Classification — elliptic, parabolic, hyperbolic (discriminant test)

4.7.1 · Maths › Partial Differential Equations

80/20 core: Do variables mein ek linear 2nd-order PDE ko discriminant $B^{2} - 4 A C$ ke sign se classify karte hain. Negative → elliptic, zero → parabolic, positive → hyperbolic. Yeh ek sign hi batata hai physics kya hai, kaun se boundary conditions chahiye, aur solution method kya hoga.

The setup

WHERE discriminant aata hai (derivation, ek dump nahi)

Hum jaanna chahte hain: kin curves ke saath solution mein "kink" (2nd derivatives mein discontinuity) ho sakti hai? Un curves ko characteristics kehte hain. Inki existence hi classification ka gehra matlab hai.

Step 1 — Characteristic curves $y = y (x)$ dhundo. Kyun? Kyunki ek PDE un special directions ke saath ek ODE ki tarah behave karti hai jahan second derivatives data se poori tarah determine nahi hote. Un directions ko dhundhna type batata hai.

Maano ek curve $ϕ (x, y) = const$ ek characteristic hai. $ξ = ϕ$ ko naye coordinate ke roop mein substitute karo. Principal part transform hota hai; $u_{ξ ξ}$ ka coefficient ban jaata hai $Q (ϕ_{x}, ϕ_{y}) = A ϕ_{x}^{2} + B ϕ_{x} ϕ_{y} + C ϕ_{y}^{2} .$ Yeh expression kyun? Kyunki variables change karne par chain rule $u_{xx} \to ϕ_{x}^{2} u_{ξ ξ} + \dots$ , $u_{x y} \to ϕ_{x} ϕ_{y} u_{ξ ξ} + \dots$ , $u_{y y} \to ϕ_{y}^{2} u_{ξ ξ} + \dots$ bhejta hai. $u_{ξ ξ}$ collect karne par exactly $Q$ milta hai.

Step 2 — Characteristic wahan hoti hai jahan yeh coefficient vanish hota hai: $Q = 0$ . $ϕ = const$ ke saath $d ϕ = ϕ_{x} d x + ϕ_{y} d y = 0$ , toh slope hai $\frac{d y}{d x} = - \frac{ϕ _{x}}{ϕ _{y}} .$ $Q = 0$ ko $ϕ_{y}^{2}$ se divide karo aur $m = ϕ_{x} / ϕ_{y}$ lo: $A m^{2} + B m + C = 0 \Rightarrow m = \frac{- B \pm B ^{2} - 4 A C}{2 A} .$

Yeh names conic sections $A x^{2} + B x y + C y^{2} = const$ se echo karte hain, jo same $B^{2} - 4 A C$ se classify hoti hain.

Figure — Classification — elliptic, parabolic, hyperbolic (discriminant test)

Worked examples

Boundary/initial conditions jo har type ko chahiye (practical payoff)

Type	Typical PDE	Domain	Correct data
Elliptic	Laplace	closed region	poori boundary par values (Dirichlet/Neumann)
Parabolic	Heat	$t$ mein open	initial $u (x, 0)$ + side conditions, $t$ mein aage badhte hain
Hyperbolic	Wave	$t$ mein open	initial $u$ aur $u_{t}$ , $t$ mein aage badhte hain

Common mistakes (steel-manned)

Active recall

Recall Pehle cover karke jawab do

$Δ$ mein kaun se teen quantities jaate hain? (coeffs of $u_{xx}, u_{x y}, u_{y y}$ )
Sign rule? (− elliptic, 0 parabolic, + hyperbolic)
Har ek ke liye kitni real characteristics? (0, 1, 2)
Har ek ke liye kaun si physical equation? (Laplace, Heat, Wave)
Linear vs quasilinear? (linear: coeffs sirf $x, y$ par depend karte hain; quasilinear: top-derivative coeffs $u$ aur lower derivatives par depend kar sakte hain)

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

Socho teen alag duniyaon mein ek kankar giraya. Wave world mein, ripples straight lines ke saath ek fixed speed se nikalti hain — tum exactly dekh sakte ho ripple front kahan hai. Yeh hai hyperbolic (do clear lines). Heat world mein, warm dye ki ek bund dheere dheere blur ho jaati hai, koi sharp edge nahi, bas time ke saath smearing hoti rehti hai. Yeh hai parabolic (ek smearing direction). Calm-pond world mein, paani pehle se bilkul still hai aur surface ka har hissa dheerey se har doosre hisse ko balance karta hai — koi fronts nahi, koi smearing nahi, bas smooth equilibrium. Yeh hai elliptic. Chhota sa number $B^{2} - 4 A C$ ek magic detector hai jo batata hai aapki equation kaun si duniya mein rehti hai: negative = calm pond, zero = blurring dye, positive = shooting ripples.

Connections

Wave Equation — d'Alembert solution (hyperbolic, characteristics $x \pm c t$ use karta hai)
Heat Equation — separation of variables (parabolic)
Laplace Equation — harmonic functions (elliptic, maximum principle)
Method of Characteristics (seedha yahan derive ki gayi slopes par built)
Linear vs Quasilinear PDEs (coefficient dependence aur yeh kya allow karta hai)
Conic Sections (same $B^{2} - 4 A C$ test!)
Well-posedness and Boundary Conditions

2nd-order PDE classify karne ke liye kaun sa discriminant use hota hai?

Δ = B^{2} - 4 A C

u_{xx}, u_{x y}, u_{y y}

ke coefficients use karke.

Elliptic, parabolic, hyperbolic ke liye

Δ

ka sign?

Δ < 0

elliptic,

Δ = 0

parabolic,

Δ > 0

hyperbolic.

Kaun sa standard PDE hyperbolic hai aur kyun?

Wave equation

u_{tt} = c^{2} u_{xx}

;

Δ = 4 c^{2} > 0

, do real characteristics

x \pm c t

Kaun sa standard PDE parabolic hai aur kyun?

Heat equation

u_{t} = k u_{xx}

;

C = 0

(koi

u_{tt}

nahi), toh

Δ = 0

Kaun sa standard PDE elliptic hai aur kyun?

Laplace

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

;

Δ = - 4 < 0

, koi real characteristics nahi.

Har type ke liye real characteristics ki number?

Hyperbolic 2, parabolic 1 (repeated), elliptic 0 (complex).

Sirf second-order terms PDE ko kyun classify karte hain?

Yeh characteristics/propagation control karte hain; lower-order terms sirf shift aur damp karte hain, type nahi badlate.

Characteristic slope formula kya hai?

d y / d x = (B \mp B^{2} - 4 A C) / (2 A)

Tricomi

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

ko classify karo.

Δ = - 4 y

: elliptic (

y > 0

), parabolic (

y = 0

), hyperbolic (

y < 0

) — mixed type.

Heat equation mein

C = 0

kyun hota hai?

u_{t}

first order hai; koi

u_{tt}

term nahi hai, toh

u_{y y}

ka coefficient zero hai.

Elliptic PDE ko kaun sa data chahiye?

Poori closed boundary par values (Dirichlet/Neumann).

Hyperbolic PDE ko kaun sa data chahiye?

Initial

u

AUR

u_{t}

, phir time mein aage badhte hain.

LINEAR 2nd-order PDE ki definition?

Saare coefficients (har derivative ke, highest ke bhi) sirf independent variables

x, y

par depend karte hain —

u

ya uske derivatives par nahi.

QUASILINEAR PDE ki definition?

Highest-order derivatives ke coefficients

u

aur uske lower-order derivatives par depend kar sakte hain, lekin highest derivatives phir bhi linearly appear karte hain.