TY - JOUR

T1 - Spherical Spline Solutions of an Inhomogeneous Biharmonic Equation

AU - Vaskevich, V. L.

N1 - This work was carried out as part of a state assignment for the Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, project no. FWNF-2022-0008.

PY - 2024/8

Y1 - 2024/8

N2 - An inhomogeneous biharmonic equation is considered on a unit sphere of three-dimensional space. The solution of this equation belonging to the spherical Sobolev space is approximated by a sequence of solutions of the same equation, but with special right-hand sides, which are linear combinations of shifts of the Dirac delta function. It is proved that, for given nodes on the sphere that determine shifts, there exist special solutions of the equation: spherical biharmonic splines, and the weights corresponding to each of them are solutions of the accompanying nondegenerate system of linear algebraic equations. A relation is established between the quality of approximation of the solution of the differential problem by spherical biharmonic splines and the problem of the convergence rate of optimal weighted spherical cubature formulas.

AB - An inhomogeneous biharmonic equation is considered on a unit sphere of three-dimensional space. The solution of this equation belonging to the spherical Sobolev space is approximated by a sequence of solutions of the same equation, but with special right-hand sides, which are linear combinations of shifts of the Dirac delta function. It is proved that, for given nodes on the sphere that determine shifts, there exist special solutions of the equation: spherical biharmonic splines, and the weights corresponding to each of them are solutions of the accompanying nondegenerate system of linear algebraic equations. A relation is established between the quality of approximation of the solution of the differential problem by spherical biharmonic splines and the problem of the convergence rate of optimal weighted spherical cubature formulas.

KW - biharmonic equation

KW - extremal functions

KW - spherical Sobolev spaces

KW - splines

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85205289609&origin=inward&txGid=0b4f503a3c7d4c2b4d18d9cf731a0d47

UR - https://www.webofscience.com/wos/woscc/full-record/WOS:001324967400005

UR - https://www.mendeley.com/catalogue/338e8214-60fc-3779-a533-e7ce5c227c18/

U2 - 10.1134/S0965542524700817

DO - 10.1134/S0965542524700817

M3 - Article

VL - 64

SP - 1765

EP - 1774

JO - Computational Mathematics and Mathematical Physics

JF - Computational Mathematics and Mathematical Physics

SN - 0965-5425

IS - 8

ER -