logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separation of Variables – BesselEquations

Bernd Schroder

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separation of Variables1. Solution technique for partial differential equations.

2. If the unknown function u depends on variables r,θ , t, weassume there is a solution of the form u = R(r)D(θ)T(t).

3. The special form of this solution function allows us toreplace the original partial differential equation withseveral ordinary differential equations.

4. Key step: If f (t) = g(r,θ), then f and g must be constant.5. Solutions of the ordinary differential equations we obtain

must typically be processed some more to give usefulresults for the partial differential equations.

6. Some very powerful and deep theorems can be used toformally justify the approach for many equations involvingthe Laplace operator.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separation of Variables1. Solution technique for partial differential equations.2. If the unknown function u depends on variables r,θ , t, we

assume there is a solution of the form u = R(r)D(θ)T(t).

3. The special form of this solution function allows us toreplace the original partial differential equation withseveral ordinary differential equations.

4. Key step: If f (t) = g(r,θ), then f and g must be constant.5. Solutions of the ordinary differential equations we obtain

must typically be processed some more to give usefulresults for the partial differential equations.

6. Some very powerful and deep theorems can be used toformally justify the approach for many equations involvingthe Laplace operator.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separation of Variables1. Solution technique for partial differential equations.2. If the unknown function u depends on variables r,θ , t, we

assume there is a solution of the form u = R(r)D(θ)T(t).3. The special form of this solution function allows us to

replace the original partial differential equation withseveral ordinary differential equations.

4. Key step: If f (t) = g(r,θ), then f and g must be constant.5. Solutions of the ordinary differential equations we obtain

must typically be processed some more to give usefulresults for the partial differential equations.

6. Some very powerful and deep theorems can be used toformally justify the approach for many equations involvingthe Laplace operator.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separation of Variables1. Solution technique for partial differential equations.2. If the unknown function u depends on variables r,θ , t, we

assume there is a solution of the form u = R(r)D(θ)T(t).3. The special form of this solution function allows us to

replace the original partial differential equation withseveral ordinary differential equations.

4. Key step: If f (t) = g(r,θ), then f and g must be constant.

5. Solutions of the ordinary differential equations we obtainmust typically be processed some more to give usefulresults for the partial differential equations.

6. Some very powerful and deep theorems can be used toformally justify the approach for many equations involvingthe Laplace operator.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separation of Variables1. Solution technique for partial differential equations.2. If the unknown function u depends on variables r,θ , t, we

assume there is a solution of the form u = R(r)D(θ)T(t).3. The special form of this solution function allows us to

replace the original partial differential equation withseveral ordinary differential equations.

4. Key step: If f (t) = g(r,θ), then f and g must be constant.5. Solutions of the ordinary differential equations we obtain

must typically be processed some more to give usefulresults for the partial differential equations.

6. Some very powerful and deep theorems can be used toformally justify the approach for many equations involvingthe Laplace operator.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separation of Variables1. Solution technique for partial differential equations.2. If the unknown function u depends on variables r,θ , t, we

assume there is a solution of the form u = R(r)D(θ)T(t).3. The special form of this solution function allows us to

replace the original partial differential equation withseveral ordinary differential equations.

4. Key step: If f (t) = g(r,θ), then f and g must be constant.5. Solutions of the ordinary differential equations we obtain

must typically be processed some more to give usefulresults for the partial differential equations.

6. Some very powerful and deep theorems can be used toformally justify the approach for many equations involvingthe Laplace operator.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

How Deep?

plus about 200 pages of reallyawesome functional analysis.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

How Deep?

plus about 200 pages of reallyawesome functional analysis.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

The Equation ∆u = k∂u∂ t

1. It’s the heat equation.2. Consideration in two dimensions may mean we analyze

heat transfer in a thin sheet of metal.3. It may also mean that we are working with a cylindrical

geometry in which there is no variation in the z-direction.(Heating a metal cylinder in a water bath.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

The Equation ∆u = k∂u∂ t

1. It’s the heat equation.

2. Consideration in two dimensions may mean we analyzeheat transfer in a thin sheet of metal.

3. It may also mean that we are working with a cylindricalgeometry in which there is no variation in the z-direction.(Heating a metal cylinder in a water bath.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

The Equation ∆u = k∂u∂ t

1. It’s the heat equation.2. Consideration in two dimensions may mean we analyze

heat transfer in a thin sheet of metal.

3. It may also mean that we are working with a cylindricalgeometry in which there is no variation in the z-direction.(Heating a metal cylinder in a water bath.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

The Equation ∆u = k∂u∂ t

1. It’s the heat equation.2. Consideration in two dimensions may mean we analyze

heat transfer in a thin sheet of metal.3. It may also mean that we are working with a cylindrical

geometry in which there is no variation in the z-direction.(Heating a metal cylinder in a water bath.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u∂ t

(Temporal Part)

∂ 2u∂ r2 +

1r

∂u∂ r

+1r2

∂ 2u∂θ 2 = k

∂u∂ t

u(r,θ , t) = R(r)D(Θ)T(t) = R ·D ·T∂ 2

∂ r2 RDT +1r

∂

∂ rRDT +

1r2

∂ 2

∂θ 2 RDT = k∂

∂ tRDT

R′′DT +1r

R′DT +1r2 RD′′T = kRDT ′

R′′D+ 1r R′D+ 1

r2 RD′′

RD= k

T ′

T= −λ

2

Constant is negative, becauseT ′

T=

ck

gives T = aeck t. Now

k > 0 forces c < 0, otherwise temperature would increaseexponentially with no energy input.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u∂ t

(Temporal Part)

∂ 2u∂ r2 +

1r

∂u∂ r

+1r2

∂ 2u∂θ 2 = k

∂u∂ t

u(r,θ , t) = R(r)D(Θ)T(t) = R ·D ·T∂ 2

∂ r2 RDT +1r

∂

∂ rRDT +

1r2

∂ 2

∂θ 2 RDT = k∂

∂ tRDT

R′′DT +1r

R′DT +1r2 RD′′T = kRDT ′

R′′D+ 1r R′D+ 1

r2 RD′′

RD= k

T ′

T= −λ

2

Constant is negative, becauseT ′

T=

ck

gives T = aeck t. Now

k > 0 forces c < 0, otherwise temperature would increaseexponentially with no energy input.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u∂ t

(Temporal Part)

∂ 2u∂ r2 +

1r

∂u∂ r

+1r2

∂ 2u∂θ 2 = k

∂u∂ t

u(r,θ , t)

= R(r)D(Θ)T(t) = R ·D ·T∂ 2

∂ r2 RDT +1r

∂

∂ rRDT +

1r2

∂ 2

∂θ 2 RDT = k∂

∂ tRDT

R′′DT +1r

R′DT +1r2 RD′′T = kRDT ′

R′′D+ 1r R′D+ 1

r2 RD′′

RD= k

T ′

T= −λ

2

Constant is negative, becauseT ′

T=

ck

gives T = aeck t. Now

k > 0 forces c < 0, otherwise temperature would increaseexponentially with no energy input.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u∂ t

(Temporal Part)

∂ 2u∂ r2 +

1r

∂u∂ r

+1r2

∂ 2u∂θ 2 = k

∂u∂ t

u(r,θ , t) = R(r)D(Θ)T(t)

= R ·D ·T∂ 2

∂ r2 RDT +1r

∂

∂ rRDT +

1r2

∂ 2

∂θ 2 RDT = k∂

∂ tRDT

R′′DT +1r

R′DT +1r2 RD′′T = kRDT ′

R′′D+ 1r R′D+ 1

r2 RD′′

RD= k

T ′

T= −λ

2

Constant is negative, becauseT ′

T=

ck

gives T = aeck t. Now

k > 0 forces c < 0, otherwise temperature would increaseexponentially with no energy input.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u∂ t

(Temporal Part)

∂ 2u∂ r2 +

1r

∂u∂ r

+1r2

∂ 2u∂θ 2 = k

∂u∂ t

u(r,θ , t) = R(r)D(Θ)T(t) = R ·D ·T

∂ 2

∂ r2 RDT +1r

∂

∂ rRDT +

1r2

∂ 2

∂θ 2 RDT = k∂

∂ tRDT

R′′DT +1r

R′DT +1r2 RD′′T = kRDT ′

R′′D+ 1r R′D+ 1

r2 RD′′

RD= k

T ′

T= −λ

2

Constant is negative, becauseT ′

T=

ck

gives T = aeck t. Now

k > 0 forces c < 0, otherwise temperature would increaseexponentially with no energy input.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u∂ t

(Temporal Part)

∂ 2u∂ r2 +

1r

∂u∂ r

+1r2

∂ 2u∂θ 2 = k

∂u∂ t

u(r,θ , t) = R(r)D(Θ)T(t) = R ·D ·T∂ 2

∂ r2 RDT +1r

∂

∂ rRDT +

1r2

∂ 2

∂θ 2 RDT = k∂

∂ tRDT

R′′DT +1r

R′DT +1r2 RD′′T = kRDT ′

R′′D+ 1r R′D+ 1

r2 RD′′

RD= k

T ′

T= −λ

2

Constant is negative, becauseT ′

T=

ck

gives T = aeck t. Now

k > 0 forces c < 0, otherwise temperature would increaseexponentially with no energy input.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u∂ t

(Temporal Part)

∂ 2u∂ r2 +

1r

∂u∂ r

+1r2

∂ 2u∂θ 2 = k

∂u∂ t

u(r,θ , t) = R(r)D(Θ)T(t) = R ·D ·T∂ 2

∂ r2 RDT +1r

∂

∂ rRDT +

1r2

∂ 2

∂θ 2 RDT = k∂

∂ tRDT

R′′DT

+1r

R′DT +1r2 RD′′T = kRDT ′

R′′D+ 1r R′D+ 1

r2 RD′′

RD= k

T ′

T= −λ

2

Constant is negative, becauseT ′

T=

ck

gives T = aeck t. Now

k > 0 forces c < 0, otherwise temperature would increaseexponentially with no energy input.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u∂ t

(Temporal Part)

∂ 2u∂ r2 +

1r

∂u∂ r

+1r2

∂ 2u∂θ 2 = k

∂u∂ t

u(r,θ , t) = R(r)D(Θ)T(t) = R ·D ·T∂ 2

∂ r2 RDT +1r

∂

∂ rRDT +

1r2

∂ 2

∂θ 2 RDT = k∂

∂ tRDT

R′′DT +1r

R′DT

+1r2 RD′′T = kRDT ′

R′′D+ 1r R′D+ 1

r2 RD′′

RD= k

T ′

T= −λ

2

Constant is negative, becauseT ′

T=

ck

gives T = aeck t. Now

k > 0 forces c < 0, otherwise temperature would increaseexponentially with no energy input.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u∂ t

(Temporal Part)

∂ 2u∂ r2 +

1r

∂u∂ r

+1r2

∂ 2u∂θ 2 = k

∂u∂ t

u(r,θ , t) = R(r)D(Θ)T(t) = R ·D ·T∂ 2

∂ r2 RDT +1r

∂

∂ rRDT +

1r2

∂ 2

∂θ 2 RDT = k∂

∂ tRDT

R′′DT +1r

R′DT +1r2 RD′′T

= kRDT ′

R′′D+ 1r R′D+ 1

r2 RD′′

RD= k

T ′

T= −λ

2

Constant is negative, becauseT ′

T=

ck

gives T = aeck t. Now

k > 0 forces c < 0, otherwise temperature would increaseexponentially with no energy input.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u∂ t

(Temporal Part)

∂ 2u∂ r2 +

1r

∂u∂ r

+1r2

∂ 2u∂θ 2 = k

∂u∂ t

u(r,θ , t) = R(r)D(Θ)T(t) = R ·D ·T∂ 2

∂ r2 RDT +1r

∂

∂ rRDT +

1r2

∂ 2

∂θ 2 RDT = k∂

∂ tRDT

R′′DT +1r

R′DT +1r2 RD′′T = kRDT ′

R′′D+ 1r R′D+ 1

r2 RD′′

RD= k

T ′

T= −λ

2

Constant is negative, becauseT ′

T=

ck

gives T = aeck t. Now

k > 0 forces c < 0, otherwise temperature would increaseexponentially with no energy input.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u∂ t

(Temporal Part)

∂ 2u∂ r2 +

1r

∂u∂ r

+1r2

∂ 2u∂θ 2 = k

∂u∂ t

u(r,θ , t) = R(r)D(Θ)T(t) = R ·D ·T∂ 2

∂ r2 RDT +1r

∂

∂ rRDT +

1r2

∂ 2

∂θ 2 RDT = k∂

∂ tRDT

R′′DT +1r

R′DT +1r2 RD′′T = kRDT ′

R′′D+ 1r R′D+ 1

r2 RD′′

RD

= kT ′

T= −λ

2

Constant is negative, becauseT ′

T=

ck

gives T = aeck t. Now

k > 0 forces c < 0, otherwise temperature would increaseexponentially with no energy input.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u∂ t

(Temporal Part)

∂ 2u∂ r2 +

1r

∂u∂ r

+1r2

∂ 2u∂θ 2 = k

∂u∂ t

u(r,θ , t) = R(r)D(Θ)T(t) = R ·D ·T∂ 2

∂ r2 RDT +1r

∂

∂ rRDT +

1r2

∂ 2

∂θ 2 RDT = k∂

∂ tRDT

R′′DT +1r

R′DT +1r2 RD′′T = kRDT ′

R′′D+ 1r R′D+ 1

r2 RD′′

RD= k

T ′

T

= −λ2

Constant is negative, becauseT ′

T=

ck

gives T = aeck t. Now

k > 0 forces c < 0, otherwise temperature would increaseexponentially with no energy input.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u∂ t

(Temporal Part)

∂ 2u∂ r2 +

1r

∂u∂ r

+1r2

∂ 2u∂θ 2 = k

∂u∂ t

u(r,θ , t) = R(r)D(Θ)T(t) = R ·D ·T∂ 2

∂ r2 RDT +1r

∂

∂ rRDT +

1r2

∂ 2

∂θ 2 RDT = k∂

∂ tRDT

R′′DT +1r

R′DT +1r2 RD′′T = kRDT ′

R′′D+ 1r R′D+ 1

r2 RD′′

RD= k

T ′

T= −λ

2

Constant is negative, becauseT ′

T=

ck

gives T = aeck t. Now

k > 0 forces c < 0, otherwise temperature would increaseexponentially with no energy input.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u∂ t

(Temporal Part)

∂ 2u∂ r2 +

1r

∂u∂ r

+1r2

∂ 2u∂θ 2 = k

∂u∂ t

u(r,θ , t) = R(r)D(Θ)T(t) = R ·D ·T∂ 2

∂ r2 RDT +1r

∂

∂ rRDT +

1r2

∂ 2

∂θ 2 RDT = k∂

∂ tRDT

R′′DT +1r

R′DT +1r2 RD′′T = kRDT ′

R′′D+ 1r R′D+ 1

r2 RD′′

RD= k

T ′

T= −λ

2

Constant is negative,

becauseT ′

T=

ck

gives T = aeck t. Now

k > 0 forces c < 0, otherwise temperature would increaseexponentially with no energy input.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u∂ t

(Temporal Part)

∂ 2u∂ r2 +

1r

∂u∂ r

+1r2

∂ 2u∂θ 2 = k

∂u∂ t

u(r,θ , t) = R(r)D(Θ)T(t) = R ·D ·T∂ 2

∂ r2 RDT +1r

∂

∂ rRDT +

1r2

∂ 2

∂θ 2 RDT = k∂

∂ tRDT

R′′DT +1r

R′DT +1r2 RD′′T = kRDT ′

R′′D+ 1r R′D+ 1

r2 RD′′

RD= k

T ′

T= −λ

2

Constant is negative, becauseT ′

T=

ck

gives T = aeck t.

Nowk > 0 forces c < 0, otherwise temperature would increaseexponentially with no energy input.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u∂ t

(Temporal Part)

∂ 2u∂ r2 +

1r

∂u∂ r

+1r2

∂ 2u∂θ 2 = k

∂u∂ t

u(r,θ , t) = R(r)D(Θ)T(t) = R ·D ·T∂ 2

∂ r2 RDT +1r

∂

∂ rRDT +

1r2

∂ 2

∂θ 2 RDT = k∂

∂ tRDT

R′′DT +1r

R′DT +1r2 RD′′T = kRDT ′

R′′D+ 1r R′D+ 1

r2 RD′′

RD= k

T ′

T= −λ

2

Constant is negative, becauseT ′

T=

ck

gives T = aeck t. Now

k > 0 forces c < 0,

otherwise temperature would increaseexponentially with no energy input.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u∂ t

(Temporal Part)

∂ 2u∂ r2 +

1r

∂u∂ r

+1r2

∂ 2u∂θ 2 = k

∂u∂ t

u(r,θ , t) = R(r)D(Θ)T(t) = R ·D ·T∂ 2

∂ r2 RDT +1r

∂

∂ rRDT +

1r2

∂ 2

∂θ 2 RDT = k∂

∂ tRDT

R′′DT +1r

R′DT +1r2 RD′′T = kRDT ′

R′′D+ 1r R′D+ 1

r2 RD′′

RD= k

T ′

T= −λ

2

Constant is negative, becauseT ′

T=

ck

gives T = aeck t. Now

k > 0 forces c < 0, otherwise temperature would increaseexponentially with no energy input.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u∂ t

(Azimuthal Part)

R′′D+ 1r R′D+ 1

r2 RD′′

RD= −λ

2

R′′D+ 1r R′D

RD+λ

2 = −1r2 RD′′

RDr2R′′ + rR′

R+ r2

λ2 = −D′′

D= ν

2

The constant is nonnegative: −D′′

D= c leads to D′′ + cD = 0.

But D must be 2π-periodic. For negative c we get nonperiodicexponential solutions. Thus c = ν2, where ν is a nonnegativeinteger, because then D(θ) = c1 cos(νθ)+ c2 sin(νθ), which is2π-periodic.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u∂ t

(Azimuthal Part)

R′′D+ 1r R′D+ 1

r2 RD′′

RD= −λ

2

R′′D+ 1r R′D

RD+λ

2 = −1r2 RD′′

RDr2R′′ + rR′

R+ r2

λ2 = −D′′

D= ν

2

The constant is nonnegative: −D′′

D= c leads to D′′ + cD = 0.

But D must be 2π-periodic. For negative c we get nonperiodicexponential solutions. Thus c = ν2, where ν is a nonnegativeinteger, because then D(θ) = c1 cos(νθ)+ c2 sin(νθ), which is2π-periodic.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u∂ t

(Azimuthal Part)

R′′D+ 1r R′D+ 1

r2 RD′′

RD= −λ

2

R′′D+ 1r R′D

RD+λ

2 = −1r2 RD′′

RD

r2R′′ + rR′

R+ r2

λ2 = −D′′

D= ν

2

The constant is nonnegative: −D′′

D= c leads to D′′ + cD = 0.

But D must be 2π-periodic. For negative c we get nonperiodicexponential solutions. Thus c = ν2, where ν is a nonnegativeinteger, because then D(θ) = c1 cos(νθ)+ c2 sin(νθ), which is2π-periodic.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u∂ t

(Azimuthal Part)

R′′D+ 1r R′D+ 1

r2 RD′′

RD= −λ

2

R′′D+ 1r R′D

RD+λ

2 = −1r2 RD′′

RDr2R′′ + rR′

R+ r2

λ2 = −D′′

D

= ν2

The constant is nonnegative: −D′′

D= c leads to D′′ + cD = 0.

But D must be 2π-periodic. For negative c we get nonperiodicexponential solutions. Thus c = ν2, where ν is a nonnegativeinteger, because then D(θ) = c1 cos(νθ)+ c2 sin(νθ), which is2π-periodic.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u∂ t

(Azimuthal Part)

R′′D+ 1r R′D+ 1

r2 RD′′

RD= −λ

2

R′′D+ 1r R′D

RD+λ

2 = −1r2 RD′′

RDr2R′′ + rR′

R+ r2

λ2 = −D′′

D= ν

2

The constant is nonnegative: −D′′

D= c leads to D′′ + cD = 0.

But D must be 2π-periodic. For negative c we get nonperiodicexponential solutions. Thus c = ν2, where ν is a nonnegativeinteger, because then D(θ) = c1 cos(νθ)+ c2 sin(νθ), which is2π-periodic.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u∂ t

(Azimuthal Part)

R′′D+ 1r R′D+ 1

r2 RD′′

RD= −λ

2

R′′D+ 1r R′D

RD+λ

2 = −1r2 RD′′

RDr2R′′ + rR′

R+ r2

λ2 = −D′′

D= ν

2

The constant is nonnegative:

−D′′

D= c leads to D′′ + cD = 0.

But D must be 2π-periodic. For negative c we get nonperiodicexponential solutions. Thus c = ν2, where ν is a nonnegativeinteger, because then D(θ) = c1 cos(νθ)+ c2 sin(νθ), which is2π-periodic.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u∂ t

(Azimuthal Part)

R′′D+ 1r R′D+ 1

r2 RD′′

RD= −λ

2

R′′D+ 1r R′D

RD+λ

2 = −1r2 RD′′

RDr2R′′ + rR′

R+ r2

λ2 = −D′′

D= ν

2

The constant is nonnegative: −D′′

D= c leads to D′′ + cD = 0.

But D must be 2π-periodic. For negative c we get nonperiodicexponential solutions. Thus c = ν2, where ν is a nonnegativeinteger, because then D(θ) = c1 cos(νθ)+ c2 sin(νθ), which is2π-periodic.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u∂ t

(Azimuthal Part)

R′′D+ 1r R′D+ 1

r2 RD′′

RD= −λ

2

R′′D+ 1r R′D

RD+λ

2 = −1r2 RD′′

RDr2R′′ + rR′

R+ r2

λ2 = −D′′

D= ν

2

The constant is nonnegative: −D′′

D= c leads to D′′ + cD = 0.

But D must be 2π-periodic.

For negative c we get nonperiodicexponential solutions. Thus c = ν2, where ν is a nonnegativeinteger, because then D(θ) = c1 cos(νθ)+ c2 sin(νθ), which is2π-periodic.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u∂ t

(Azimuthal Part)

R′′D+ 1r R′D+ 1

r2 RD′′

RD= −λ

2

R′′D+ 1r R′D

RD+λ

2 = −1r2 RD′′

RDr2R′′ + rR′

R+ r2

λ2 = −D′′

D= ν

2

The constant is nonnegative: −D′′

D= c leads to D′′ + cD = 0.

But D must be 2π-periodic. For negative c we get nonperiodicexponential solutions.

Thus c = ν2, where ν is a nonnegativeinteger, because then D(θ) = c1 cos(νθ)+ c2 sin(νθ), which is2π-periodic.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u∂ t

(Azimuthal Part)

R′′D+ 1r R′D+ 1

r2 RD′′

RD= −λ

2

R′′D+ 1r R′D

RD+λ

2 = −1r2 RD′′

RDr2R′′ + rR′

R+ r2

λ2 = −D′′

D= ν

2

The constant is nonnegative: −D′′

D= c leads to D′′ + cD = 0.

But D must be 2π-periodic. For negative c we get nonperiodicexponential solutions. Thus c = ν2, where ν is a nonnegativeinteger,

because then D(θ) = c1 cos(νθ)+ c2 sin(νθ), which is2π-periodic.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u∂ t

(Azimuthal Part)

R′′D+ 1r R′D+ 1

r2 RD′′

RD= −λ

2

R′′D+ 1r R′D

RD+λ

2 = −1r2 RD′′

RDr2R′′ + rR′

R+ r2

λ2 = −D′′

D= ν

2

The constant is nonnegative: −D′′

D= c leads to D′′ + cD = 0.

But D must be 2π-periodic. For negative c we get nonperiodicexponential solutions. Thus c = ν2, where ν is a nonnegativeinteger, because then D(θ) = c1 cos(νθ)+ c2 sin(νθ), which is2π-periodic.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u∂ t

(Radial Part)

r2R′′ + rR′

R+ r2

λ2 = ν

2

r2R′′ + rR′ + r2λ

2R = ν2R

r2R′′ + rR′ +(

λ2r2 −ν

2)

R = 0

and the last equation is called the parametric Bessel equation.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u∂ t

(Radial Part)

r2R′′ + rR′

R+ r2

λ2 = ν

2

r2R′′ + rR′ + r2λ

2R = ν2R

r2R′′ + rR′ +(

λ2r2 −ν

2)

R = 0

and the last equation is called the parametric Bessel equation.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u∂ t

(Radial Part)

r2R′′ + rR′

R+ r2

λ2 = ν

2

r2R′′ + rR′ + r2λ

2R = ν2R

r2R′′ + rR′ +(

λ2r2 −ν

2)

R = 0

and the last equation is called the parametric Bessel equation.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u∂ t

(Radial Part)

r2R′′ + rR′

R+ r2

λ2 = ν

2

r2R′′ + rR′ + r2λ

2R = ν2R

r2R′′ + rR′ +(

λ2r2 −ν

2)

R = 0

and the last equation is called the parametric Bessel equation.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations

logo1

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u∂ t

(Radial Part)

r2R′′ + rR′

R+ r2

λ2 = ν

2

r2R′′ + rR′ + r2λ

2R = ν2R

r2R′′ + rR′ +(

λ2r2 −ν

2)

R = 0

and the last equation is called the parametric Bessel equation.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Bessel Equations