Publication: Lie theory and separation of variables. 11. The EPD equation
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Copyright 1976 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The following article appeared in the Journal of Mathematical Physics and may be found at http://jmp.aip.org/jmp/top.jsp
Abstract
We show that the Euler–Poisson–Darboux equation {∂tt -∂rr – [(2m+1)/r]∂r}Ө=0
separates in exactly nine coordinate systems corresponding to nine orbits of symmetric second-order operators in the enveloping algebra of SL(2,R), the symmetry group of this equation. We employ techniques developed in earlier papers from this series and use the representation theory of SL(2,R) to derive special function identities relating the separated solutions. We also show that the complex EPD equation separates in exactly five coordinate systems corresponding to five orbits of symmetric second-order operators in the enveloping algebra of SL(2,C).
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Kalnins, E.G. & Miller, W., Jr. (1976). Lie theory and separation of variables. 11. The EPD equation. Journal of Mathematical Physics, 17, 369.