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Introduction Application Properties Bessels Function A Touch of Magic Fayez Karoji 1 Casey Tsai 1 Rachel Weyrens 2 1 Department of Mathematics Louisiana State University 2 Department of Mathematics University of Arkansas SMILE REU Summer

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Introduction Application Properties

Bessel’s Function

A Touch of Magic Fayez Karoji1 Casey Tsai1 Rachel Weyrens2

1Department of Mathematics

Louisiana State University

2Department of Mathematics

University of Arkansas

SMILE REU Summer 2010

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties

Outline

Introduction Bessel Functions Terminology Application Drum Example Properties Orthogonality

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Bessel Functions Terminology

General Form

Bessel’s differential equation is x2y′′ + xy′ + (x2 − n2)y = 0

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Bessel Functions Terminology

General Form

Bessel’s differential equation is x2y′′ + xy′ + (x2 − n2)y = 0 The linearly independent solutions are Jn and Yn.

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Bessel Functions Terminology

General Form

Bessel’s differential equation is x2y′′ + xy′ + (x2 − n2)y = 0 The linearly independent solutions are Jn and Yn. The zeros are jn and yn.

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Bessel Functions Terminology

Bessel Functions of Order Zero

/Bessel/j0.pdf

Figure: Bessel Function of the First Kind, J0

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Bessel Functions Terminology

Bessel Functions of Order Zero

/Bessel/y0.pdf

Figure: Bessel Function of the Second Kind, Y0

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Bessel Functions Terminology

Key Terms

▶ Separation of variables

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Bessel Functions Terminology

Key Terms

▶ Separation of variables ▶ Regular singular

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Bessel Functions Terminology

Key Terms

▶ Separation of variables ▶ Regular singular ▶ Superposition

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Drum Example

Physical Description

▶ Radially symmetric

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Drum Example

Physical Description

▶ Radially symmetric ▶ Radius r = 1

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Drum Example

Physical Description

▶ Radially symmetric ▶ Radius r = 1 ▶ Beginning at rest

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Drum Example

Physical Description

▶ Radially symmetric ▶ Radius r = 1 ▶ Beginning at rest ▶ Edges fixed

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Drum Example

Boundary Valued Problem

This physical problem can be represented by the following boundary valued problem:

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Drum Example

Boundary Valued Problem

This physical problem can be represented by the following boundary valued problem:

▶ utt = urr + 1 r ur

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Drum Example

Boundary Valued Problem

This physical problem can be represented by the following boundary valued problem:

▶ utt = urr + 1 r ur ▶ u(r, 0) = f(r)

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Drum Example

Boundary Valued Problem

This physical problem can be represented by the following boundary valued problem:

▶ utt = urr + 1 r ur ▶ u(r, 0) = f(r) ▶ ut(r, 0) = 0

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Drum Example

Boundary Valued Problem

This physical problem can be represented by the following boundary valued problem:

▶ utt = urr + 1 r ur ▶ u(r, 0) = f(r) ▶ ut(r, 0) = 0 ▶ u(1, t) = 0

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Drum Example

Separation of Variables

We have u(r, t) = R(r)T(t)

▶ T ′′ + 휇T = 0 ▶ R′′ + 1 r R′ + 휇R = 0

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Drum Example

Separation of Variables

We have u(r, t) = R(r)T(t)

▶ T ′′ + 휇T = 0 ▶ R′′ + 1 r R′ + 휇R = 0 ▶ 휇 = 훼2 > 0

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Drum Example

Solutions

The solutions of the given ODE’s are

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Drum Example

Solutions

The solutions of the given ODE’s are

▶ T(t) = c1 cos(훼t) + c2 sin(훼t)

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Drum Example

Solutions

The solutions of the given ODE’s are

▶ T(t) = c1 cos(훼t) + c2 sin(훼t) ▶ R(r) = c3J0(훼r) + c4Y0(훼r)

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Drum Example

Evaluation

Using initial and boundary conditions, we have un(r, t) = AnJ0(jnr) cos(jnt)

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Drum Example

Evaluation

Using initial and boundary conditions, we have un(r, t) = AnJ0(jnr) cos(jnt) General solution

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Drum Example

Evaluation

Using initial and boundary conditions, we have un(r, t) = AnJ0(jnr) cos(jnt) General solution u(r, t) =

∞

∑

n=1

AnJ0(jnr) cos(jnt)

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Drum Example

Amplitude

The amplitude of displacement, from u(r, 0) = f(r) is:

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Drum Example

Amplitude

The amplitude of displacement, from u(r, 0) = f(r) is: An = ∫ 1

0 rJ0(jnr)f(r)dr

∫ 1

0 rJ0(jnr)J0(jnr)dr

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Drum Example

Frequencies

Fundamental pitch

j1 2휋

First overtone

j2 2휋

Second overtone

j3 2휋

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Orthogonality

Orthogonality Property of Bessel Functions

/Bessel/jnspdf.pdf

Figure: Bessel Functions of the First Kind

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Orthogonality

Problems in Mathematical Physics

▶ PDE’s model physical phenomena. ▶ Example: Steady Temperatures in Circular Cylinder

(Laplacian in Cylindrical Coordinates).

▶ Example: The Vibrating Drumhead (Wave Equation in

Polar Coordinates).

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Orthogonality

Methods of Solution

▶ PDE’s are difficult to solve. ▶ Fourier’s Method: Linear and homogeneous PDE’s with

homogeneous boundary conditions.

▶ Also known as Separation of Variables.

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Orthogonality

Fourier’s Method: PDE − → ODE’s

▶ PDE: Wave Equation in Polar Coordinates ▶ Apply Fourier’s Method ▶ Two second order ODE’s

▶ Simple Harmonic Motion T ′′ + 휇T = 0 ▶ Bessel’s Equation R′′ + 1

r R′ + R = 0

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Orthogonality

Orthogonal Functions

▶ Analysis of solutions to ODE’s ▶ Underlying Theme: Orthogonal Functions ▶ Examples:

▶ Sine and Cosine Functions ▶ Legendre Polynomials (Special Function) ▶ Bessel Functions (A "Very" Special Function) Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Orthogonality

What is Orthogonality?

▶ Dot Product or Inner Product in ℝn

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Orthogonality

What is Orthogonality?

▶ Dot Product or Inner Product in ℝn ▶ Given x, y ∈ ℝn

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Orthogonality

What is Orthogonality?

▶ Dot Product or Inner Product in ℝn ▶ Given x, y ∈ ℝn ▶ Define x ⋅ y = ∑n i=1 xiyi

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Orthogonality

What is Orthogonality?

▶ Dot Product or Inner Product in ℝn ▶ Given x, y ∈ ℝn ▶ Define x ⋅ y = ∑n i=1 xiyi ▶ x and y are orthogonal when ∑n i=1 xiyi = 0

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Orthogonality

Generalize Orthogonality

▶ Inner Product in ℛ[a, b]

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Orthogonality

Generalize Orthogonality

▶ Inner Product in ℛ[a, b] ▶ Given f, g ∈ ℛ[a, b]

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Orthogonality

Generalize Orthogonality

▶ Inner Product in ℛ[a, b] ▶ Given f, g ∈ ℛ[a, b] ▶ Define ⟨f, g⟩ =

∫ b

a f(x)g(x) dx

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Orthogonality

Generalize Orthogonality

▶ Inner Product in ℛ[a, b] ▶ Given f, g ∈ ℛ[a, b] ▶ Define ⟨f, g⟩ =

∫ b

a f(x)g(x) dx ▶ f and g are orthogonal when

∫ b

a f(x)g(x) dx = 0

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Orthogonality

Example: Simple Harmonic Motion

▶ Consider T ′′ + n2T = 0, (휇 = n2)

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Orthogonality

Example: Simple Harmonic Motion

▶ Consider T ′′ + n2T = 0, (휇 = n2) ▶ Solutions are sin(nx) and cos(nx)

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Orthogonality

Example: Simple Harmonic Motion

▶ Consider T ′′ + n2T = 0, (휇 = n2) ▶ Solutions are sin(nx) and cos(nx) ▶ Easy to show that

∫ 휋

−휋 sin(mx) cos(nx) dx = 0

for any n, m ∈ ℤ

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Orthogonality

Example: Legendre Polynomials

▶ It was shown that the Legendre Polynomials satisfy

∫ 1

−1 Pn(x)Pm(x) dx = 0 for n, m ∈ ℤ, n ∕= m

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Orthogonality

Example: Bessel Functions

▶ Orthogonality property of Jn(휆x) and Jn(휇x)

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Orthogonality

Example: Bessel Functions

▶ Orthogonality property of Jn(휆x) and Jn(휇x) ▶ Bessel Functions of the First Kind of Order n

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Orthogonality

Example: Bessel Functions

▶ Orthogonality property of Jn(휆x) and Jn(휇x) ▶ Bessel Functions of the First Kind of Order n ▶ 휆 and 휇 are distinct positive roots of Jn(x) = 0

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Orthogonality

Example: Bessel Functions

▶ Orthogonality property of Jn(휆x) and Jn(휇x) ▶ Bessel Functions of the First Kind of Order n ▶ 휆 and 휇 are distinct positive roots of Jn(x) = 0 ▶ Will show:

∫ 1

0 xJn(휆x)Jn(휇x) dx = 0

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Orthogonality

Theorem

Theorem

If 휆 and 휇 are distinct positive roots of Jn(x) = 0 then ∫ 1 xJn(휆x)Jn(휇x) dx = { 0, if 휆 ∕= 휇

1 2J2 n+1(휆),

if 휆 = 휇

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Orthogonality

Proof

Proof. Suppose 휆 ∕= 휇, then 휆 and 휇 are distinct positive roots of Jn(x) = 0. Since Jn(휆x) and Jn(휇x) are solutions of the Bessel equation in parametric form, we can write x2J′′

n(휆x) + xJ′ n(휆x) + (휆2x2 − n2)Jn(휆x) = 0

(1) and x2J′′

n(휇x) + xJ′ n(휇x) + (휇2x2 − n2)Jn(휇x) = 0

(2) Equations (1) and (2) may be written in the form

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Orthogonality

Proof

x d dx [ x d dx Jn(휆x) ] + (휆2x2 − n2)Jn(휆x) = 0 (3) and x d dx [ x d dx Jn(휇x) ] + (휇2x2 − n2)Jn(휇x) = 0 (4) Multiplying (3) by Jn(휇x)

x

and (4) by Jn(휆x)

x

we get

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Orthogonality

Proof

Jn(휇x) d dx [ x d dx Jn(휆x) ] + 1 x (휆2x2 − n2)Jn(휆x)Jn(휇x) = 0 (5) and Jn(휆x) d dx [ x d dx Jn(휇x) ] + 1 x (휇2x2 − n2)Jn(휇x)Jn(휆x) = 0 (6) Then subtracting, (5) - (6) we get

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Orthogonality

Proof

Jn(휇x) d dx [ x d dx Jn(휆x) ] − Jn(휆x) d dx [ x d dx Jn(휇x) ] + (휆2 − 휇2)xJn(휆x)Jn(휇x) = 0 (7) With some more manipulation, equation (7) may be written as

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Orthogonality

Proof

d dx [ Jn(휇x)x d dx Jn(휆x) ] − d dx [ Jn(휆x)x d dx Jn(휇x) ] + (휆2 − 휇2)xJn(휆x)Jn(휇x) = 0 (8) Finally integrating (8) from 0 to 1 noting that Jn(휆) = Jn(휇) = 0, we get

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Orthogonality

Proof

(휆2 − 휇2) ∫ 1 xJn(휆x)Jn(휇x) dx = 0 And since 휆 ∕= 휇, then we may divide to get the desired result ∫ 1 xJn(휆x)Jn(휇x) dx = 0 (9)

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Orthogonality

Coefficients

Theorem

If 휆 and 휇 are distinct positive roots of Jn(x) = 0 then ∫ 1 xJn(휆x)Jn(휇x) dx = { 0, if 휆 ∕= 휇

1 2J2 n+1(휆),

if 휆 = 휇

▶ ∫ 1 0 xJn(휆x)Jn(휆x) dx = 1 2J2 n+1(휆) ∕= 0

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Orthogonality

Coefficients

Theorem

If 휆 and 휇 are distinct positive roots of Jn(x) = 0 then ∫ 1 xJn(휆x)Jn(휇x) dx = { 0, if 휆 ∕= 휇

1 2J2 n+1(휆),

if 휆 = 휇

▶ ∫ 1 0 xJn(휆x)Jn(휆x) dx = 1 2J2 n+1(휆) ∕= 0 ▶ ∫ 1 0 rJ0(jnr)J0(jnr) dr = 1 2J2 1(jn) ∕= 0

Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Orthogonality

Coefficients

Theorem

If 휆 and 휇 are distinct positive roots of Jn(x) = 0 then ∫ 1 xJn(휆x)Jn(휇x) dx = { 0, if 휆 ∕= 휇

1 2J2 n+1(휆),

if 휆 = 휇

▶ ∫ 1 0 xJn(휆x)Jn(휆x) dx = 1 2J2 n+1(휆) ∕= 0 ▶ ∫ 1 0 rJ0(jnr)J0(jnr) dr = 1 2J2 1(jn) ∕= 0 ▶ An = ∫ 1

0 rJ0(jnr)f(r) dr

∫ 1

0 rJ0(jnr)J0(jnr) dr Karoji, Tsai, Weyrens Bessel Functions

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Introduction Application Properties Orthogonality

Thank You!

Karoji, Tsai, Weyrens Bessel Functions