Multidimensional generalizations of atomic radial basis functions
Abstract
Possible approaches to generalizing multidimensional atomic radial basis functions are presented. The functions of mathematical physics are used in solving two-dimensional and three-dimensional boundary value problems with partial derivatives. Depending on the problem, the functions of different dimensions are used, that is, the functions generated by various differential operators. The functional-differential equations that generate those functions are considered and families of finite solutions for those equations generated by the differential operators of Laplace, Helmholtz, etc. have been constructed according to the given scheme. The results are presented in the form of theorems. In order to expand the class of functions and improve their properties, the construction of the atomic radial basis function family of three independent variables on the example of a functional-differential equation of the appropriate type is considered. The solution methods refer to seedless schemes and combine the possibilities of constructing the boundaries of regions by using R-functions. Atomic functions are convenient for implementing computational algorithms for constructing approximate solutions of boundary value problems in 2D and 3D domains by using meshless schemes. The properties of these functions make it possible to use them as basic functions in solving boundary value problems by meshless methods based on collocation methods. For atomic functions, the dependence on the compression ratio is provided, which is specified in the process of constructing the solution of the boundary value problem as necessary to ensure certain properties of the functions. The scheme for constructing solutions of heat conduction problems by using a gridless scheme is provided.
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Lisina O.Yu. Modeling thermal fields in non-canonical technical products // Journal of Problems of mechanical engineering, Vol. 14, No. 6, 2011. – P. 57–64. (in Russian).
Protektor, D.O., Kolodyazhny, V.M., Lisin, D.O, Lisina O.Yu. A Meshless Method of Solving Three-Dimensional Nonstationary Heat Conduction Problems in Anisotropic Materials // Cybern Syst Anal. 2021. Vol. 57, Issue 3.
Gorin E.A. On finite solutions of some functional differential equations. // UMN, 36, No. 4, 1981. – P. 211–212. (in Russian)
Kolodyazhniy V.M., Rvachev V.A. Finite functions generated by the Laplace operator // Proceedings of the National Academy of Sciences of Ukraine. No. 4, 2004. – P. 17–22. (in Russian)
Theory of R-functions and topical problems of applied mathematics. / Stoyan Yu.G., Protsenko V.S., Manko G.P., Goncharyuk I.V., Kurpa L.V., Rvachev V.A., Sinekop N.S., Sirodzha I.B., Shevchenko A.N., Sheiko V.I. – K.: Nauk. dumka, 1986. – 262 p. (in Russian)
Sigmund A. Trigonometric series. / A. Sigmund. – V. 1, Moscow: Fizmatiz, 1965. – 616 p. V. 2, 1965. – 538 p. (in Russian)
Kolodyazhniy V.M., Rvachov V.O. Finite functions generated by a harmonious operator // Proceedings of the National Academy of Sciences of Ukraine. - 2006. – No. 2. – P. 23–30. (in Ukrainian)
Kolodyazhniy V.M., Rvachov V.O. Finite functions, which are generated by the operator Laplace // Proceedings of the National Academy of Sciences of Ukraine. – 2004. – No. 4. – P. 17–22. (in Ukrainian)
Kolodyazhny V.M., Rvachov V.A. Atomic functions of three variables invariant with respect to the rotation group // Cybernetics and Systems Analysis. – 2004. – No. 6 – P. 118–130. (in Russian)
Kolodyazhny V.M., Rvachov V.A. Atomic functions. Generalizations to the case of many variables and promising directions of practical applications // Cybernetics and System Analysis. – 2007. Vol. 43, No. 6. – P. 155–177. (in Russian)
Lisina O.Yu. Modeling thermal fields in non-canonical technical products // Journal of Problems of mechanical engineering, Vol. 14, No. 6, 2011. – P. 57–64. (in Russian).
Protektor, D.O., Kolodyazhny, V.M., Lisin, D.O, Lisina O.Yu. A Meshless Method of Solving Three-Dimensional Nonstationary Heat Conduction Problems in Anisotropic Materials // Cybern Syst Anal. 2021. Vol. 57, Issue 3.
