# kelvin-package: Fundamental and equivalent solutions to the Kelvin... In kelvin: Calculate 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} - ≤ft(i x^2 + ν^2\right) y = 0

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

\mathcal{B}_ν \equiv \mathrm{Ber}_ν (x) + i \mathrm{Bei}_ν (x)

= J_ν ≤ft(x \cdot \exp(3 π i / 4)\right)

= \exp(ν π i) \cdot J_ν ≤ft(x \cdot \exp(-π i / 4)\right)

= \exp(ν π i / 2) \cdot I_ν ≤ft(x \cdot \exp(π i / 4)\right)

= \exp(3 ν π i / 2) \cdot I_ν ≤ft(x \cdot \exp(-3 π i / 4)\right)

where J_ν is a Bessel function of the first kind, and I_ν is a modified Bessel function of the first kind.

Similarly, the complementary solutions, \mathcal{K}_ν, are defined as

\mathcal{K}_ν \equiv \mathrm{Ker}_ν (x) + i \mathrm{Kei}_ν (x)

= \exp(- ν π i / 2) \cdot K_ν ≤ft(x \cdot \exp(π i / 4)\right)

where K_ν is a modified Bessel function of the second kind.

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

y = \mathrm{Ber}_ν (x) + i \mathrm{Bei}_ν (x)

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

= \mathrm{Ker}_ν (x) + i \mathrm{Kei}_ν (x)

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

In the case where ν=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 ≤ft(i √{i} \cdot x\right)

= J_0 ≤ft(√{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}_ν and \mathcal{K}_ν.

## 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

Fundamental solution: Beir
Equivalent solution: Keir