Heat equation (parabolic) 1D — derivation from Fourier's law: Visual walkthrough

4.7.8 · D2 · Maths › Partial Differential Equations › Heat equation (parabolic) 1D — derivation from Fourier's law

Kisi bhi symbol se pehle, hamari poori duniya yeh hai: ek patli rod, aur uske upar ek tedhi-medhi line jo batati hai ki har jagah kitni garmi hai.

Step 1 — Temperature profile kya hota hai?

KYA HAI. Socho ek patli metal ki rod seedhi padi hui hai. Hum uski position ko ek number $x$ se measure karte hain (left end se kitna door, metres mein). Har jagah hum temperature measure kar sakte hain. Hum us temperature ko rod ke upar ek height ki tarah draw karte hain. Woh curve hai $u (x, t)$ .

KYUN. Baad mein jo bhi symbol use karenge woh sab is curve ke baare mein facts hain. Isse pehle ki hum "kitna curved hai" ya "kaise badal raha hai" ki baat karein, pehle yeh agree karna zaroori hai ki temperature bas har position ke upar ek height hai — ek aisi shape jise hum dekh sakte hain.

$u$ ::: temperature (ek height), kelvin $K$ mein measure ki gayi
$x$ ::: rod ke saath position, metres mein
$t$ ::: time; yeh curve time ke saath hilti-dulti reh sakti hai, isliye hum $u (x, t)$ likhte hain

PICTURE. Rod horizontal line hai; colored curve uske upar float karti hai. Zyada upar = zyada garam, neeche = thanda.

Figure — Heat equation (parabolic) 1D — derivation from Fourier's law

Step 2 — Slope $u_{x}$ : profile kitna steep hai?

KYA HAI. Curve ke ek point par zoom karo aur wahan ek seedha ruler tangent ki tarah rakh do. Us ruler ka slope $u_{x}$ likha jaata hai (chhota $x$ matlab "jab $x$ -direction mein step lete ho toh $u$ kitni jaldi badalta hai"). Tezi se upar = bada positive $u_{x}$ ; tezi se neeche = bada negative $u_{x}$ ; seedha flat = zero.

KYUN. Heat ko is baat ki parwah hogi ki neeche ki taraf kaunsi disha hai, kyunki heat garam se thande ki taraf neeche girती hai. "Neeche" precisely kehne ke liye slope chahiye. Derivative yahan sirf isi wajah se aata hai — yeh woh machine hai jo "wahan zyada garam hai" ko ek number mein ek sign ke saath badal deti hai.

$u_{x} = temperature curve ka slope \frac{\partial u}{\partial x}$

$\partial u$ ::: temperature mein ek bahut chhota change
$\partial x$ ::: position mein woh chhota step jo yeh change laya
unka ratio ::: rise over run = steepness, bilkul kisi pahad ki grade ki tarah

PICTURE. Teen jagahon par green tangent lines: upar (positive), flat (zero), neeche (negative).

Step 3 — Fourier's Law: heat neeche girती hai

KYA HAI. Heat flux $q$ define karo = har second mein kitni thermal energy ek point ko cross karti hai, per unit area, $+ x$ direction mein move karti hui. Fourier's law kehta hai ki $q$ negative slope ke proportional hai:

$q = material & sign - k steepness u_{x}$

Minus sign kyun. Figure mein red arrows dekho. Agar curve daayein taraf upar jaati hai ( $u_{x} > 0$ , daayein taraf zyada garam), toh heat baayein flow karni chahiye — thande ki taraf — isliye $q$ negative hai. Minus sign hi woh cheez hai jo "garam → thanda" ko force karti hai. Iske bina, heat garam jagah ki taraf chadhti, jo kabhi nahi hota.

$k$ ::: thermal conductivity ( $W/m\cdotK$ ) — yeh material kitni aasaani se heat pass karta hai
$-$ ::: direction flip kar deta hai taaki flow hamesha slope se neeche ho
$q > 0$ ::: energy daayein jaati hai; $q < 0$ ::: energy baayein jaati hai

PICTURE. Upar-daayein slope waali pahadi par, flux arrow baayein (neeche) jaata hai. Upar-baayein slope par, daayein jaata hai. Hamesha thande ki taraf.

Step 4 — Control slab: energy ki bookkeeping

KYA HAI. Rod ka ek patla slice $x$ aur $x + Δ x$ ke beech kaato, face area $A$ ke saath. Yeh hamara control slab hai. Ab saari physics bas yeh hai: energy in − energy out = energy jo andar jama ho gayi. Yeh conservation of energy hai.

KYUN. Hum har atom track nahi kar sakte. Iske bajaye hum ek chhote box ko dekhte hain aur uske do faces par energy count karte hain. Ek chhota box exactly wahi cheez hai jo baad mein $Δ x \to 0$ shrink karne aur derivative lene mein help karta hai — geometry khud hamare liye calculus kar deta hai.

Slab ke andar stored energy: $E (t) = \int_{x}^{x + Δ x} energy per volume ρ c u volume slice A d s$

$ρ$ ::: density ( $kg/m^{3}$ )
$c$ ::: specific heat ( $J/kg\cdotK$ ) — 1 kg ko 1 K warm karne ke liye energy
$ρ c u$ ::: isliye temperature $u$ par stored energy per cubic metre
$A d s$ ::: position $s$ par ek kaagaz-jaisi patli sub-slice ka volume

PICTURE. Slab, uska left face $x$ par flux $q (x)$ andar aata hua, uska right face $x + Δ x$ par flux $q (x + Δ x)$ bahar jaata hua.

Step 5 — In minus out: flux imbalance

KYA HAI. Left face se har second enter hoti energy hai $A q (x)$ . Right face se nikalne wali energy hai $A q (x + Δ x)$ . Toh slab ko har second mila net energy hai

$\frac{d E}{d t} = left mein aayi A q (x) - right se gayi A q (x + Δ x)$

KYUN. Andar koi energy create ya destroy nahi hoti (abhi koi source nahi hai), isliye slab ki stored energy badalne ka ek hi tarika hai — do faces se leak hona. Agar left se zyada aaye aur right se kam nikle, toh slab garm ho jaata hai. Yeh derivation ka poora dil hai: flux ka mismatch hi cheezein garm karta hai.

$A q (x)$ ::: rate jis par energy left wall se andar chalti hai
$A q (x + Δ x)$ ::: rate jis par yeh right wall se bahar chalti hai
unka fark ::: net pile-up (positive = warming)

PICTURE. Do faces par alag-alag lambaai ke do arrows; bacha hua (shaded) woh hai jo peeche rehta hai aur slab ko garm karta hai.

Step 6 — Slab ko shrink karo: flux difference derivative ban jaata hai

KYA HAI. Stored energy $E$ ko differentiate karo (limits space mein fixed hain, isliye $\frac{d}{d t}$ andar ghus jaata hai) aur Step 5 se match karo:

$\int_{x}^{x + Δ x} ρ c u_{t} A d s = A (q (x) - q (x + Δ x))$

Dono sides ko $A Δ x$ se divide karo aur $Δ x \to 0$ hone do:

$ρ c u_{t} = = - q_{x} Δ x \to 0 lim \frac{q ( x ) - q ( x + Δ x )}{Δ x} = - \frac{\partial q}{\partial x}$

KYUN. Us fraction ko dhyan se dekho. Yeh exactly (minus) ek derivative ki definition hai: (yahan value − ek step aage value) ÷ (step ka size). Jab box shrink hota hai, andar ka average temperature-change theek us point ki value ban jaata hai, $u_{t}$ , aur flux difference flux ka slope ban jaata hai, $q_{x}$ . Hum sirf isi liye chhota slab liye the taaki yeh derivative summon ho sake.

$u_{t}$ ::: is point par temperature time mein kitni jaldi badhti hai ( $K/s$ )
$- q_{x}$ ::: flux daayein jaate hue kitni jaldi girti hai = net inflow rate

PICTURE. Slab ek point mein collapse hota hai; do arrows ek number mein merge hote hain — $q$ ka slope.

Step 7 — Fourier's law substitute karo aur equation se milo

KYA HAI. Ab hamare paas do facts hain: $ρ c u_{t} = - q_{x}$ (Step 6) aur $q = - k u_{x}$ (Step 3). Doosre ko pehle mein plug karo:

$ρ c u_{t} = - \frac{\partial}{\partial x} (q - k u_{x}) = k u_{xx}$

$ρ c$ se divide karo aur constant ko $α = \frac{k}{ρ c}$ naam do:

$u_{t} = diffusivity α curvature u_{xx}$

KYUN. Do minus signs (ek Fourier se, ek "in minus out" se) milkar plus dete hain. Yahi plus hai jis ki wajah se heat spread hoti aur smooth hoti hai blast hone ki jagah. Second derivative $u_{xx}$ isliye aaya kyunki humne slope ( $u_{x}$ ) ko ek baar aur differentiate kiya — isliye equation curvature se govern hoti hai, slope se nahi. Dekho kyun $\alpha$, $k$ nahi, pace set karta hai.

$u_{xx}$ ::: curvature — curve kitna muda hua hai (cup up vs bump)
$α = k / (ρ c)$ ::: "kitna curved" ko "kitna jaldi" mein badalta hai ( $m^{2} / s$ )

PICTURE. Ek dip (cup-up, $u_{xx} > 0$ ) upar-arrow ke saath (warming); ek bump (cup-down, $u_{xx} < 0$ ) neeche-arrow ke saath (cooling).

Step 8 — Degenerate cases: har sign padho

KYA HAI. Equation ko har situation mein sense banana chahiye, toh aao extremes test karein.

Profile shape	$u_{xx}$	$u_{t} = α u_{xx}$	Kya hota hai
Bump / hill	$< 0$	$< 0$	point cool hota hai
Dip / valley	$> 0$	$> 0$	point warm hota hai
Straight line	$= 0$	$= 0$	koi change nahi — steady
Flat (constant)	$= 0$	$= 0$	already uniform, waisai rehta hai

KYUN. Do "no change" rows dhyan se dekhne laayak hain. Ek straight-line profile mein slope hai lekin koi curvature nahi: heat downhill side se utni hi tezi se nikalta hai jitni tezi se uphill side se aata hai, isliye point break-even karta hai — yeh 1D mein steady state hai. Ek constant profile mein koi slope hi nahi hai: kuch bhi kahan bhi flow nahi karta. Dono $u_{t} = 0$ dete hain, lekin subtly alag wajahon se — aur equation dono ko automatically capture kar leti hai kyunki yeh curvature dekhti hai, slope nahi.

PICTURE. Teen panels: bump (down-arrow), valley (up-arrow), straight line (flat arrows in = out, net zero).

Ek-picture summary

Upar sab kuch, ek frame mein: ek curve, uska slope flux deta hai (pahadi se neeche), ek shrinking slab mein flux mismatch $u_{t}$ deta hai, aur do minus signs milkar $u_{t} = α u_{xx}$ ban jaate hain.

Recall Poore walkthrough ki Feynman-style retelling

Ek garam rod ko temperature ka pahadi landscape socho. Slope batata hai neeche kaunsi disha hai (Step 2). Heat aalsi hoti hai — yeh hamesha slope se neeche girती hai, aur Fourier's law mein minus sign yeh bake in karta hai (Step 3). Ab ek chhota box kaat lo (Step 4) aur bas count karo: energy left face se andar, right face se bahar (Step 5). Agar andar zyada aaye aur bahar kam nikle, toh box garm hota hai. Box ko shrink karo aur "in minus out" ban jaata hai "flux kitni jaldi change ho raha hai," jo ek derivative hai (Step 6). Fourier's law feed karo, aur do minus signs khushi se cancel hokar plus ban jaate hain (Step 7). Jo bachta hai woh beautiful hai: second derivative — curvature — kyunki humne slope ka slope liya. Hills (neeche curve karti hain) heat lose karti hain aur cool hoti hain; valleys (upar curve karti hain) heat gain karti hain aur warm hoti hain; straight ramps break even karte hain (Step 8). Woh ek sentence — "time-change equals diffusivity times curvature" — hi $u_{t} = α u_{xx}$ hai.

Connections

Heat equation (parabolic) 1D — derivation from Fourier's law
Fourier's Law of Conduction
Conservation Laws and Continuity Equation
Classification of PDEs (elliptic, parabolic, hyperbolic)
Separation of Variables for the Heat Equation
Thermal Diffusivity and Material Properties
Laplace Equation (steady-state heat)
Wave Equation (hyperbolic) 1D