Package 'kelvin'

Fundamental and equivalent solutions to the Kelvin differential equation using Bessel functions

Description

The functions here use Bessel functions to calculate the analytic solutions to the Kelvin differential equation, namely the fundamental (Be) and equivalent (Ke) complex functions.

Details

The complex second-order ordinary differential equation, known as the Kelvin differential equation, is defined as

$x^2 \ddot{y} + x \dot{y} - \left(i x^2 + \nu^2\right) y = 0$

and has a suite of complex solutions. One set of solutions, $\mathcal{B}_\nu$, is defined in the following manner:

$\mathcal{B}_\nu \equiv \mathrm{Ber}_\nu (x) + i \mathrm{Bei}_\nu (x)$

$= J_\nu \left(x \cdot \exp(3 \pi i / 4)\right)$

$= \exp(\nu \pi i) \cdot J_\nu \left(x \cdot \exp(-\pi i / 4)\right)$

$= \exp(\nu \pi i / 2) \cdot I_\nu \left(x \cdot \exp(\pi i / 4)\right)$

$= \exp(3 \nu \pi i / 2) \cdot I_\nu \left(x \cdot \exp(-3 \pi i / 4)\right)$

where $J_\nu$ is a Bessel function of the first kind, and $I_\nu$ is a modified Bessel function of the first kind.

Similarly, the complementary solutions, $\mathcal{K}_\nu$, are defined as

$\mathcal{K}_\nu \equiv \mathrm{Ker}_\nu (x) + i \mathrm{Kei}_\nu (x)$

$= \exp(- \nu \pi i / 2) \cdot K_\nu \left(x \cdot \exp(\pi i / 4)\right)$

where $K_\nu$ is a modified Bessel function of the second kind.

The relationships between $y$ in the differential equation, and the solutions $\mathcal{B}_\nu$ and $\mathcal{K}_\nu$ are as follows

$y = \mathrm{Ber}_\nu (x) + i \mathrm{Bei}_\nu (x)$

$= \mathrm{Ber}_{-\nu} (x) + i \mathrm{Bei}_{-\nu} (x)$

$= \mathrm{Ker}_\nu (x) + i \mathrm{Kei}_\nu (x)$

$= \mathrm{Ker}_{-\nu} (x) + i \mathrm{Kei}_{-\nu} (x)$

In the case where $\nu=0$, the differential equation reduces to

$x^2 \ddot{y} + x \dot{y} - i x^2y = 0$

which has the set of solutions:

$J_0 \left(i \sqrt{i} \cdot x\right)$

$= J_0 \left(\sqrt{2} \cdot (i-1) \cdot x / 2\right)$

$= \mathrm{Ber}_0 (x) + i \mathrm{Bei}_0 (x) \equiv \mathcal{B}_0$

This package has functions to calculate $\mathcal{B}_\nu$ and $\mathcal{K}_\nu$.

Author(s)

Andrew Barbour <[email protected]>

References

Abramowitz, M. and Stegun, I. A. (Eds.). "Kelvin Functions." $\S 9.9$ in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 379-381, 1972.

Kelvin functions: http://mathworld.wolfram.com/KelvinFunctions.html

Bessel functions: http://mathworld.wolfram.com/BesselFunction.html

See Also

Fundamental solution: Beir

Equivalent solution: Keir

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Fundamental solution to the Kelvin differential equation (J)

Description

This function calculates the complex solution to the Kelvin differential equation using modified Bessel functions of the first kind, specifically those produced by BesselJ.

Usage

Beir(x, ...)

## Default S3 method:
Beir(x, nu. = 0, nSeq. = 1, return.list = FALSE, ...)

Bei(...)

Ber(...)

Arguments

x	numeric; values to evaluate the complex solution at
...	additional arguments passed to BesselK or Beir
nu.	numeric; value of $\nu$ in $\mathcal{B}_\nu$ solutions
nSeq.	positive integer; equivalent to nSeq in BesselJ
return.list	logical; Should the result be a list instead of matrix?

Details

Ber and Bei are wrapper functions which return the real and imaginary components of Beir, respectively.

Value

If return.list==FALSE (the default), a complex matrix with as many columns as using nSeq. creates. Otherwise the result is a list with matrices for Real and Imaginary components.

Author(s)

Andrew Barbour

References

http://mathworld.wolfram.com/KelvinFunctions.html

Imaginary: http://mathworld.wolfram.com/Bei.html

Real: http://mathworld.wolfram.com/Ber.html

See Also

kelvin-package, Keir, BesselJ

Examples

Beir(1:10)    # defaults to nu.=0
Beir(1:10, nu.=2)
Beir(1:10, nSeq.=2)
Beir(1:10, nSeq.=2, return.list=TRUE)


# Imaginary component only
Bei(1:10)

# Real component only
Ber(1:10)

---

Complementary solution to the Kelvin differential equation (K)

Description

This function calculates the complex solution to the Kelvin differential equation using modified Bessel functions of the second kind, specifically those produced by BesselK.

Usage

Keir(x, ...)

## Default S3 method:
Keir(
  x,
  nu. = 0,
  nSeq. = 1,
  add.tol = TRUE,
  return.list = FALSE,
  show.scaling = FALSE,
  ...
)

Kei(...)

Ker(...)

Arguments

x	numeric; values to evaluate the complex solution at
...	additional arguments passed to BesselK or Keir
nu.	numeric; value of $\nu$ in $\mathcal{K}_\nu$ solutions
nSeq.	positive integer; equivalent to nSeq in BesselK
add.tol	logical; Should a fudge factor be added to prevent an error for zero-values?
return.list	logical; Should the result be a list instead of matrix?
show.scaling	logical; Should the normalization values be given as a message?

Details

Ker and Kei are wrapper functions which return the real and imaginary components of Keir,, respectively.

Value

If return.list==FALSE (the default), a complex matrix with as many columns as using nSeq. creates. Otherwise the result is a list with matrices for Real and Imaginary components.

Author(s)

Andrew Barbour

References

http://mathworld.wolfram.com/KelvinFunctions.html

Imaginary: http://mathworld.wolfram.com/Kei.html

Real: http://mathworld.wolfram.com/Ker.html

See Also

kelvin-package, Beir, BesselK

Examples

Keir(1:10)    # defaults to nu.=0, nSeq=1
Keir(1:10, nu.=2)
Keir(1:10, nSeq=2)
Keir(1:10, nSeq=2, return.list=TRUE)


# Imaginary component only
Kei(1:10)

# Real component only
Ker(1:10)

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