Application of spherical pseudo-differential operators and spherical wavelets for numerical solutions of the fixed altimetry-gravimetry boundary value problem

Grebenitcharsky, R.; Sideris, Michael G.

doi:10.1007/978-3-662-10735-5_31

Cited by 3 publications

(3 citation statements)

References 13 publications

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“…Some further examples of wavelet application to ODEs are presented in [29,20], and to PDEs in [19,30,38,21,15,41,4]. Contrary to the articles listed so far, the paper [16] describes a method for numerical solving of PDEs on the sphere. Recent years brought a number of papers in which wavelet methods were involved to solving fractional differential equations [18,40,32,1].…”

Section: Introductionmentioning

confidence: 99%

Wavelet methods in partial differential equations on spheres

Iglewska-Nowak,

Stefaniak

2021

Preprint

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Section: Introductionmentioning

confidence: 99%

Wavelet methods in partial differential equations on spheres

Iglewska-Nowak,

Stefaniak

2021

Preprint

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“…These preliminary numerical results encourage a further theoretical investigation for finding solutions of AGBVPs based on the combination of spherical harmonics, spherical pseudo-differential operators, and spherical wavelets for solving differential equations; see Grebenitcharsky and Sideris (2002a).…”

Section: Introductionmentioning

confidence: 94%

“…4, L and M are extensions from the land and the sea part of the sphere to the space outside the sphere (see Grebenitcharsky and Sideris, 2002a), S is Stokes's operator and 7\ is the first degree term of T. On the sphere 5/2 with radius R, the Laplacean will have the following representation in spherical coordinates:…”

Section: Introductionmentioning

confidence: 99%

A numerical study of solving the altimetry-gravimetry boundary value problem in coastal regions

Grebenitcharsky

Sideris

International Association of Geodesy Symposia

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Abstract. The paper presents numerical results for the solution of the altimetry-gravimetry boundary value problem with compatibility conditions along the coastline. The solution is based on the application of spherical Shannon wavelets based on the AbelPoisson kernel and pseudo-diffferential operators. The orthogonality of Shannon wavelets can be used for the numerical solution overcoming some technical problems with generally non-orthogonal spherical wavelets.A numerical experiment has been conducted in eastern Canada and the results have been compared with the geoid heights computed from gravity anomalies by applying ID FFT spherical Stokes convolution with 50 km integration radius. The effect of compatibility conditions along the coastline on the solution are discussed and the following conclusions are drawn: spherical Shannon wavelets can be very successfully applied in the numerical solution of the AGBVP II and the smoothing effect of spherical wavelets is much more effective than the classical planar type of wavelets.

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Wavelet Based Solutions to the Poisson and the Helmholtz Equations on the n-Dimensional Unit Sphere

Iglewska-Nowak

Stefaniak

2023

J Fourier Anal Appl

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We present a method of solving partial differential equations on the n-dimensional unit sphere using methods based on the continuous wavelet transform derived from approximate identities. We give an explicit analytical solution to the Poisson equation and to the Helmholtz equations. For the first one and for some special values of the parameter in the latter one, we derive a closed formula for the generalized Green function.

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Application of spherical pseudo-differential operators and spherical wavelets for numerical solutions of the fixed altimetry-gravimetry boundary value problem

Cited by 3 publications

References 13 publications

Wavelet methods in partial differential equations on spheres

Wavelet methods in partial differential equations on spheres

A numerical study of solving the altimetry-gravimetry boundary value problem in coastal regions

Wavelet Based Solutions to the Poisson and the Helmholtz Equations on the n-Dimensional Unit Sphere

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