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

4.7.8 · D1 · Maths › Partial Differential Equations › Heat equation (parabolic) 1D — derivation from Fourier's law

Is page par assume kiya gaya hai ki aapne pehle kuch nahi dekha. Hum har woh symbol collect karte hain jo parent note parent topic use karta hai, aur har ek ko ek picture se build karte hain — pehle picture, phir symbol.

0. Stage: ek rod aur do rulers

Kisi bhi symbol se pehle, us object ki picture banao jise hum study kar rahe hain.

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

1. $x$ — rod ke along position

Picture: ek point jo rod ke along left–right slide karta hai. Topic ko kyun chahiye: heat rod ke along flow karti hai, isliye hume yeh naam dena hoga ki rod par kuch kahan hota hai. $x$ ke bina hum "right side par zyada hot" nahi keh sakte.

2. $t$ — time

Picture: ek stopwatch jo tab chal rahi hai jab rod wahin baitha hai. Topic ko kyun chahiye: temperature change hota hai — heat equation ka poora point yahi hai ki future predict karo. Jo quantity change hoti hai use ek clock chahiye.

3. $u (x, t)$ — temperature field

Topic ko kyun chahiye: $u$ literally woh unknown hai jise hum solve karte hain. PDE ek equation hai $u$ ke baare mein.

4. Derivative — "rate of change" as a slope

Aage aane wale do symbols ( $u_{t}$ aur $u_{x}$ ) dono derivatives hain, isliye pehle yeh idea banana zaroori hai.

Topic ko kyun chahiye: heat flow change ke baare mein hai — time mein change (kya yeh spot warm up ho raha hai?) aur space mein change (kya right side par zyada hot hai?). Dono slopes hain.

5. $u_{t}$ aur $\frac{\partial u}{\partial t}$ — time mein change

Curly $\partial$ (kaho "partial") ka matlab sirf yeh hai: "doosre input ko freeze rakh ke derivative lena." Kyunki $u$ ke do inputs hain, hume batana hoga ki hum kaun sa wala wiggle kar rahe hain — yahi ek reason hai plain $d$ ki jagah is naye symbol ka.

Picture: rod ke ek point par khade raho; ek thermometer par uska temperature badhte ya girte dekho. Topic ko kyun chahiye: $u_{t} = α u_{xx}$ ka left-hand side yahi quantity hai. Yeh woh answer hai jo hum chahte hain: ab se future kaise alag hai.

6. $u_{x}$ — temperature gradient (space mein slope)

Picture: snapshot figure mein temperature curve ki tilt. Right ki taraf uphill matlab $u_{x} > 0$ (right side par zyada hot).

Topic ko kyun chahiye: Fourier's law kehta hai heat is slope ke neeche flow karti hai. Slope nahi, flow nahi. Gradient hi heat ko drive karta hai.

7. $u_{xx}$ — curvature (slope ka slope)

Yeh poore parent page par sabse important symbol hai, isliye hum ise ek figure dete hain.

Topic ko kyun chahiye: yeh heat equation ka poora right-hand side hai. Curvature hi engine hai.

8. $q (x, t)$ — heat flux

Picture: rod ke cross-section se ek arrow; uska size $q$ hai, uski direction sign hai. Topic ko kyun chahiye: heat "move karna" wahi hai jo flux measure karta hai. Fourier's law flux ko gradient se link karta hai; conservation flux ko storage se link karta hai.

9. Material constants: $ρ$ , $c$ , $k$ , aur $α$

Picture: $ρ c$ = har point par ek bucket ka size; $k$ = buckets ke beech pipes ki width; $α$ = paani ka level kitni tezi se even out hota hai. Topic ko kyun chahiye: derivation mein storage term $ρ c u$ aur Fourier's $k$ milke $u_{xx}$ ke aage single constant $α$ dete hain.

10. $\int$ , $Δ x$ , aur $A$ — control-slab tools

Picture: rod ka ek patla coin-shaped slice; hum energy ko uske left face mein andar aur right face se bahar track karte hain. Topic ko kyun chahiye: derivation "slab mein stored energy" ko "uske faces par flow ho rahi heat" se balance karta hai, phir $Δ x \to 0$ shrink karta hai. Dekho Conservation Laws and Continuity Equation.

11. "Parabolic" — classification word

Picture: Wave Equation (hyperbolic) 1D (cheezein travel karti hain aur bounce karti hain) aur Laplace Equation (steady-state heat) (time mein kuch nahi badalta) se contrast karo. Parabolic beech mein baitha hai: yeh steady state ki taraf relax karta hai.

Prerequisite map

Equipment checklist

Reveal karne se pehle har ek ka plain-words mein matlab do.

x

kya measure karta hai aur kis unit mein?

Rod ke along left end se position, metres mein.

t

kya measure karta hai?

Time — kaun sa moment (movie frame) hum dekh rahe hain, seconds mein.

u (x, t)

kya hai?

Position

x

aur time

t

par temperature (K); do variables ka function —

(x, t)

plane ke upar ek surface.

Derivative geometrically kya hoti hai?

Graph ka slope: rise over run jab run zero ho jaye — rate of change.

Plain

d

ki jagah curly

\partial

kyun?

Kyunki

u

ke do inputs hain;

\partial

ka matlab hai "doosre input ko fixed rakhte hue differentiate karo."

u_{t}

physically kya batata hai?

Ek fixed point par time mein temperature kitni tezi se change hoti hai;

u_{t} > 0

warm hota hai,

u_{t} < 0

cool hota hai.

u_{x}

(gradient) kya represent karta hai?

Temperature curve ka spatial slope — right ki taraf step karne par yeh kitni steeply hot ya thandi hoti hai.

u_{xx}

(curvature) kya represent karta hai, aur uska sign kya karta hai?

Temperature curve ka bend;

u_{xx} > 0

(dip) point ko warm karta hai,

u_{xx} < 0

(bump) use cool karta hai,

u_{xx} = 0

(straight) matlab koi change nahi.

Heat flux

q

kya hai aur

q = - k u_{x}

mein minus kyun hai?

Per area per time energy crossing; minus heat ko gradient ke neeche, hot se cold ki taraf flow karata hai.

ρ c

aur

k

mein fark karo aur

α

define karo.

ρ c

per volume per degree stored energy hai;

k

yeh hai ki heat kitni achhi tarah se travel karti hai;

α = k / (ρ c)

hai ki temperature patterns kitni tezi se smooth out hote hain (m²/s).

Derivation mein

Δ x

A

, aur

\int

kya contribute karte hain?

Ek patli slab width, uska cross-section area, aur slab par ek sum — stored energy ko flow ke against balance karne ka bookkeeping.

Equation "parabolic" kyun hai?

Iska discriminant

B^{2} - A C = 0

hai; physically yeh pure diffusion/smoothing hai.

Connections

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