kelvin-package {kelvin}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

x2y¨+xy˙(ix2+ν2)y=0x^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, Bν\mathcal{B}_\nu, is defined in the following manner:

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

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

=exp(νπi)Jν(xexp(πi/4))= \exp(\nu \pi i) \cdot J_\nu \left(x \cdot \exp(-\pi i / 4)\right)

=exp(νπi/2)Iν(xexp(πi/4))= \exp(\nu \pi i / 2) \cdot I_\nu \left(x \cdot \exp(\pi i / 4)\right)

=exp(3νπi/2)Iν(xexp(3πi/4))= \exp(3 \nu \pi i / 2) \cdot I_\nu \left(x \cdot \exp(-3 \pi i / 4)\right)

where JνJ_\nu is a Bessel function of the first kind, and IνI_\nu is a modified Bessel function of the first kind.

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

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

=exp(νπi/2)Kν(xexp(πi/4))= \exp(- \nu \pi i / 2) \cdot K_\nu \left(x \cdot \exp(\pi i / 4)\right)

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

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

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

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

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

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

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

x2y¨+xy˙ix2y=0x^2 \ddot{y} + x \dot{y} - i x^2y = 0

which has the set of solutions:

J0(iix) J_0 \left(i \sqrt{i} \cdot x\right)

=J0(2(i1)x/2) = J_0 \left(\sqrt{2} \cdot (i-1) \cdot x / 2\right)

=Ber0(x)+iBei0(x)B0 = \mathrm{Ber}_0 (x) + i \mathrm{Bei}_0 (x) \equiv \mathcal{B}_0

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

Author(s)

Andrew Barbour <andy.barbour@gmail.com>

References

Abramowitz, M. and Stegun, I. A. (Eds.). "Kelvin Functions." §9.9\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


[Package kelvin version 2.0-2 Index]