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Introduction in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates Separation of Variables Bessel Equations Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College of Engineering and Science

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SLIDE 1

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separation of Variables – Bessel Equations

Bernd Schr¨

  • der

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

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SLIDE 2

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separation of Variables

  • 1. Solution technique for partial differential equations.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

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SLIDE 3

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separation of Variables

  • 1. 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).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

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SLIDE 4

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separation of Variables

  • 1. 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 with several ordinary differential equations.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

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SLIDE 5

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separation of Variables

  • 1. 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 with several ordinary differential equations.

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

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

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SLIDE 6

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separation of Variables

  • 1. 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 with several 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 useful results for the partial differential equations.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

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SLIDE 7

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separation of Variables

  • 1. 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 with several 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 useful results for the partial differential equations.

  • 6. Some very powerful and deep theorems can be used to

formally justify the approach for many equations involving the Laplace operator.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

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SLIDE 8

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

How Deep?

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

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SLIDE 9

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

How Deep?

plus about 200 pages of really awesome functional analysis.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

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SLIDE 10

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

The Equation ∆u = k∂u ∂t

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

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SLIDE 11

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

The Equation ∆u = k∂u ∂t

  • 1. It’s the heat equation.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

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SLIDE 12

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.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

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SLIDE 13

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 Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

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SLIDE 14

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u ∂t (Temporal Part)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

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SLIDE 15

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u ∂t (Temporal Part)

∂ 2u ∂r2 + 1 r ∂u ∂r + 1 r2 ∂ 2u ∂θ 2 = k∂u ∂t

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

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SLIDE 16

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u ∂t (Temporal Part)

∂ 2u ∂r2 + 1 r ∂u ∂r + 1 r2 ∂ 2u ∂θ 2 = k∂u ∂t u(r,θ,t)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

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SLIDE 17

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u ∂t (Temporal Part)

∂ 2u ∂r2 + 1 r ∂u ∂r + 1 r2 ∂ 2u ∂θ 2 = k∂u ∂t u(r,θ,t) = R(r)D(Θ)T(t)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

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SLIDE 18

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u ∂t (Temporal Part)

∂ 2u ∂r2 + 1 r ∂u ∂r + 1 r2 ∂ 2u ∂θ 2 = k∂u ∂t u(r,θ,t) = R(r)D(Θ)T(t) = R·D·T

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

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SLIDE 19

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u ∂t (Temporal Part)

∂ 2u ∂r2 + 1 r ∂u ∂r + 1 r2 ∂ 2u ∂θ 2 = k∂u ∂t u(r,θ,t) = R(r)D(Θ)T(t) = R·D·T ∂ 2 ∂r2RDT + 1 r ∂ ∂rRDT + 1 r2 ∂ 2 ∂θ 2RDT = k ∂ ∂tRDT

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

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SLIDE 20

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u ∂t (Temporal Part)

∂ 2u ∂r2 + 1 r ∂u ∂r + 1 r2 ∂ 2u ∂θ 2 = k∂u ∂t u(r,θ,t) = R(r)D(Θ)T(t) = R·D·T ∂ 2 ∂r2RDT + 1 r ∂ ∂rRDT + 1 r2 ∂ 2 ∂θ 2RDT = k ∂ ∂tRDT R′′DT

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

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SLIDE 21

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u ∂t (Temporal Part)

∂ 2u ∂r2 + 1 r ∂u ∂r + 1 r2 ∂ 2u ∂θ 2 = k∂u ∂t u(r,θ,t) = R(r)D(Θ)T(t) = R·D·T ∂ 2 ∂r2RDT + 1 r ∂ ∂rRDT + 1 r2 ∂ 2 ∂θ 2RDT = k ∂ ∂tRDT R′′DT + 1 r R′DT

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

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SLIDE 22

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u ∂t (Temporal Part)

∂ 2u ∂r2 + 1 r ∂u ∂r + 1 r2 ∂ 2u ∂θ 2 = k∂u ∂t u(r,θ,t) = R(r)D(Θ)T(t) = R·D·T ∂ 2 ∂r2RDT + 1 r ∂ ∂rRDT + 1 r2 ∂ 2 ∂θ 2RDT = k ∂ ∂tRDT R′′DT + 1 r R′DT + 1 r2RD′′T

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

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SLIDE 23

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u ∂t (Temporal Part)

∂ 2u ∂r2 + 1 r ∂u ∂r + 1 r2 ∂ 2u ∂θ 2 = k∂u ∂t u(r,θ,t) = R(r)D(Θ)T(t) = R·D·T ∂ 2 ∂r2RDT + 1 r ∂ ∂rRDT + 1 r2 ∂ 2 ∂θ 2RDT = k ∂ ∂tRDT R′′DT + 1 r R′DT + 1 r2RD′′T = kRDT′

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

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SLIDE 24

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u ∂t (Temporal Part)

∂ 2u ∂r2 + 1 r ∂u ∂r + 1 r2 ∂ 2u ∂θ 2 = k∂u ∂t u(r,θ,t) = R(r)D(Θ)T(t) = R·D·T ∂ 2 ∂r2RDT + 1 r ∂ ∂rRDT + 1 r2 ∂ 2 ∂θ 2RDT = k ∂ ∂tRDT R′′DT + 1 r R′DT + 1 r2RD′′T = kRDT′ R′′D+ 1

r R′D+ 1 r2RD′′

RD

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

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SLIDE 25

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u ∂t (Temporal Part)

∂ 2u ∂r2 + 1 r ∂u ∂r + 1 r2 ∂ 2u ∂θ 2 = k∂u ∂t u(r,θ,t) = R(r)D(Θ)T(t) = R·D·T ∂ 2 ∂r2RDT + 1 r ∂ ∂rRDT + 1 r2 ∂ 2 ∂θ 2RDT = k ∂ ∂tRDT R′′DT + 1 r R′DT + 1 r2RD′′T = kRDT′ R′′D+ 1

r R′D+ 1 r2RD′′

RD = kT′ T

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

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SLIDE 26

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u ∂t (Temporal Part)

∂ 2u ∂r2 + 1 r ∂u ∂r + 1 r2 ∂ 2u ∂θ 2 = k∂u ∂t u(r,θ,t) = R(r)D(Θ)T(t) = R·D·T ∂ 2 ∂r2RDT + 1 r ∂ ∂rRDT + 1 r2 ∂ 2 ∂θ 2RDT = k ∂ ∂tRDT R′′DT + 1 r R′DT + 1 r2RD′′T = kRDT′ R′′D+ 1

r R′D+ 1 r2RD′′

RD = kT′ T = −λ 2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

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SLIDE 27

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u ∂t (Temporal Part)

∂ 2u ∂r2 + 1 r ∂u ∂r + 1 r2 ∂ 2u ∂θ 2 = k∂u ∂t u(r,θ,t) = R(r)D(Θ)T(t) = R·D·T ∂ 2 ∂r2RDT + 1 r ∂ ∂rRDT + 1 r2 ∂ 2 ∂θ 2RDT = k ∂ ∂tRDT R′′DT + 1 r R′DT + 1 r2RD′′T = kRDT′ R′′D+ 1

r R′D+ 1 r2RD′′

RD = kT′ T = −λ 2 Constant is negative,

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

slide-28
SLIDE 28

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u ∂t (Temporal Part)

∂ 2u ∂r2 + 1 r ∂u ∂r + 1 r2 ∂ 2u ∂θ 2 = k∂u ∂t u(r,θ,t) = R(r)D(Θ)T(t) = R·D·T ∂ 2 ∂r2RDT + 1 r ∂ ∂rRDT + 1 r2 ∂ 2 ∂θ 2RDT = k ∂ ∂tRDT R′′DT + 1 r R′DT + 1 r2RD′′T = kRDT′ R′′D+ 1

r R′D+ 1 r2RD′′

RD = kT′ T = −λ 2 Constant is negative, because T′ T = c k gives T = ae

c kt.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

slide-29
SLIDE 29

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u ∂t (Temporal Part)

∂ 2u ∂r2 + 1 r ∂u ∂r + 1 r2 ∂ 2u ∂θ 2 = k∂u ∂t u(r,θ,t) = R(r)D(Θ)T(t) = R·D·T ∂ 2 ∂r2RDT + 1 r ∂ ∂rRDT + 1 r2 ∂ 2 ∂θ 2RDT = k ∂ ∂tRDT R′′DT + 1 r R′DT + 1 r2RD′′T = kRDT′ R′′D+ 1

r R′D+ 1 r2RD′′

RD = kT′ T = −λ 2 Constant is negative, because T′ T = c k gives T = ae

c

  • kt. Now

k > 0 forces c < 0,

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

slide-30
SLIDE 30

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u ∂t (Temporal Part)

∂ 2u ∂r2 + 1 r ∂u ∂r + 1 r2 ∂ 2u ∂θ 2 = k∂u ∂t u(r,θ,t) = R(r)D(Θ)T(t) = R·D·T ∂ 2 ∂r2RDT + 1 r ∂ ∂rRDT + 1 r2 ∂ 2 ∂θ 2RDT = k ∂ ∂tRDT R′′DT + 1 r R′DT + 1 r2RD′′T = kRDT′ R′′D+ 1

r R′D+ 1 r2RD′′

RD = kT′ T = −λ 2 Constant is negative, because T′ T = c k gives T = ae

c

  • kt. Now

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

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

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SLIDE 31

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u ∂t (Azimuthal Part)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

slide-32
SLIDE 32

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u ∂t (Azimuthal Part)

R′′D+ 1

r R′D+ 1 r2RD′′

RD = −λ 2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

slide-33
SLIDE 33

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u ∂t (Azimuthal Part)

R′′D+ 1

r R′D+ 1 r2RD′′

RD = −λ 2 R′′D+ 1

r R′D

RD +λ 2 = −

1 r2RD′′

RD

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

slide-34
SLIDE 34

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u ∂t (Azimuthal Part)

R′′D+ 1

r R′D+ 1 r2RD′′

RD = −λ 2 R′′D+ 1

r R′D

RD +λ 2 = −

1 r2RD′′

RD r2R′′ +rR′ R +r2λ 2 = −D′′ D

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

slide-35
SLIDE 35

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u ∂t (Azimuthal Part)

R′′D+ 1

r R′D+ 1 r2RD′′

RD = −λ 2 R′′D+ 1

r R′D

RD +λ 2 = −

1 r2RD′′

RD r2R′′ +rR′ R +r2λ 2 = −D′′ D = ν2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

slide-36
SLIDE 36

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u ∂t (Azimuthal Part)

R′′D+ 1

r R′D+ 1 r2RD′′

RD = −λ 2 R′′D+ 1

r R′D

RD +λ 2 = −

1 r2RD′′

RD r2R′′ +rR′ R +r2λ 2 = −D′′ D = ν2 The constant is nonnegative:

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

slide-37
SLIDE 37

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u ∂t (Azimuthal Part)

R′′D+ 1

r R′D+ 1 r2RD′′

RD = −λ 2 R′′D+ 1

r R′D

RD +λ 2 = −

1 r2RD′′

RD r2R′′ +rR′ R +r2λ 2 = −D′′ D = ν2 The constant is nonnegative: −D′′ D = c leads to D′′ +cD = 0.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

slide-38
SLIDE 38

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u ∂t (Azimuthal Part)

R′′D+ 1

r R′D+ 1 r2RD′′

RD = −λ 2 R′′D+ 1

r R′D

RD +λ 2 = −

1 r2RD′′

RD r2R′′ +rR′ R +r2λ 2 = −D′′ D = ν2 The constant is nonnegative: −D′′ D = c leads to D′′ +cD = 0. But D must be 2π-periodic.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

slide-39
SLIDE 39

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u ∂t (Azimuthal Part)

R′′D+ 1

r R′D+ 1 r2RD′′

RD = −λ 2 R′′D+ 1

r R′D

RD +λ 2 = −

1 r2RD′′

RD r2R′′ +rR′ R +r2λ 2 = −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 nonperiodic exponential solutions.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

slide-40
SLIDE 40

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u ∂t (Azimuthal Part)

R′′D+ 1

r R′D+ 1 r2RD′′

RD = −λ 2 R′′D+ 1

r R′D

RD +λ 2 = −

1 r2RD′′

RD r2R′′ +rR′ R +r2λ 2 = −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 nonperiodic exponential solutions. Thus c = ν2, where ν is a nonnegative integer,

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

slide-41
SLIDE 41

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u ∂t (Azimuthal Part)

R′′D+ 1

r R′D+ 1 r2RD′′

RD = −λ 2 R′′D+ 1

r R′D

RD +λ 2 = −

1 r2RD′′

RD r2R′′ +rR′ R +r2λ 2 = −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 nonperiodic exponential solutions. Thus c = ν2, where ν is a nonnegative integer, because then D(θ) = c1 cos(νθ)+c2 sin(νθ), which is 2π-periodic.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

slide-42
SLIDE 42

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u ∂t (Radial Part)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

slide-43
SLIDE 43

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u ∂t (Radial Part)

r2R′′ +rR′ R +r2λ 2 = ν2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

slide-44
SLIDE 44

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u ∂t (Radial Part)

r2R′′ +rR′ R +r2λ 2 = ν2 r2R′′ +rR′ +r2λ 2R = ν2R

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

slide-45
SLIDE 45

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u ∂t (Radial Part)

r2R′′ +rR′ R +r2λ 2 = ν2 r2R′′ +rR′ +r2λ 2R = ν2R r2R′′ +rR′ +

  • λ 2r2 −ν2

R =

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

slide-46
SLIDE 46

logo1 Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates

Separating the Equation ∆u = k∂u ∂t (Radial Part)

r2R′′ +rR′ R +r2λ 2 = ν2 r2R′′ +rR′ +r2λ 2R = ν2R r2R′′ +rR′ +

  • λ 2r2 −ν2

R = and the last equation is called the parametric Bessel equation.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations