Question bank — Classification — elliptic, parabolic, hyperbolic (discriminant test)

False — sirf principal (second-order) part $A{u}_{xx}+B{u}_{xy}+C{u}_{yy}$ hi $\Delta$ mein enter hoti hai; lower-order terms $D{u}_x,E{u}_y$ solutions ko shift aur damp karti hain lekin kabhi nahi badlti ki real characteristics exist karte hain ya nahi.

False — agar $A,B,C$, $x,y$ par depend karte hain (jaise Tricomi $y{u}_{xx}+u_{yy}=0$), to $\Delta$ plane mein sign change karta hai, isliye type poori equation ki nahi balki point ki property hai.

False — yeh parabolic hai; koi $u_{tt}$ term nahi hai isliye $C=0$, jo deta hai $\Delta =0-4(-k)(0)=0$. Iska type exactly ek real characteristic hone se aata hai.

True — $\Delta <0$ hone se dono characteristic slopes $m=\frac{-B\pm \sqrt{\Delta }}{2A}$ complex ho jaate hain (negative number ka square root), isliye koi real curves nahi hain jinke saath information travel kare; instead har boundary point har interior point ko influence karta hai.

True — top-derivative coefficients $u$ par hi depend kar sakte hain, isliye ek baar solution plug karne par $\Delta$ ek region mein elliptic aur doosre mein hyperbolic nikl sakta hai, us $u$ par depend karte hue. Dekho Linear vs Quasilinear PDEs.

False — woh single sign flip $\Delta$ ko positive se negative kar deta hai, finite-speed wave propagation (hyperbolic) ko smooth all-at-once equilibrium (elliptic) mein badal deta hai. Ek choti algebraic change total physical change hai.

False — repeated root sirf ek real characteristic direction deta hai; parabolic problems data ko us single family of lines ke saath smear karte hain, jaise heat diffusion mein.

False — $A,B,C$ ko $-1$ se scale karne par $(-B{)}^2-4(-A)(-C)=B^2-4AC$ milta hai, same $\Delta$; discriminant ka sign equation ko scale karne ke invariant hai.

True — koi characteristics nahi hain jo information andar le jaayein, isliye solution pin down karne ka ek hi tarika hai ki values (ya normal derivatives) sab taraf prescribe karo; dekho Well-posedness and Boundary Conditions.

Conclusion sahi hai. Deeper reason ki koi fark nahi padta tum kis second derivative ko $A$ ya $C$ kaho yeh hai ki $\Delta =B^2-4AC$, $A$ aur $C$ mein symmetric hai: inhe swap karne se $B^2-4AC$ unchanged rehta hai, isliye koi bhi labelling $\Delta =4c^2>0$ deti hai.

Mixed conventions ki galti. $\Delta =B^2-AC$ form assume karta hai ki cross term $2B{u}_{xy}$ likha gaya hai, isliye wahan $B=\frac{3}{2}$ hai. Ek convention chuno: $B{u}_{xy}$ ke saath $B^2-4AC$ use karo; sign dono mein same aata hai.

$u_x^2$ ek squared first derivative hai — yeh na second-order hai na linear. Yeh term $A,B,C$ mein kuch contribute nahi karta (jo second-order coefficients hain), aur $u_x^2$ ki presence poori equation ko nonlinear bana deti hai.

Koi $u_{tt}$ nahi hai, isliye $C=0$ hai, $1$ nahi; ek $C$ term invent karna galti hai. Sahi tarike se $\Delta =0-4(-k)(0)=0$, parabolic — coefficient assign karne se pehle har term ka order check karo.

Tum actually sahi ho — $\Delta =16>0$ hai hyperbolic. Trap yeh hai ki sahi computation par shak karna; Tricomi genuinely $y<0$ ke liye hyperbolic, $y>0$ ke liye elliptic, aur $y=0$ par parabolic hai.

Discriminant test second-order PDEs ke liye bana hai. Ek pure first-order transport equation ka koi principal second-order part nahi hota; ise Method of Characteristics se directly classify karte hain, $\Delta$ se nahi.

Character — diffuse, propagate, ya equilibrate — solution ke fastest-varying part se set hota hai; lower-order terms shift ya damp kar sakte hain lekin sharp lines (characteristics) ko create ya destroy nahi kar sakte jinke saath information travel karta hai.

Characteristic slopes $A{m}^2+Bm+C=0$ solve karte hain, slope $m$ mein ek quadratic; positive discriminant do distinct real roots deta hai, isliye do real slope families jinke saath second derivatives jump kar sakti hain.

Hyperbolic (wave) ke do real characteristics $x\pm ct=\text{const}$ hain jo bound karte hain ki disturbance kahan tak pahunch sakti hai; parabolic (heat) mein aise bounding lines nahi hain, isliye ek point par change instantly sab jagah feel hoti hai, chahe kitni bhi faintly. Dekho Wave Equation — d'Alembert solution aur Heat Equation — separation of variables.

Kyunki dono ek identical quantity $B^2-4AC$ se classify hote hain: PDE ke principal part ki algebra wahi quadratic form hai jo conic ki hai, isliye ellipse/parabola/hyperbola → elliptic/parabolic/hyperbolic map hote hain. Dekho Conic Sections.

Koi real characteristics nahi hain isliye solution "march" karne ki koi direction nahi hai; har interior value poore boundary par depend karti hai, isliye boundary data complete aur closed hona chahiye — ek Laplace/harmonic property, dekho Laplace Equation — harmonic functions.

Ek open surface par Cauchy data march karna elliptic equations ke liye ill-posed hai — data mein tiny high-frequency errors exponentially blow up ho jaati hain. Elliptic equilibrium ko closed boundary chahiye, forward evolution nahi.

Yahan $\varphi (x,y)=\text{const}$ candidate characteristic curve hai, aur hum naya coordinate $\xi =\varphi$ adopt karte hain; iske partial derivatives ${\varphi }_x,{\varphi }_y$ wo rates hain jis par $\varphi$, $x$ aur $y$ mein change hota hai. Chain rule second derivatives ko ek coefficient $Q=A{\varphi }_x^2+B{\varphi }_x{\varphi }_y+C{\varphi }_y^2$ multiplying $u_{\xi \xi }$ mein badal deta hai; jahan $Q=0$, woh second derivative unconstrained aur jump karne ke liye free hai — exactly yahi characteristic define karta hai.

$A=0$ ke saath characteristic equation $A{m}^2+Bm+C=0$ linear $Bm+C=0$ mein reduce ho jaati hai, ek finite slope $m=-C/B$ deti hai; "missing" doosra root ek vertical characteristic (infinite slope $dy/dx$, yaani $dx/dy=0$) correspond karta hai. Dono count karo to do real characteristics hain aur $\Delta =B^2-4AC=B^2>0$, isliye hyperbolic — standard quadratic ko sirf degenerate (ek root at infinity) case ki tarah padhna padta hai.

Characteristic equation $A{m}^2+Bm+C=0$ collapse hokar $C=0$ ban jaati hai, jiske koi finite root nahi — instead dono roots infinite slope par hain, matlab characteristics vertical lines $x=\text{const}$ hain (ek repeated family). Meanwhile $\Delta =B^2-4AC=0-0=0$, isliye parabolic: ek real (repeated) characteristic direction, $u_{yy}=0$ ke consistent jo kehta hai $u$, $y$ mein linear hai har vertical line ke saath.

$y=0$ par, $\Delta =-4y=0$, isliye exactly us line par parabolic hai — elliptic region ($y>0$) aur hyperbolic region ($y<0$) ko separate karne wali degenerate frontier.

Koi principal (second-order) part nahi hai, isliye discriminant test apply nahi hota; yeh actually ek first-order PDE hai aur Method of Characteristics se handle karna padega.

Type pointwise defined hai: elliptic jahan $\Delta <0$, hyperbolic jahan $\Delta >0$, aur exactly zero-set curve par parabolic. Aise "mixed-type" equations (transonic flow) genuinely us curve ke across character change karte hain.

Nahi — swapping $A$ aur $C$ exchange karta hai, aur $\Delta =B^2-4AC=B^2-4CA$ unchanged rehta hai. Classification coordinate relabelling ke invariant hai.

Sirf us solution-dependent curve ke saath us instant par; elsewhere yeh elliptic ya hyperbolic ho sakta hai. Quasilinear equations ke liye parabolic frontier ki jagah khud solution ke saath chalti hai, isliye type ek local, solution-dependent statement hai.