Email this article 
SPHERICAL SPLINE SOLUTIONS OF THE INHOMOGENEOUS BIHARMONIC EQUATION
- Authors: Vaskevich V.L1,2
-
Affiliations:
- Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
- Novosibirsk State University
- Issue: Vol 64, No 8 (2024)
- Pages: 1456-1465
- Section: Partial Differential Equations
- URL: https://rjeid.com/0044-4669/article/view/665035
- DOI: https://doi.org/10.31857/S0044466924080107
- EDN: https://elibrary.ru/YAMREM
- ID: 665035
Cite item
Open Access
Access granted
Subscription or Fee Access
Abstract
An inhomogeneous biharmonic equation is considered on the unit sphere in three-dimensional space. The solution of this equation, belonging to the Sobolev space on the sphere, is approximated by a sequence of solutions of the same equation but with specific right-hand sides, represented as linear combinations of shifts of the Dirac delta function. It is proven that, given specified nodes on the sphere determining the shifts, special solutions of the equation — spherical biharmonic splines — exist, and the weights corresponding to each are solutions of an associated non-degenerate system of linear algebraic equations. The connection between the approximation quality of the differential problem solution by spherical biharmonic splines and the problem of the convergence rate of optimal weighted spherical cubature formulas is established.
About the authors
V. L Vaskevich
Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences; Novosibirsk State University
Email: v.vaskevich@g.nsu.ru
Novosibirsk, Russia
References
- Васкевич В.Л. Полигармоническое уравнение на сфере трехмерного пространства // Матем. заметки СВФУ. 2022. Т. 29. № 3. C. 22-30.
- Соболев С.Л. Введение в теорию кубатурных формул. М.: Наука, 1974. 808 с.
- Михлин С.Г. Многомерные сингулярные интегралы и интегральные уравнения. М.: ГИФМЛ, 1962. 256 с.
- Игнатов М.И., Певный А.Б. Натуральные сплайны многих переменных. Л.: Наука, 1991. 125 с.
- Васкевич В.Л. Константы и функции вложения пространств соболевского типа на единичной сфере // Докл. АН. 2010. Т 433. № 4. С. 441-446.
- Васкевич В.Л. Константы вложения периодических пространств Соболева дробного порядка // Сиб. ма-тем. ж. 2008. Т. 49. № 5. С. 1019-1027.
- Васкевич В.Л. Сферические кубатурные формулы в пространствах Соболева // Сиб. матем. ж. 2017. Т. 58. № 3. С. 530-542.
- Muller C. Spherical harmonics. Berlin: Springer-Verlag. 1966. 46 p.
- Соболев С.Л., Васкевич В.Л. Кубатурные формулы. Новосибирск: Изд-во Ин-та математики, 1996. 484 с.
- Sobolev S. L., Vaskevich V. L. The Theory of Cubature Formulas. Dordrecht: Kluwer Academic Publishers. 1997. XXII+416 pp.
Supplementary files
