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=0
and has a suite of complex solutions. One set of solutions,
Bν, is defined in the following manner:
Bν≡Berν(x)+iBeiν(x)
=Jν(x⋅exp(3πi/4))
=exp(νπi)⋅Jν(x⋅exp(−πi/4))
=exp(νπi/2)⋅Iν(x⋅exp(πi/4))
=exp(3νπi/2)⋅Iν(x⋅exp(−3πi/4))
where
Jν is a Bessel function of the first kind, and
Iν is a modified Bessel function of the first kind.
Similarly, the complementary solutions, Kν,
are defined as
Kν≡Kerν(x)+iKeiν(x)
=exp(−νπi/2)⋅Kν(x⋅exp(πi/4))
where Kν is a modified Bessel function of the second kind.
The relationships between y in the differential equation, and
the solutions Bν and Kν are as follows
y=Berν(x)+iBeiν(x)
=Ber−ν(x)+iBei−ν(x)
=Kerν(x)+iKeiν(x)
=Ker−ν(x)+iKei−ν(x)
In the case where ν=0, the differential equation reduces to
x2y¨+xy˙−ix2y=0
which has the set of solutions:
J0(ii⋅x)
=J0(2⋅(i−1)⋅x/2)
=Ber0(x)+iBei0(x)≡B0
This package has functions to calculate
Bν and Kν.
Author(s)
Andrew Barbour <andy.barbour@gmail.com>
References
Abramowitz, M. and Stegun, I. A. (Eds.). "Kelvin Functions."
§9.9 in Handbook of Mathematical Functions with Formulas, Graphs,
and Mathematical Tables, 9th printing. New York: Dover, pp. 379-381, 1972.