Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Separation of Laplace's equation

Authors: N. Levinson, B. Bogert and R. M. Redheffer
Journal: Quart. Appl. Math. 7 (1949), 241-262
MSC: Primary 36.0X
DOI: https://doi.org/10.1090/qam/32091
MathSciNet review: 32091
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Abstract: The following results are established in this paper:

(I)$ ^{**}$ For the Laplace equation $ \Delta \theta = 0$ in curvilinear co-ordinates $ \left( {u,v,w} \right)$ in Euclidean space to be directly separableinto two equations, one for $ S$ and one for $ Z$, when the solution is $ \theta = R\left( {u,v,w} \right)S\left( {u,v} \right)Z\left( w \right)$ with fixed $ R$, it is necessary and sufficient that the surfaces $ {w}$ = constant (1) be orthogonal to the surfaces $ {u}$ = constant, $ {v}$ = constant and (2) be parallel planes, planes with a common axis, concentric spheres, spheres tangent at a common point, or one of the two sets of spheres generated by the co-ordinate lines when bicircular co-ordinates are rotated about the line joining the poles or about its perpendicular bisector.

(II) We have $ R = 1$ always and only in the first three cases, namely, when the surfaces $ {w}$ = constant are parallel planes, planes with a common axis, or concentric spheres.

(III) In these three cases, but only these, the wave equation separates in the sense $ RSZ$, and hence, for the wave equation, $ R = 1$ automatically.

(IV) For further separation of the equation found above for $ S$, when $ S = X\left( u \right)Y\left( v \right)$ so that the solution is now $ RXYZ$, it is necessary and sufficient that the co-ordinates be toroidal, or such that the wave equation so separates, or any inversions of these.

(V) The co-ordinates where the wave equation so separates, that is, admits solutions $ RX\left( u \right)Y\left( v \right)Z\left( w \right)$, are only the well-known cases where this happens with $ R = 1$, namely, degenerate ellipsoidal or paraboloidal co-ordinates (but see Sec. 8.2).

(VI). In these cases, but only these, $ R = 1$ for the Laplace equation too.

(VII) Co-ordinates for $ RSZ$ or $ RXYZ$ separation of the Laplace equation have the group property under inversion.

(VIII) In all cases $ R$ can be found by inspection of the linear element.

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DOI: https://doi.org/10.1090/qam/32091
Article copyright: © Copyright 1949 American Mathematical Society

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