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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 <andy.barbour@gmail.com>
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
Useful links:
Fundamental solution: Beir
Equivalent solution: Keir
kelvin documentation built on April 14, 2025, 9:07 a.m.
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| kelvin-package | R Documentation |
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 <andy.barbour@gmail.com>
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
Useful links:
Fundamental solution: Beir
Equivalent solution: Keir
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