Sturm-Liouville theory — eigenvalue problems, orthogonality of eigenfunctions

4.7.17 · Maths › Partial Differential Equations

1. Sturm–Liouville problem kya hota hai?

HAR piece ka MATLAB:

$p (x)$ — "stiffness"/diffusion coefficient.
$q (x)$ — ek potential/source-jaisa term.
$w (x)$ — weight; yeh inner product define karta hai (orthogonality ke liye CRUCIAL).
$λ$ — eigenvalue (e.g. $λ_{n} = (nπ / L)^{2}$ heat equation mein).

Kyun KISI BHI linear 2nd-order ODE ko SL form mein daala ja sakta hai (derivation)

General form se shuru karo: $a_{2} (x) y^{''} + a_{1} (x) y^{'} + a_{0} (x) y + λ y = 0.$ Hum chahte hain $(p y^{'})^{'} = p y^{''} + p^{'} y^{'}$ . $a_{2}$ se divide karo aur ek integrating factor $μ (x)$ se multiply karo: $μ y^{''} + μ \frac{a _{1}}{a _{2}} y^{'} + \dots$ Yeh $(μ y^{'})^{'} = μ y^{''} + μ^{'} y^{'}$ ke barabar ho iske liye humein chahiye $μ^{'} = μ \frac{a _{1}}{a _{2}}$ , toh $μ (x) = exp (\int \frac{a _{1}}{a _{2}} d x)$ Phir $p = μ$ , $q = μ a_{0} / a_{2}$ , $w = μ / a_{2}$ . Yeh step kyun? Integrating factor exactly wahi cheez hai jo do alag terms ko ek perfect derivative mein convert karta hai — same trick jaise first-order linear ODEs mein hota hai.

2. Boundary conditions jo isse "regular" banate hain

3. Main Theorem (80/20 core)

3.1 Orthogonality SCRATCH se derive karo (key result)

Maano $y_{m}, y_{n}$ SL equation solve karte hain eigenvalues $λ_{m}, λ_{n}$ ke saath: $(p y_{m}^{'})^{'} + (q + λ_{m} w) y_{m} = 0, (p y_{n}^{'})^{'} + (q + λ_{n} w) y_{n} = 0.$ Pehle ko $y_{n}$ se multiply karo, doosre ko $y_{m}$ se multiply karo, subtract karo: $y_{n} (p y_{m}^{'})^{'} - y_{m} (p y_{n}^{'})^{'} + (λ_{m} - λ_{n}) w y_{m} y_{n} = 0.$ Subtract kyun? $q y_{m} y_{n}$ wale terms cancel ho jaate hain, eigenvalue difference isolate ho jaata hai.

Ab note karo Lagrange's identity: $y_{n} (p y_{m}^{'})^{'} - y_{m} (p y_{n}^{'})^{'} = \frac{d}{d x} [p (y_{n} y_{m}^{'} - y_{m} y_{n}^{'})] .$ Yeh step kyun? RHS expand karo: $p^{'} (y_{n} y_{m}^{'} - y_{m} y_{n}^{'}) + p (y_{n} y_{m}^{''} - y_{m} y_{n}^{''})$ — exactly LHS hi hai. Yeh sab kuch ek total derivative mein pack kar deta hai taaki hum cleanly integrate kar sakein.

$a$ se $b$ tak integrate karo: $[p (y_{n} y_{m}^{'} - y_{m} y_{n}^{'})]_{a}^{b} + (λ_{m} - λ_{n}) \int_{a}^{b} w y_{m} y_{n} d x = 0.$ Boundary term Wronskian-jaisa bracket hai: $[p W (y_{m}, y_{n})]_{a}^{b}$ .

Boundary term zero kyun hota hai: $x = a$ par, separated BC $α_{1} y_{m} (a) + α_{2} y_{m}^{'} (a) = 0$ aur $α_{1} y_{n} (a) + α_{2} y_{n}^{'} (a) = 0$ do equations hain $(α_{1}, α_{2}) \neq = (0, 0)$ ke liye. Nonzero solution ka matlab determinant $y_{m} (a) y_{n}^{'} (a) - y_{n} (a) y_{m}^{'} (a) = 0$ hona chahiye, yani $W (y_{m}, y_{n}) ∣_{a} = 0$ . Same $b$ par. Isliye bracket $0$ hai.

Therefore $(λ_{m} - λ_{n}) \int_{a}^{b} w y_{m} y_{n} d x = 0.$ Kyunki $λ_{m} \neq = λ_{n}$ , integral zero hai. $■$ Weight $w$ ke saath Orthogonality.

3.2 Eigenvalues real kyun hote hain (same machinery)

Complex $y$ lo eigenvalue $λ$ ke saath, aur uska conjugate $\overset{y}{ˉ}$ $\overset{ˉ}{λ}$ ke saath (kyunki $p, q, w$ real hain). Same subtraction se milta hai $(λ - \overset{ˉ}{λ}) \int w ∣ y ∣^{2} d x = 0$ . Kyunki $\int w ∣ y ∣^{2} > 0$ , hume milta hai $λ = \overset{ˉ}{λ}$ , yani real.

4. Eigenfunction expansion — coefficients kaise nikaalte hain

Yeh kyun kaam karta hai: orthogonality har term ko kill kar deta hai sivaaye $m = n$ ke — exactly jaise ek vector ko orthogonal axis par project karna: $c_{m} = \frac{⟨ f , y _{m} ⟩ _{w}}{⟨ y _{m} , y _{m} ⟩ _{w}}$ .

Figure — Sturm-Liouville theory — eigenvalue problems, orthogonality of eigenfunctions

5. Worked Examples

6. Common Mistakes (Steel-man + Fix)

Recall Feynman: ek 12-saal ke bachche ko samjhao

Ek guitar ki string imagine karo. Jab tum use pluck karte ho, toh woh sirf special shapes mein vibrate karti hai — ek bump, do bumps, teen bumps. "Dhai bumps" nahi mil sakta. Woh allowed shapes eigenfunctions hain, aur har ek kitni tezi se hilti hai woh eigenvalue hai. Cool trick yeh hai: yeh shapes itni "independent" hain ki string ki koi bhi movement bas ek recipe hai — shape-1 ka kuch, shape-2 ka kuch, etc. Yeh pata karne ke liye ki shape-2 ka kitna hai, tum apni movement ko sirf shape-2 se "compare" karte ho; baaki saari shapes politely zero deti hain. Woh polite-zero property orthogonality hai. Sturm–Liouville theory woh rulebook hai jo promise karta hai ki yeh special shapes hamesha exist karengi aur hamesha itni acchi tarah behave karengi.

7. Flashcards

Ek 2nd-order ODE ka standard Sturm–Liouville form kya hai?

(p (x) y^{'})^{'} + (q (x) + λ w (x)) y = 0

jahan

p, w > 0

aur homogeneous BCs hoon.

Weight function

w (x)

kya determine karta hai?

Orthogonality ke liye inner product:

\int_{a}^{b} y_{m} y_{n} w d x

SL eigenfunctions ka orthogonality relation batao.

\int_{a}^{b} y_{m} (x) y_{n} (x) w (x) d x = 0

for

m \neq = n

a_{2} y^{''} + a_{1} y^{'} + a_{0} y + λ y = 0

ko SL form mein convert karne ka integrating factor kya hai?

μ (x) = exp (\int \frac{a _{1}}{a _{2}} d x)

; phir

p = μ

q = μ a_{0} / a_{2}

w = μ / a_{2}

SL eigenvalues real kyun hote hain?

y

aur

\overset{y}{ˉ}

ke equations subtract karne par milta hai

(λ - \overset{ˉ}{λ}) \int w ∣ y ∣^{2} d x = 0

; kyunki integral

> 0

hai,

λ = \overset{ˉ}{λ}

Kaunsi identity

y_{n} (p y_{m}^{'})^{'} - y_{m} (p y_{n}^{'})^{'}

ko ek derivative mein pack karti hai?

Lagrange's identity:

\frac{d}{d x} [p (y_{n} y_{m}^{'} - y_{m} y_{n}^{'})]

Orthogonality proof mein boundary term kyun vanish hota hai?

Separated BCs Wronskian

y_{m} y_{n}^{'} - y_{n} y_{m}^{'} = 0

ko har endpoint par force karte hain (ya wahan

p = 0

hota hai).

f = \sum c_{n} y_{n}

mein expansion coefficient

c_{m}

ka formula?

c_{m} = \frac{\int _{a}^{b} f y _{m} w d x}{\int _{a}^{b} y _{m}^{2} w d x}

y^{''} + λ y = 0

y (0) = y (L) = 0

ke eigenvalues/eigenfunctions kya hain?

λ_{n} = (nπ / L)^{2}

y_{n} = sin (nπ x / L)

n = 1, 2, \dots

, weight

w = 1

Periodic SL problems mein multiplicity-2 eigenvalues kyun ho sakte hain?

Yeh regular SL problems nahi hote; simplicity sirf regular separated-BC problems ke liye guaranteed hai.

8. Connections

Separation of Variables — PDE se SL problem produce karta hai.
Fourier Series — special case $w = 1$ , $sin / cos$ basis.
Bessel Functions — singular SL, weight $w = x$ ke saath.
Legendre Polynomials — singular SL on $[- 1, 1]$ , $w = 1$ .
Hilbert Space and Inner Products — orthogonality as projection.
Heat Equation / Wave Equation — solutions ke eigenfunction expansions.