhi kaam karta hai, kyunki curve ko scale karne se slope bhi same factor se scale hoti hai. Section 3 ka starting value $y_{0}$ fix karta hai kaun sa multiple:

Do decaying curves: $λ = - 1$ (dheemi, yellow) aur $λ = - 100$ (ek cliff, red). Dono zero ki taraf jaati hain, lekin red wali almost instantly khatam ho jaati hai. Ek stiff problem mein dono shapes ek saath hoti hain — aur speeds ka woh clash hi poori mushkil hai.

5. Complex $λ$ aur " $Re λ$ "

Kabhi kabhi $λ$ ek complex number hota hai (iska ek "real part" aur ek "imaginary part" hota hai) kyunki oscillations iske andar chhupe hote hain. Aapko yahan complex analysis ki zaroorat nahi — sirf ek word ki.

Is topic ko yeh kyun chahiye: real systems (springs, circuits) decay bhi karte hain aur oscillate bhi. $Re λ < 0$ likhne se ek clean inequality sab cover kar leti hai.

6. Discrete world: $h$ , $t_{n}$ , $y_{n}$

Computer smooth continuum handle nahi kar sakta. Woh fixed hops mein aage koodta hai.

Smooth blue curve sach hai; yellow dots computer ke samples $y_{0}, y_{1}, y_{2}, \dots$ hain jo $h$ door door hain. Sabse pehla dot $y_{0}$ hume initial condition deta hai; numerical methods recipes hain next dot nikalne ki current dot se.

7. Approximation $y^{'} (t_{n}) \approx \frac{y _{n + 1} - y _{n}}{h}$

Yahan "slope" "hops" se milti hai. Slope rise over run hoti hai. Ek hop mein, run $h$ hai aur rise $y$ mein change hai.

Yeh exact world (Section 2) se computable world (Section 6) ka bridge hai. Yeh rise-over-run kaun se dot ki slope represent karta hai — purana dot $t_{n}$ ya naya dot $t_{n + 1}$ — woh single choice hai jo forward Euler (explicit) ko backward Euler (implicit) se alag karti hai. Yahi choice parent topic ka dil hai; ab aapke paas ise samajhne ke liye har symbol hai.

8. Absolute value $∣ \cdot ∣$ — woh size jo stability decide karti hai

Is topic ko yeh kyun chahiye: ek method stable hota hai jab, step after step, woh value ko zero ki taraf shrink karta hai, na ki usse blow up karta hai. "Kya woh shrink karta hai?" bilkul yehi hai "kya multiplier ki magnitude $1$ se neeche hai?" — likha jaata hai $∣ multiplier ∣ < 1$ . Sign matter nahi karta (value har step par sign flip kar sakti hai aur phir bhi shrink ho sakti hai); sirf size karti hai. Isliye absolute value, plain number nahi, stability govern karti hai.

9. Prerequisite map

Har foundation agla feed karta hai: time aur value milke curve banate hain, curve ki slope hoti hai, slope-rule plus starting value $y_{0}$ ek IVP hai, IVP ka answer ek scaled exponential hai, fast aur slow exponentials milake stiffness banate hain, aur finite-difference slope plus size-test decide karta hai ki koi method survive karta hai ya nahi.

Equipment checklist

Self-test: har ek reveal karne se pehle answer kar sako?

Single value

y (t)

ka kya matlab hai, versus function

y (\cdot)

y (t)

ek number hai — us ek time

t

par height; function

y (\cdot)

poori curve hai, har time par har value.

Derivative

y^{'}

kya hai?

Ek point par curve ki slope/steepness — tiny bit of time mein

y

kitni tezi se badlti hai.

y^{'} = f (t, y)

kya kehta hai?

Ek rule jo current time aur current value se current slope deta hai;

f

slope-rule hai.

Slope-rule ko IVP mein kya extra ingredient banata hai, aur kyun?

Initial condition

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

; yeh pin down karta hai ki curve kahan se shuru hoti hai, us slope-rule wali infinite family mein se ek curve select karke.

y_{0}

kya hai?

Start time

t_{0}

par

y

ki starting value — woh ek number jo hume exactly pata hai, aur pehla sample.

y^{'} = λ y

with

y (0) = y_{0}

ko kaun si curve solve karti hai?

y (t) = y_{0} e^{λ t}

, jiski slope hamesha apni height ka

λ

times hoti hai aur jo

y_{0}

se shuru hoti hai.

λ < 0

solution ka kya karta hai?

Ise zero ki taraf decay karwata hai — zyada negative matlab tezi se decay.

Re λ < 0

ka matlab kya hai?

λ

ka real part negative hai, yaani system decay karta hai (hidden oscillation allow karte hue).

h

t_{n}

, aur

y_{n}

kya hain?

Step size,

n

hops ke baad time

nh

, aur wahan

y

ka computed estimate.

Hum samples se

y^{'} (t_{n})

approximate kaise karte hain?

Finite difference

(y_{n + 1} - y_{n}) / h

se —

t_{n}

aur

t_{n + 1}

par dots ke beech rise over run.

Stability

∣ \cdot ∣

use kyun karta hai, plain value nahi?

Kyunki stability poochti hai "kya value har step shrink hoti hai?", jo multiplier ki size par depend karta hai, sign par nahi.

y_{n}

aur

y (t_{n})

mein farq?

y_{n}

computed dot hai (err ya explode ho sakta hai);

y (t_{n})

exact value hai; inka gap error hai.