Ellipsoid Interface Neumann Boundary Value Problem Of The Integral Solution

This paper makes a comprehensive and systematic study of the establishment ofthe earth's ellipsoidal gravity model and the solution of the boundary value problemunder ellipsoidal boundary.Firstly, according to the Laplace equation of elIipsoidal coordinates and theel1ipsoidal harmonic functions in common use, the ellipsoidal series expansion ofharmonic functions outside the referential e1lipsoidal is introduced. However, thesecond associated Legendre function is too comp1icated to compute, so a simple andpractical ellipsoida1 harmonic series expansion is introduced, and the series solutionand integral solution of the Dirichlet boundary value problem under ellipsoidalboundary are given. Then, the solution of the Neumann boundary value problemunder ellipsoidal boundary is discussed in detail, a1ong with its series solution, theintegral expression is mainly studied.,As al1 the solutions reserve an accuracy of s:, the O(r.S') can thus beacquired. That is, the relative accuracy could approximate l0-9 to l0-,o. Once theaccuracy of model linearization is reserved, a cm accuracy of geoid is guaranteedtheoretically.At the end of this papeY, the transformation between spherical coefficients ande1lipsoidal coefficients is introduced, from which the series solution of the boundaryvalue problem under ellipsoidal boundary can be derived easily. And the integralsolution is generalized to that of the non-linear fined gravity boundary valueproblem.