Method of Applied Math

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Lecture 4: Bessel Equation

Sujin Khomrutai, Ph.D.

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Bessel Equation

Bessel eqn EX 1

Use Frobenius Bessel 1st kind EX 2.

EX 3.

Bessel 2nd kind EX 4.

EX 5.

Complete solution EX 6.

EX 7.

Let ν be a non-negative constant. The equation of the form x²y^′′ + xy^′ + (x² − ν²)y = 0

is called the Bessel equation of order ν.

Note that a = 0 is a regular singular point. Expressing in the standard form

y^′′ + 1

xy^′ + x² − ν²

x² y = 0 ⇒ p(x) = 1

x, q(x) = x² − ν² x² we have

p0 = lim xp(x) = 1, q0 = lim x²q(x) = −ν².

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Bessel Equation

Bessel eqn EX 1

EX 3.

EX 5.

EX 7.

The indicial equation is

r(r − 1) + r − ν² = 0 or

r² − ν² = 0 ⇒ r1 = ν, r2 = −ν.

The larger root is r1 = ν.

So by Frobenius’ theorem, the Bessel equation has a series solution y = x^ν(b0 + b1x + b2x² + · · · )

= b0x^ν + b1x^ν⁺¹ + b2x^ν⁺² + · · · .

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Example 1

Bessel eqn EX 1

EX 3.

EX 5.

EX 7.

EX. Find a Frobenius series solution for the Bessel equation of order 1:

x²y^′′ + xy^′ + (x² − 1)y = 0,

by calculating the first four coefficients.

Sol. y = b0x + b1x² + b2x³ + b3x⁴ + b4x⁵ + · · ·

xy^′ = b0x + 2b1x² + 3b2x³ + 4b3x⁴ + 5b4x⁵ + · · · x²y^′′ = 2b1x² + 6b2x³ + 12b3x⁴ + 20b4x⁵ + · · · x¹ : b0 − b0 = 0

x² : 2b1 + 2b1 − b1 = 0 ⇒ b1 = 0

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Example 1

Bessel eqn EX 1

EX 3.

EX 5.

EX 7.

x³ : 6b2 + 3b2 + b0 − b2 = 0 ⇒ b2 = −1 8b0

x⁴ : 12b3 + 4b3 + b1 − b3 = 0 ⇒ b3 = 0 x⁵ : 20b4 + 5b4 + b2 − b4 = 0 ⇒ b4 = − 1

24b2 = 1 192b0

So we get y = b0

x − 1

8x³ + 1

192x⁵ + · · ·

. Taking b0 = ¹₂, we can write the solution as

y =

∞

X

k=0

(−1)^k

k!Γ(ν + k + 1)

x 2

²k+ν

(ν = 1).

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Solving Bessel equation

Bessel eqn EX 1

EX 3.

EX 5.

EX 7.

Generally, we have the following solution for the Bessel equation of order ν.

Theorem. The function J_ν(x) =

∞

X

k=0

(−1)^k

k!Γ(ν + k + 1)

x 2

²^k⁺^ν , is a solution of the Bessel equation of order ν.

This solution is obtained by taking the larger root of the indicial equation.

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Solving Bessel equation

Bessel eqn EX 1

EX 3.

EX 5.

EX 7.

Frobenius’ method. Choose a = 0 and use the larger root r = ν of the indicial equation.

Step 1 Set

y =

∞

X

k=0

b_kx^k⁺^ν

y^′ =

∞

X

k=0

b_k(k + ν)x^k⁺^ν⁻¹

y^′′ =

∞

X

k=0

b_k(k + ν)(k + ν − 1)x^k⁺^ν⁻².

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Solving Bessel equation

Bessel eqn EX 1

EX 3.

EX 5.

EX 7.

Step 2. Recurrence equations:

k = 0 : ν(ν − 1)b0 + νb0 − ν²b0 = 0

k = 1 : (ν + 1)νb1 + (ν + 1)b1 − ν²b1 = 0

k ≥ 2 : (k + ν)(k + ν − 1)b_k + (k + ν)b_k + b_k⁻2 − ν²b_k = 0 Step 3 Solve the recurrence equations to get

b1 = b3 = b5 = · · · = 0 and

b2n = (−1)ⁿb0

2²ⁿn!(ν + 1)(ν + 2) · · · (ν + n), n = 1, 2, . . .

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Solving Bessel equation

Bessel eqn EX 1

EX 3.

EX 5.

EX 7.

Step 4 Let

b0 = 1

2^νΓ(ν + 1) ⇒ b2n = (−1)ⁿ

2²ⁿ⁺^νn!Γ(ν + n + 1) The solution is

J_ν(x) =

∞

X

n=0

(−1)ⁿ

n!Γ(ν + n + 1)

x 2

²n+ν

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Bessel Functions of the First Kind

Bessel eqn EX 1

EX 3.

EX 5.

EX 7.

Definition. The function J_ν(x) =

∞

X

k=0

(−1)^k

k!Γ(ν + k + 1)

x 2

²k+ν

is called the Bessel function of the first kind of order ν. Recall. Γ is the Gamma function. It has the properties that

Γ(n + 1) = n! n ∈ {0,1, 2, . . .}, and

Γ(x + n + 1) = (x + n)(x + n − 1) · · · xΓ(x).

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Example 2

Bessel eqn EX 1

EX 3.

EX 5.

EX 7.

EX. Use the solution formula to find a solution of the equation xy^′′ + y^′ + xy = 0 (x > 0).

Sol.

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Example 3

Bessel eqn EX 1

EX 3.

EX 5.

EX 7.

EX. Show that the following property hold:

(x^νJ_ν(x))^′ = x^νJ_ν⁻1(x).

Sol.

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Bessel Functions of the Second Kind

Bessel eqn EX 1

EX 3.

EX 5.

EX 7.

The indicial equation of the Bessel equation of order ν has two roots r1 = ν, r2 = −ν.

The larger root r1 = ν leads to a solution y1 = J_ν(x). For the root r2 = −ν, one get another solution

J⁻_ν(x) =

∞

X

n=0

(−1)^k

k!Γ(−ν + k + 1)

x 2

²k−ν

.

A difficulty: {J_ν, J−ν} are linearly dependent for ν = 0,1, 2, . . . In fact, J⁻_ν(x) = (−1)^νJ_ν(x).

If ν 6∈ {0, 1,2, . . .} then the Bessel equation has a general solution y = C1J_ν(x) + C2J⁻_ν(x).

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Example 4

Bessel eqn EX 1

EX 3.

EX 5.

EX 7.

EX. Show that J−1(x) = −J1(x). Sol.

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Example 5

Bessel eqn EX 1

EX 3.

EX 5.

EX 7.

EX. Find the general solution of the Bessel equation x²y^′′ + xy^′ +

x² − 4 9

y = 0. Sol.

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Bessel Functions of the Second Kind

Bessel eqn EX 1

EX 3.

EX 5.

EX 7.

Definition. We define Y_ν(x) = 1

sin νπ [J_ν(x) cosνπ − J⁻_ν(x)] , ν 6∈ {0,1, 2, . . .}

and

Y_ν(x) = lim

µ→ν Y_µ(x), ν ∈ {0,1, 2, . . .}.

Y_ν is another solution to the Bessel equation of order ν and {J_ν, Y_ν} are fundamental solutions.

Y_ν(x) is called Bessel function of the second kind order ν.

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Complete Solution of Bessel Equation

Bessel eqn EX 1

EX 3.

EX 5.

EX 7.

Theorem. The complete solution of the Bessel equation x²y^′′ + xy^′ + (x² − ν²)y = 0

is

y = C1J_ν(x) + C2Y_ν(x) where C1, C2 are constants.

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Example 6

Bessel eqn EX 1

EX 3.

EX 5.

EX 7.

EX. Find the complete solution of the Bessel equation x²y^′′ + xy^′ + x² − 5

y = 0.

Sol.

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Example 7

Bessel eqn EX 1

EX 3.

EX 5.

EX 7.

EX. Find the complete solution of the equation y^′′ +

e⁻²^x − 1 9

y = 0

by changing the variable z = e⁻^x. EX.