4.7.8
Intuition Ek-sentence mein pictureHeat, hot se cold ki taraf flow karti hai, aur ek chhoti si slab kitni tezi se heat hoti hai yeh depend karta hai ki uska temperature profile kitna curved hai. Ek bump (concave-down) cool hota hai; ek dip (concave-up) warm hota hai. Yeh "curvature drives change in time" wali baat exactly u t = α u xx
Ek patli insulated rod jo x
u ( x , t ) x t temperature (units K),ρ c k
Hum u ( x , t ) parabolic hai kyunki A u xx + 2 B u x t + C u tt B 2 − A C = 0 C = 0 B = 0
x x + Δ x A
Worked example Step-by-step energy balanceStep 1 — Slab mein stored energy.
E ( t ) = ∫ x x + Δ x ρ c u ( s , t ) A d s . Kyun? Energy density ρ c u A d s
Step 2 — Stored energy ke change ki rate.
d t d E = ∫ x x + Δ x ρ c u t A d s . Kyun? Integral ke andar differentiate karo (limits space mein fixed hain).
Step 3 — Net heat jo ANDAR aa rahi hai. Flux left face se enter karta hai, right face se leave karta hai:
net in = A q ( x , t ) − A q ( x + Δ x , t ) . Kyun? q ( x ) + x + A q ( x ) + A q ( x + Δ x )
Step 4 — Equate karo (conservation).
∫ x x + Δ x ρ c u t A d s = A ( q ( x , t ) − q ( x + Δ x , t ) ) .
Step 5 — A Δ x Δ x → 0
Left side mean value → ρ c u t Δ x q ( x ) − q ( x + Δ x ) Δ x → 0 − ∂ x ∂ q . Kyun? Yeh q ρ c u t = − ∂ x ∂ q .
Step 6 — Fourier's law q = − k u x ρ c u t = − ∂ x ∂ ( − k u x ) = k u xx ( k constant ) .
Step 7 — Diffusivity define karo α = ρ c k u t = α u xx
Intuition "Parabolic" / infinite speed kyunHeat eq ko 0 ⋅ u tt + 0 + ( − α ) u xx + u t = 0 A u tt + 2 B u x t + C u xx B 2 − A C = 0 parabolic . Physically iska matlab hai smoothing/diffusion ke saath infinite signal speed (ek disturbance ek dam se har jagah feel hoti hai, wave equation ke unlike).
Worked example Kya yeh point heat hoga ya cool?Ek rod ka ek instant par u ( x , t ) = 20 + 5 sin ( π x ) α = 2 x = 1/2 u t Step 1: u xx = ∂ x 2 [ 20 + 5 sin π x ] = − 5 π 2 sin π x Kyun? Do baar differentiate karo; constants vanish ho jaate hain.
Step 2: x = 1/2 sin ( π /2 ) = 1 u xx = − 5 π 2 Step 3: u t = α u xx = 2 ( − 5 π 2 ) = − 10 π 2 ≈ − 98.7 Interpretation: bump ki peak tezi se cool hoti hai — intuition ke consistent hai.
Worked example Ek known solution check karoDikhao ki u = e − α π 2 t sin ( π x ) u t = α u xx Step 1: u t = − α π 2 e − α π 2 t sin π x Kyun? Exponential par chain rule.
Step 2: u x = π e − α π 2 t cos π x u xx = − π 2 e − α π 2 t sin π x Step 3: α u xx = − α π 2 e − α π 2 t sin π x = u t .
Recall Check karne se pehle predict karo
Agar α Forecast…
Ek linear initial profile u = a + b x u t Forecast…
Verify: (1) Jaldi — u t α u xx = 0 ⇒ u t = 0
Common mistake "Heat equation mein waves jaisau tt Kyun sahi lagta hai: wave equation u tt = c 2 u xx Fix: energy storage ρ c u t ek time derivative. Yeh time-reversal symmetry tod deta hai (heat spread hoti hai, kabhi un-spread nahi hoti).
Common mistake Fourier's law mein minus sign bhool jana.Kyun sahi lagta hai: "flux proportional to gradient" clean lagta hai. Fix: minus ke bina, heat hotter regions ki taraf flow karegi, jo thermodynamics violate karta hai. Minus final equation ko + α u xx u t = − α u xx
α = k Kyun sahi lagta hai: k Fix: PDE ka time scale diffusivity α = k / ( ρ c ) k ρ c α
Recall Ek 12-saal ke bachche ko samjhaoSocho ek metal ruler, beech mein hot, ends par cold. Heat ek bheed ki tarah hai jo hamesha ek bheed waali (hot) jagah se khaali (cold) jagah ki taraf push karti hai. Ek tiny spot ka temperature kitni tezi se badlega yeh depend karta hai ki woh temperature map par ek "hill" hai ya "valley": hills apne neeche wale neighbours ko heat lose karti hain aur cool hoti hain; valleys heat gain karti hain aur warm hoti hain. Equation u t = α u xx
"Time-change equals C urvature × D iffusivity" → u t = α u xx "flux flows down-hill" → q = − k u x
Fourier's law of heat conduction (1D) batao aur minus sign explain karo. q = − k u x
Kaun se do physical principles heat equation derive karte hain? Fourier's law (flux ∝ −gradient) aur control slab par conservation of energy.
1D heat equation likho aur α u t = α u xx α = k / ( ρ c )
Heat equation ko parabolic kyun kaha jaata hai? Iska second-order classification discriminant B 2 − A C = 0
Derivation mein ρ c u Thermal energy density (J/m³): density × specific heat × temperature.
Sirf ek time derivative kyun hai (u tt Energy storage aur Fourier flux dono time mein first order hain, toh balance sirf ek u t
Kisi point par u xx > 0 Profile concave up hai (ek dip); u t > 0
1D mein steady-state temperature profile ki shape (source ke bina)? Linear, kyunki u xx = 0
Kya α Speed up hoti hai; u t α
Internal source f ρ c u t = k u xx + f
Stored energy E over slab
dE/dt = integral rho c u_t
Net heat in = A q x minus q x+dx
Heat equation u_t = alpha u_xx
