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

4.6.19 · D5 · Maths › Ordinary Differential Equations › Bessel's equation and Bessel functions (intro, physical rele

Shuru karne se pehle, ek reminder ki har symbol kya hai, taaki koi line undefined term use na kare:

$u (r, θ)$ = disc par physical field (drum displacement, temperature, ...); $u_{r}$ matlab $\partial u / \partial r$ , radial direction mein iska slope.
$r$ = actual radius (centre se doori); $a$ = disc ka rim radius, jahan edge clamped hai.
$k$ = wave-number, ek fixed positive constant jo wave ka scale set karta hai; ye substitution $x = k r$ ke zariye frequency ko radial equation mein pack karta hai.
$x$ = independent variable (ek stretched radius, $x = k r$ ). Ye move karta hai.
$ν$ (Greek "nu") = order, ek fixed number jo aap ek baar choose karte ho. Ye move nahi karta.
$J_{ν} (x)$ = Bessel function of the first kind — $x = 0$ par finite, "round chezon ke liye sine."
$Y_{ν} (x)$ = Bessel function of the second kind — $x = 0$ par blow up karta hai.
$α_{ν, k}$ = $J_{ν}$ ka $k$ -th positive zero, yaani $x$ ki $k$ -th value (baahir ki taraf count karte hue, $k = 1, 2, 3, \dots$ ) jahan $J_{ν} (x) = 0$ . Yahan $k$ ek counting index hai, upar wale wave-number $k$ se unrelated — letter tradition se overloaded hai, isliye context se padhein.
$c_{1}, c_{2}$ = arbitrary constants of integration: wo do free constants jo kisi bhi second-order linear ODE ke general solution mein hote hain, baad mein boundary conditions se fix hote hain.
Drum problem: rim radius $a$ wali ek circular membrane, edge par clamped, isliye boundary condition hai $u = 0$ (equivalently $R (a) = 0$ ) $r = a$ par, jabki $u$ centre $r = 0$ par finite rehta hai.

True or false — justify

TF1. Bessel's equation ek linear ODE hai.

True —

y

y^{'}

y^{''}

har ek pehli power mein appear karte hain, coefficients sirf

x

par depend karte hain; koi

y^{2}

y y^{'}

kabhi nahi aata, isliye solutions superpose karte hain.

TF2. Kyunki

J_{ν} (x)

oscillate karta hai aur zero cross karta hai, ye

cos x

jaisa periodic hai.

False — iske zeros equally spaced nahi hain aur iska amplitude

1/ x

ki tarah shrink karta hai. Ye ek stretched, damped cosine hai, kabhi exactly periodic nahi.

TF3. Solid disc par general solution mein

J_{ν}

aur

Y_{ν}

dono rakhte hain.

False — solid disc mein

r = 0

shamil hai jahan

Y_{ν} \to \pm \infty

; physical displacement wahan finite hai, isliye iska coefficient

0

forced ho jaata hai. Dono sirf annulus (beech mein hole) par survive karte hain.

TF4.

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

mein, term

- ν^{2}

ek constant hai,

x

ka function nahi.

True —

ν

fixed order hai, isliye

- ν^{2}

sirf ek number hai jo

x^{2}

se subtract hota hai; ise variable samajhna classic slip hai.

TF5. Har

J_{n}

x = 0

par height

1

se start karta hai.

False — sirf

J_{0} (0) = 1

;

n \geq 1

ke liye,

J_{n} (0) = 0

kyunki

J_{ν}

ki series power

(x /2)^{ν}

se start hoti hai, jo

0

par vanish ho jaata hai jab tak

ν = 0

na ho.

TF6. Point

x = 0

ek ordinary point hai, isliye ek plain Taylor series Bessel's equation solve karta hai.

False —

x^{2}

se divide karne par coefficients blow up karte hain, isliye

x = 0

ek regular singular point hai; aapko extra power

x^{s}

ke saath Frobenius method chahiye.

TF7. Do indicial roots hamesha

+ ν

aur

- ν

hote hain chahe

ν

integer ho ya na ho.

True — indicial equation

s^{2} - ν^{2} = 0

har

ν

ke liye hai; lekin jab

ν

integer hota hai to do roots integer se differ karte hain, isliye second solution ko

ln x

term chahiye.

TF8. Half-integer orders jaise

ν = \frac{1}{2}

genuinely naye "special" functions dete hain jinका koi closed form nahi.

False — half-integer orders elementary functions mein collapse ho jaate hain, jaise

J_{1/2} (x) = 2/ (π x) sin x

. Ye sanity check ke liye favourite hain precisely kyunki ye elementary hain.

TF9. Drum ke overtones apne fundamental ke integer multiples hote hain, jaise guitar string.

False — frequencies zeros

α_{0, k}

(

2.405, 5.520, 8.654, \dots

) ke saath scale karte hain, jinke ratios irrational hain. Wahi inharmonicity hai jis wajah se drum drum jaisa lagta hai, string jaisa nahi.

TF10. Agar hum straight string par wave equation set up karte to bhi

x y^{'}

term wahan hoti.

False —

x y^{'}

term polar Laplacian mein

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

(radial slope) se aata hai, jo isliye exist karta hai kyunki area

r

ki tarah badhta hai. Straight string mein aisi koi geometric spreading nahi, isliye koi first-derivative term nahi aur ek clean sine aata hai.

Edge cases

EC1.

ν = 0

par kya hota hai jo

ν > 0

par nahi hota?

Do indicial roots

\pm ν

0

par coincide karte hain, ek repeated root dete hain; tab second solution

Y_{0}

necessarily

ln x

contain karta hai aur origin par logarithmically diverge karta hai.

EC2.

x \to 0^{+}

par

ν > 0

ke liye

J_{ν} (x)

ka behaviour kya hai?

0

ki taraf

(x /2)^{ν} /Γ (ν + 1)

ki tarah approach karta hai — series ki leading power — isliye ye finite aur vanishing hai, bade

ν

ke liye flatter start ke saath.

EC3.

x \to \infty

par

J_{ν} (x)

ki limiting shape kya hai?

2/ (π x) cos (x - \frac{ν π}{2} - \frac{π}{4})

jaisa lagta hai: ek ever-slower-shrinking cosine jiske zeros almost evenly spaced ho jaate hain (spacing

\to π

EC4. Annular region

b \leq r \leq a

(washer) par kitne Bessel functions survive karte hain aur kyun?

J_{ν}

aur

Y_{ν}

dono survive karte hain, kyunki centre

r = 0

exclude hai — domain par kuch unbounded nahi, isliye dono constants

c_{1}, c_{2}

rehte hain aur do boundary conditions unhe fix karti hain.

EC5. Agar ek problem

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

(plus

ν^{2}

ke saath) deta hai, to kya ye Bessel's equation hai?

Standard wali nahi —

+ ν^{2}

ab bhi order

i ν

mein reduce hoga, lekin sign flip ke baare mein zyada commonly dhyan dene wali baat

x^{2} \to - x^{2}

hai, jo modified Bessel equation deta hai non-oscillating solutions

I_{ν}, K_{ν}

ke saath.

EC6.

J_{1/2}

family ke "half-integer" edge ke baare mein kya batata hai?

Ki family smoothly elementary trigonometry se connect karti hai:

J_{1/2} (x) = 2/ (π x) sin x

ek exact closed form hai, jo

1/ x

decay ko explicitly confirm karta hai.

Recall Ek-line survival guide

" $x$ move karta hai, $ν$ fixed hai; $J$ rehta hai ( $0$ par finite), $Y$ udta hai ( $0$ par infinite); ye ek damped, unevenly-spaced cosine hai, periodic nahi." Upar ka har trap in teen facts mein se ek hi disguise mein hai.

Connections

Bessel's equation and Bessel functions (intro, physical relevance) — parent topic jo ye bank drill karta hai.
Frobenius method — series (aur $Y_{n}$ mein log) kyun appear karte hain.
Regular singular points — wajah ki $x = 0$ par Taylor fail karta hai.
Separation of variables — jahan integer- $ν$ constraint paida hoti hai.
Gamma function — $J_{ν}$ ke andar factorial extension.

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

True or false — justify

Spot the error

Why questions

Edge cases

Connections