World economy. Optimism and problems

Komarov, V.V.; Litvina, N.I.; Ananieva, E.V.

doi:10.20339/am.06-20.076

Cited by 2 publications

(5 citation statements)

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“…(A59) Both these identities lead, in the respective regions, to the identity (see Eq. (F24) and (F44) of Supplement F of [4]) (A60) which is valid in both regions (small-ν, large-ν) when the proper series for the functions hR, hν (see (7), ( 8)) are used for the respective regions. Then (see (8) and (A35)) (A61) in the small-ν region, while…”

Section: A42 Derivative Of Hrmentioning

confidence: 99%

Interior solution of azimuthally symmetric case of Laplace equation in orthogonal similar oblate spheroidal coordinates

Strunz

2023

Preprint

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Curvilinear coordinate systems distinct from the rectangular Cartesian coordinate system are particularly valuable in the field calculations as they facilitate the expression of boundary conditions of differential equations in a reasonably simple way when the coordinate surfaces fit the physical boundaries of the problem. The recently finalized orthogonal similar oblate spheroidal (SOS) coordinate system can be particularly useful for a physical processes description inside or in the vicinity of the bodies with the geometry of an oblate spheroid. Such shape is aproximating well objects investigated within astrophysics. The solution of the azimuthally symmetric case of the Laplace equation was found for the interior space in the orthogonal SOS coordinates. In the frame of the derivation of the harmonic functions, the Laplace equation was separated by a special separation procedure. A generalized Legendre equation was introduced as the equation for the angular part of the separated Laplace equation. The harmonic functions were determined as relations involving generalized Legendre functions of the first and of the second kind. Several lower-degree functions are reported. Recursion formula facilitating determination of the higher-degree harmonic functions was found. The general solution of the azimuthally symmetric Laplace equation for the interior space in the SOS coordinates is reported.

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Section: A42 Derivative Of Hrmentioning

confidence: 99%

Interior solution of azimuthally symmetric case of Laplace equation in orthogonal similar oblate spheroidal coordinates

Strunz

2023

Preprint

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“…Several relations between the metric scale factors and the generalized sine and cosine (see (7) and ( 14), ( 15)) are needed for the Laplace equation solution. An important relation is…”

Section: A22 Relations Between Metric Scale Factor Hr and The General...mentioning

confidence: 99%

“…(A59) Both these identities lead, in the respective regions, to the identity (see Eq. (F24) and (F44) of Supplement F of [4]) (A60) which is valid in both regions (small-ν, large-ν) when the proper series for the functions h R , h ν (see (7), ( 8)) are used for the respective regions. Then (see (8) and (A35))…”

Section: A42 Derivative Of H Rmentioning

confidence: 99%

“…The similar oblate spheroidal coordinates are distinct from all the standard orthogonal coordinate systems [2,7], as one coordinate surfaces family is not of the second degree (and even not of the fourth degree), but it is formed by general power functions 𝑧 ≈ 𝑥 1+𝜇 (i.e. with a real-number exponent 1+μ) rotated around the z-axis [3,4].…”

Section: Introductionmentioning

confidence: 99%

“…Recently, the earlier suggested [3] similar oblate spheroidal (SOS) orthogonal coordinate system was finalized [4,5]. The similar oblate spheroidal coordinates are distinct from all the well known standard orthogonal coordinate systems [2,6], the confocal oblate spheroidal (COS) system not excluding. The SOS system can be a powerful tool for a description of field or physical processes inside or in the vicinity of the bodies or particles with an oblate spheroidal geometry.…”

Section: Introductionmentioning

confidence: 99%

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Analytical solution of orthogonal similar oblate spheroidal coordinate system

Strunz¹

2022

Celest Mech Dyn Astron

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Curvilinear coordinate systems distinct from the rectangular Cartesian coordinate system are particularly valuable in the field calculations as they facilitate the expression of boundary conditions of differential equations in a reasonably simple way when the coordinate surfaces fit the physical boundaries of the problem. The recently finalized orthogonal similar oblate spheroidal (SOS) coordinate system can be particularly useful for a physical processes description inside or in the vicinity of the bodies with the geometry of an oblate spheroid. Such shape is aproximating well objects investigated within astrophysics. The solution of the azimuthally symmetric case of the Laplace equation was found for the interior space in the orthogonal SOS coordinates. In the frame of the derivation of the harmonic functions, the Laplace equation was separated by a special separation procedure. A generalized Legendre equation was introduced as the equation for the angular part of the separated Laplace equation. The harmonic functions were determined as relations involving generalized Legendre functions of the first and of the second kind. Several lower-degree functions are reported. Recursion formula facilitating determination of the higher-degree harmonic functions was found.The general solution of the azimuthally symmetric Laplace equation for the interior space in the SOS coordinates is reported.

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World economy. Optimism and problems

Cited by 2 publications

References 0 publications

Interior solution of azimuthally symmetric case of Laplace equation in orthogonal similar oblate spheroidal coordinates

Interior solution of azimuthally symmetric case of Laplace equation in orthogonal similar oblate spheroidal coordinates

Analytical solution of orthogonal similar oblate spheroidal coordinate system

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