Tensor calculus and functional analysis in the iteration solution of the geodetic boundary value problem

Kód výsledku v IS VaVaI

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Výsledek na webu

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DOI - Digital Object Identifier

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Jazyk výsledku

angličtina

Název v původním jazyce

Tensor calculus and functional analysis in the iteration solution of the geodetic boundary value problem

Popis výsledku v původním jazyce

An alternative between the boundary complexity and the complexity of the coefficients of the partial differential equation governing the solution is discussed in treating the geodetic boundary value problem. The Laplace operator has a relatively simple structure in terms of spherical or ellipsoidal coordinates frequently used in geodesy. However, the physical surface of the Earth substantially differs from a sphere or an oblate ellipsoid of revolution, even if optimally fitted. The same holds true for the solution domain and the exterior of a sphere or of an oblate ellipsoid of revolution. The situation is more convenient in a system of general curvilinear coordinates such that the physical surface of the Earth (smoothed to a certain degree) is imbedded in the family of coordinate surfaces. Therefore, a transformation of coordinates is applied in treating the geodetic boundary value problem. The idea is generally close to concepts followed also in other branches of engineering and mathematical physics. In our case tensor calculus is used and the Laplace operator is expressed in the new coordinates. Clearly, its structure is more complicated now. In a sense it represents the topography of the physical surface of the Earth. For this reason the Green’s function method is used together with the method of successive approximations in the solution of the geodetic boundary value problem expressed in terms of the new coordinates. The structure of iteration steps is analyzed and if possible, it is modified by means of integration by parts. Subsequently, the iteration steps and their convergence are discussed and interpreted, numerically as well as in terms of functional analysis.

Název v anglickém jazyce

Tensor calculus and functional analysis in the iteration solution of the geodetic boundary value problem

Popis výsledku anglicky

An alternative between the boundary complexity and the complexity of the coefficients of the partial differential equation governing the solution is discussed in treating the geodetic boundary value problem. The Laplace operator has a relatively simple structure in terms of spherical or ellipsoidal coordinates frequently used in geodesy. However, the physical surface of the Earth substantially differs from a sphere or an oblate ellipsoid of revolution, even if optimally fitted. The same holds true for the solution domain and the exterior of a sphere or of an oblate ellipsoid of revolution. The situation is more convenient in a system of general curvilinear coordinates such that the physical surface of the Earth (smoothed to a certain degree) is imbedded in the family of coordinate surfaces. Therefore, a transformation of coordinates is applied in treating the geodetic boundary value problem. The idea is generally close to concepts followed also in other branches of engineering and mathematical physics. In our case tensor calculus is used and the Laplace operator is expressed in the new coordinates. Clearly, its structure is more complicated now. In a sense it represents the topography of the physical surface of the Earth. For this reason the Green’s function method is used together with the method of successive approximations in the solution of the geodetic boundary value problem expressed in terms of the new coordinates. The structure of iteration steps is analyzed and if possible, it is modified by means of integration by parts. Subsequently, the iteration steps and their convergence are discussed and interpreted, numerically as well as in terms of functional analysis.

Druh

O - Ostatní výsledky

CEP obor

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OECD FORD obor

10102 - Applied mathematics

Projekt

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Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2021

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů