Neumann and Dirichlet boundary conditions — Hinglish notes

4.7.16 · Maths › Partial Differential Equations

YE HAIN KYA

BOUNDARY CONDITIONS KYUN CHAHIYE (derivation from scratch)

Worked example Sirf PDE kaafi nahi kyun — ek chhoti si derivation

Sabse simple steady-state 1D Laplace equation lo $[0, L]$ par: $u^{''} (x) = 0.$ Step 1 — ek baar integrate karo. $u^{'} (x) = A$ . Kyun? Ek constant ki derivative zero hoti hai, isliye $u^{''} = 0$ force karta hai ki $u^{'}$ constant ho. Step 2 — dobara integrate karo. $u (x) = A x + B$ . Kyun? Constant $A$ ka antiderivative $A x$ hota hai, plus constant $B$ . Step 3 — freedom count karo. Do unknown constants $A, B$ hain. PDE ne humein ek family di, single answer nahi. Hume exactly do extra facts chahiye — boundary conditions — $A$ aur $B$ fix karne ke liye.

Dirichlet $u (0) = α, u (L) = β$ : deta hai $B = α$ aur $A = (β - α) / L$ . Unique solution $u = α + \frac{β - α}{L} x$ .
Neumann $u^{'} (0) = p, u^{'} (L) = q$ : lekin $u^{'} = A$ constant hai, isliye consistency ke liye $p = q$ chahiye, aur phir $B$ free rehta hai — solution unique hai sirf ek additive constant tak.

Figure — Neumann and Dirichlet boundary conditions

EIGENFUNCTIONS KAISE BADLE JAATE HAIN (separation of variables)

Jab aap heat equation $u_{t} = k u_{xx}$ ko $[0, L]$ par separation se solve karte ho, toh $X^{''} + λ X = 0$ milta hai aur boundary condition allowed modes decide karta hai.

Aur worked examples

Common mistake Common errors ko steel-man karo

Galti 1: "Neumann ka matlab hai ki boundary par $u = 0$ ." Kyun sahi lagta hai: dono "boundary = 0" type statements hain. Fix: Neumann slope $\partial u / \partial n$ fix karta hai, value nahi. Insulated ka matlab hai $\partial u / \partial n = 0$ , flux vanish hoti hai, temperature nahi.

Galti 2: Neumann ke liye $n = 0$ mode bhool jaana. Kyun sahi lagta hai: Dirichlet ke liye aap sahi taur par $n = 0$ drop karte ho, aur wahi habit copy ho jaati hai. Fix: $X_{0} = cos 0 = 1$ satisfy karta hai $X^{'} (0) = X^{'} (L) = 0$ — yeh ek valid nontrivial mode hai. Isse drop karne par average-temperature ki physics kho jaati hai.

Galti 3: 2D/3D mein $\partial u / \partial n$ ki jagah $\partial u / \partial x$ use karna. Kyun sahi lagta hai: 1D mein outward normal sirf $\pm \overset{x}{^}$ hai, isliye dono agree karte hain (sign tak). Fix: Higher dimensions mein aapko outward unit normal ke saath dot karna padta hai; $x = 0$ par outward normal $- \overset{x}{^}$ direction mein point karta hai, isliye $\partial u / \partial n = - u_{x} (0)$ .

Galti 4: Pure Neumann ke liye unique solution expect karna. Kyun sahi lagta hai: Dirichlet problems unique hote hain, isliye aap assume karte ho ki sab hote hain. Fix: Pure-Neumann solutions unique hote hain sirf ek additive constant tak, aur exist karne ke liye compatibility condition $\oint_{\partial Ω} h d S = 0$ chahiye (steady state ke liye net flux balance hona chahiye).

Recall Feynman: ek 12-saal ke bacche ko explain karo

Ek metal rod imagine karo. Dirichlet rule aise hai jaise kehna "main dono ends pakad raha hoon aur force kar raha hoon ki exactly itni garmi ho." Neumann rule aise hai jaise kehna "main ends ko mote kambal mein wrap kar raha hoon taaki koi heat ends se bahar na nikle (slope = 0), ya exactly itni heat bahar leak hone do." Equation heat ko batata hai ki andar kaise move karo; boundary rule batata hai ki edges par kya allowed hai. Edge rule ke bina zillions of possible answers hain — edge rule asli wala chunti hai.

Active recall

Dirichlet condition boundary par kaun si quantity fix karta hai?

u

ki value khud:

u = g

\partial Ω

Neumann condition boundary par kaun si quantity fix karta hai?

Normal derivative / flux:

\partial u / \partial n = h

\partial Ω

Neumann normal (tangential nahi) derivative kyun use karta hai?

\nabla u

ka sirf woh component jo boundary ke perpendicular hai woh flow represent karta hai across it; tangential flow andar hi rehti hai.

Homogeneous Neumann (

\partial u / \partial n = 0

) ka physical matlab kya hai?

Insulated / no-flux boundary — kuch bhi edge cross nahi karta.

[0, L]

par Dirichlet kaun se eigenfunctions deta hai?

X_{n} = sin (nπ x / L)

n = 1, 2, \dots

(sine series), jahan

λ_{n} = (nπ / L)^{2}

[0, L]

par Neumann kaun se eigenfunctions deta hai?

X_{n} = cos (nπ x / L)

n = 0, 1, 2, \dots

(cosine series, constant

n = 0

include hai).

Neumann ke liye

n = 0

mode kyun rakha jaata hai lekin Dirichlet ke liye drop kiya jaata hai?

cos 0 = 1

satisfy karta hai

X^{'} = 0

dono ends par (valid hai), lekin

sin 0 = 0

trivial zero solution deta hai.

Initial

f (x)

wale insulated rod ka long-run temperature kya hoga?

Average

\frac{1}{L} \int_{0}^{L} f d x

— sirf non-decaying

λ_{0} = 0

constant mode survive karta hai.

Pure-Neumann solution non-unique kyun hota hai?

Sirf derivatives constrained hain; koi bhi constant add karne se saari normal derivatives unchanged rehti hain.

Compatibility condition for a steady pure-Neumann problem?

Net flux vanish hona chahiye:

\oint_{\partial Ω} h d S = 0

(aur Laplace's equation ke liye

\int_{Ω} f = \oint h

Connections

Separation of Variables — boundary conditions eigenfunctions select karte hain.
Heat Equation — Dirichlet → $0$ tak cooling, Neumann → average par settle.
Laplace Equation — Neumann ko compatibility (zero-net-flux) condition chahiye.
Sturm-Liouville Theory — dono conditions operator ko self-adjoint banate hain, real $λ_{n}$ aur orthogonal modes milte hain.
Fourier Series — Dirichlet ↔ sine series, Neumann ↔ cosine series.
Robin Boundary Conditions — mixed $α u + β \partial u / \partial n = g$ generalisation.