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Date: 25-7-2018
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Date: 13-7-2018
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In toroidal coordinates, Laplace's equation becomes
(1) |
Attempt separation of variables by plugging in the trial solution
(2) |
then divide the result by to obtain
(3) |
The function then separates with
(4) |
giving solution
(5) |
Plugging back in and dividing by gives
(6) |
The function then separates with
(7) |
giving solution
(8) |
Plugging back in and multiplying by gives
(9) |
which can also be written
(10) |
(Arfken 1970, pp. 114-115). Laplace's equation is partially separable, although the Helmholtz differential equation is not.
Solutions to the differential equation for are known as toroidal functions.
REFERENCES:
Arfken, G. "Toroidal Coordinates ." §2.13 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 112-115, 1970.
Byerly, W. E. An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, pp. 264-266, 1959.
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 666, 1953.
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