Bessel's equation and Bessel functions (intro, physical relevance)

4.6.19 · Maths › Ordinary Differential Equations

Bessel's equation kya hai?

KYA matlab hai in pieces ka:

$x^{2} y^{''}$ aur $x y^{'}$ milke polar/cylindrical coordinates mein Laplacian se aate hain.
$- ν^{2}$ term angular separation constant se aata hai (circle ke around kitni waves fit hoti hain).
$+ x^{2}$ term woh hai jo ise oscillate karata hai — iske bina aapko ek power-law (Euler) equation milti.

YEH KAHAN se aata hai (derive karo, memorize mat karo)

2-D Helmholtz / vibrating membrane equation $\nabla^{2} u + k^{2} u = 0$ lo polar coordinates mein: $u_{r r} + \frac{1}{r} u_{r} + \frac{1}{r ^{2}} u_{θ θ} + k^{2} u = 0.$

Step 1 — Variables separate karo. $u (r, θ) = R (r) Θ (θ)$ try karo. Yeh step kyun? Circular symmetry suggest karti hai ki radial aur angular behaviour independent hain.

Substitute karo aur $R Θ/ r^{2}$ se divide karo: $\frac{r ^{2} R ^{''} + r R ^{'}}{R} + r^{2} k^{2} = - \frac{Θ ^{''}}{Θ} .$

Step 2 — Har side ko constant $ν^{2}$ set karo. Kyun? Left sirf $r$ par depend karta hai, right sirf $θ$ par; sab $r, θ$ ke liye equal ⇒ dono constant hain.

Angular: $Θ^{''} + ν^{2} Θ = 0 \Rightarrow Θ = cos ν θ, sin ν θ$ . Single-valuedness ( $Θ (θ + 2 π) = Θ (θ)$ ) force karti hai ki $ν = n$ integer ho.

Radial: $r^{2} R^{''} + r R^{'} + (r^{2} k^{2} - ν^{2}) R = 0.$

Step 3 — $x = k r$ se rescale karo. Tab $\frac{d}{d r} = k \frac{d}{d x}$ , aur $k$ cleanly cancel ho jaate hain: $x^{2} R^{''} + x R^{'} + (x^{2} - ν^{2}) R = 0.$ Yahi hai Bessel's equation. $\frac{1}{r} u_{r}$ term $x R^{'}$ term ka source hai — geometry hai, magic nahi.

Ise solve karna: Frobenius series (first principles)

Kyunki $x = 0$ ek regular singular point hai, hum Frobenius series $y = \sum_{m \geq 0} a_{m} x^{m + s}$ use karte hain.

Step 1 — Indicial equation. Lowest power plug in karo. Substitute karne par lowest term ke liye milta hai, $[s (s - 1) + s - ν^{2}] a_{0} = (s^{2} - ν^{2}) a_{0} = 0 \Rightarrow s = \pm ν .$ Yeh step kyun? Sabse chhota power apne aap vanish hona chahiye; yeh leading exponent fix karta hai.

Step 2 — Recurrence. $x^{m + s}$ collect karo: $[(m + s)^{2} - ν^{2}] a_{m} + a_{m - 2} = 0 \Rightarrow a_{m} = - \frac{a _{m - 2}}{( m + s ) ^{2} - ν ^{2}} .$ Kyun? $+ x^{2} y$ index ko 2 se shift karta hai, $a_{m}$ ko $a_{m - 2}$ se link karta hai. Odd terms vanish ho jaate hain ( $a_{1} = 0$ ).

Step 3 — $s = ν$ lo, normalize karo. $a_{0} = \frac{1}{2 ^{ν} Γ ( ν + 1 )}$ choose karne par standard Bessel function of the first kind milta hai:

Doosra solution ( $s = - ν$ se, ya jab $ν$ integer ho tab ek log term se) Bessel function of the second kind $Y_{ν} (x)$ hai, jo $x = 0$ par $ln x$ ya $x^{- ν}$ ki tarah blow up karta hai.

Key behaviours (graphs kya batate hain)

Worked examples

Common mistakes

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho jab tum ek round taalaab mein pathar giraate ho toh ripples banti hain. Guitar string par wiggles even aur tidy hoti hain — woh sine hai. Lekin round taalaab par ripple ko center se door jaate waqt ek bade aur bade ring par spread karna padta hai, toh woh jitni bahar jaati hai utni kamzor hoti jaati hai. Bessel functions woh special "ripple shapes" hain jo exactly ise describe karte hain: yeh wave ki tarah upar-neeche wiggle karte hain, lekin har wiggle pichli se thodi chhoti hoti hai kyunki circle bada hota ja raha hai. Drums, pipes, aur lenses sab in round-shaped waves use karte hain.

Active-recall flashcards

Bessel's equation of order

ν

x^{2} y^{''} + x y^{'} + (x^{2} - ν^{2}) y = 0

x y^{'}

term physically kahan se aata hai?

Polar Laplacian mein

\frac{1}{r} u_{r}

se (area

r

ki tarah badhta hai).

Bessel's equation ki indicial roots

s = \pm ν

(

s^{2} - ν^{2} = 0

se).

Coefficients ke liye recurrence relation

a_{m} = - a_{m - 2} / [(m + ν)^{2} - ν^{2}]

; odd terms vanish ho jaate hain.

J_{ν} (x)

ki series

\sum_{m \geq 0} \frac{( - 1 ) ^{m}}{m ! Γ ( m + ν + 1 )} (x /2)^{2 m + ν}

Solid disc par

Y_{ν}

kyun drop karte hain?

Y_{ν}

x = 0

par diverge karta hai; physical displacement finite rehni chahiye.

J_{0} (0)

ki value aur

n \geq 1

ke liye

J_{n} (0)

J_{0} (0) = 1

J_{n} (0) = 0

J_{ν}

ka large-

x

behaviour

2/ (π x) cos (x - ν π /2 - π /4)

— decaying oscillation.

Drum ki allowed frequencies kya set karta hai?

J_{ν}

ke zeros

α_{ν, k}

J_{ν} (k a) = 0

ke through.

J_{1/2} (x)

elementary form mein

2/ (π x) sin x

Drum ke overtones inharmonic kyun hote hain?

J_{0}

ke zeros ke ratios (2.405, 5.520, …) integers nahi hain.

Connections

Frobenius method — woh engine jo $J_{ν}$ series produce karta hai.
Regular singular points — kyun $x = 0$ ko Frobenius chahiye, Taylor nahi.
Separation of variables — kaise PDEs se Bessel's eqn appear hoti hai.
Laplacian in polar and cylindrical coordinates — $1/ r$ ka origin.
Gamma function — non-integer orders ke liye $m!$ ko generalize karta hai.
Sturm-Liouville theory — Fourier-Bessel series mein $J_{ν}$ ki orthogonality.
Legendre's equation — Bessel ka spherical cousin.