Laplace's equation (elliptic) — physical meaning (steady-state)

4.7.13 · Maths › Partial Differential Equations

Laplace's equation HAI kya?

Yeh prototype elliptic PDE hai. "Elliptic" naam aata hai $A u_{xx} + B u_{x y} + C u_{y y} + \dots = 0$ ko discriminant $B^{2} - 4 A C$ se classify karne se. Laplace ke liye, $A = C = 1, B = 0$ , toh $B^{2} - 4 A C = - 4 < 0$ → elliptic (jaise ellipse ki equation $x^{2} + y^{2} = 1$ ).

"Steady-state" se $\nabla^{2} u = 0$ kyun milta hai?

Sabse saaf derivation heat (diffusion) equation se shuru hoti hai aur poochti hai "jab time matter karna band kar de toh kya hoga?"

Formula Heat equation se derivation — scratch se

Step 1. Heat, garm se thande ki taraf flow karti hai, temperature gradient ke proportional (Fourier's law): $q = - k \nabla u$ Yeh step kyun? Heat flux $q$ temperature ke neeche ki taraf point karta hai; $- \nabla u$ thande regions ki taraf point karta hai; $k > 0$ conductivity hai.

Step 2. Energy conservation: ek tiny volume mein stored heat ke change ki rate, bahar nikalne wali net heat ka minus hoti hai: $ρ c \frac{\partial u}{\partial t} = - \nabla \cdot q$ Yeh step kyun? $\nabla \cdot q$ net outflow per unit volume hai (miniature divergence theorem). Agar zyada bahar jaaye toh temperature girta hai.

Step 3. Step 1 ko Step 2 mein substitute karo: $ρ c \frac{\partial u}{\partial t} = \nabla \cdot (k \nabla u) = k \nabla^{2} u$ heat equation milti hai $u_{t} = α \nabla^{2} u$ jahan $α = k / (ρ c)$ . Yeh step kyun? Yeh pura time-dependent law hai; Laplace iska long-time limit hai.

Step 4. Steady state ka matlab hai kuch time ke saath nahi badalta: $\frac{\partial u}{\partial t} = 0$ . Tab $0 = k \nabla^{2} u \Rightarrow \nabla^{2} u = 0$ Yeh step kyun? Jab system settle ho jaata hai, har jagah heat-in = heat-out; koi bhi point energy gain ya lose nahi kar raha.

Ise kaise picture karo: mean-value property

Yeh sach kyun hai (sketch). $\overset{u}{ˉ} (r) =$ radius $r$ ke circle par $u$ ka average define karo. Differentiate karo aur divergence theorem use karo: $\frac{d u ˉ}{d r} = \frac{1}{2 π r} \oint \frac{\partial u}{\partial n} d s = \frac{1}{2 π r} \iint_{d i s k} \nabla^{2} u d A = 0.$ Toh $\overset{u}{ˉ} (r)$ $r$ mein constant hai; $r \to 0$ shrink karne par $\overset{u}{ˉ} = u (x_{0})$ milta hai.

Figure — Laplace's equation (elliptic) — physical meaning (steady-state)

Boundary conditions — aur yeh well-posed kyun banate hain

Elliptic problems ko poori closed boundary par conditions chahiye (time mein initial conditions nahi, kyunki time hai hi nahi!).

Dirichlet: boundary par $u$ specified (edge temperature fix karo).
Neumann: $\partial u / \partial n$ specified (edge ke through heat flux fix karo; insulated ⇒ $= 0$ ).

Worked examples

Common mistakes (Steel-manned)

Recall Feynman: 12-saal ke bachche ko explain karo

Ek metal plate imagine karo. Tum edges ko fixed temperatures par pakdte ho — ek edge garam, ek thandi — aur bahut, bahut lambe time tak wait karte ho. Aakhir mein heat idhar-udhar hona band ho jaati hai; har spot ek aisi temperature par settle ho jaata hai jo bas apne aas-paas ki temperatures ka average hai. Woh settled picture hi hai jo Laplace's equation describe karta hai. Kyunki har point apne neighbours ka average hai, tum kabhi beech mein ek akela super-garam dot nahi rakh sakte jo thande surroundings se ghira ho — woh instantly apni heat share karna shuru kar deta aur "settled" rehna band kar deta. Toh sabse garam aur thande jagah hamesha un edges par hote hain jo tum pakde ho.

Active-recall flashcards

Symbols mein Laplace's equation

\nabla^{2} u = 0

(the Laplacian of

u

is zero)

\nabla^{2} u = 0

satisfy karne wale function ko kya kehte hain?

harmonic

Laplace's equation kaun si physical situation describe karta hai?

Ek steady-state (time-independent) field, jaise equilibrium temperature, electrostatic potential, ya incompressible irrotational flow

Steady-state heat flow se

\nabla^{2} u = 0

kyun milta hai?

u_{t} = 0

set karne par heat equation

u_{t} = α \nabla^{2} u

mein

\nabla^{2} u = 0

bachta hai

Laplace's equation ka PDE-classification type kya hai?

Elliptic (

B^{2} - 4 A C < 0

jisme

A = C = 1, B = 0

)

Mean-value property state karo

Ek harmonic function ki kisi point par value us point ke surrounding circle/sphere par uski values ke average ke barabar hoti hai

Maximum principle kya kehta hai?

Ek harmonic function ke koi interior maxima ya minima nahi hote; extremes sirf boundary par aate hain

Ek elliptic PDE ko kis tarah ki conditions chahiye?

Poori closed boundary par boundary conditions (Dirichlet ya Neumann), initial conditions nahi

Dirichlet vs Neumann condition

Dirichlet, boundary par

u

fix karta hai; Neumann, normal derivative

\partial u / \partial n

(flux) fix karta hai

Kya

u = x^{2} - y^{2}

harmonic hai?

Haan:

u_{xx} + u_{y y} = 2 - 2 = 0

u^{''} (x) = 0

u (0) = 10, u (L) = 30

ke saath solve karo

u (x) = 10 + \frac{20}{L} x

(ek straight line)

Kisi point par

\nabla^{2} u > 0

ka sign interpretation

Point apne neighbours ke average se neeche hai; heat andar aati hai, yeh warm hone ki tendency hai

Connections

Heat equation (parabolic) — diffusion — Laplace iska $t \to \infty$ steady limit hai.
Poisson's equation — $\nabla^{2} u = f$ , source term ke saath Laplace.
Classification of second-order PDEs (elliptic-parabolic-hyperbolic)
Boundary value problems and well-posedness
Separation of variables for Laplace's equation
Harmonic functions and complex analysis — analytic functions ke real/imaginary parts.
Maximum principle
Electrostatics — potential and Gauss's law — charge-free regions mein $\nabla^{2} ϕ = 0$ .