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Cybernetics and Systems Analysis

, Volume 44, Issue 4, pp 603–615 | Cite as

Atomic radial basis functions in numerical algorithms for solving boundary-value problems for the Laplace equation

  • V. M. Kolodyazhny
  • V. A. Rvachov
Software-Hardware Systems

Abstract

A numerical method for solution of boundary-value problems of mathematical physics is described that is based on the use of radial atomic basis functions. Atomic functions are compactly supported solutions of functional-differential equations of special form. The convergence of this numerical method is investigated for the case of using an atomic function in solving the Dirichlet boundary-value problem for the Laplace equation.

Keywords

atomic function functional-differential equation harmonic function radial basis function approximation error 

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References

  1. 1.
    V. L. Rvachev and V. A. Rvachev, Nonclassical Methods of Approximation Theory in Boundary-Value Problems [in Russian], Naukova Dumka, Kiev (1979).Google Scholar
  2. 2.
    V. A. Rvachev, “Compactly-supported solutions of functional-differential equations and their applications,” UMN, 45, No. 1 (271), 77–103 (1990).MathSciNetGoogle Scholar
  3. 3.
    Yu. G. Stoyan, V. S. Protsenko, G. P. Man’ko, et al., Theory of R-Functions and Current Problems of Applied Mathematics [in Russian], Naukova Dumka, Kiev (1986).Google Scholar
  4. 4.
    V. M. Kolodyazhny and V. O. Rvachov, “Compactly supported functions generated by the Laplace operator,” Dop. NAN Ukr., No. 4, 17–22 (2004).Google Scholar
  5. 5.
    V. M. Kolodyazhny and V. O. Rvachov, “Using the atomic function Plop(x, y) in solving boundary-value problems for the Laplace equation,” Dop. NAN Ukr., No. 9, 16–21 (2006).Google Scholar
  6. 6.
    M. D. Buhmann, Radial basis Functions: Theory and Implementations, University Press, Cambridge, UK (2004).Google Scholar
  7. 7.
    H. Wendland, “Piecewise polynomial, positive definite, and compactly supported radial functions of minimal degree,” Adv. Comp. Math., No. 4, 389–396 (1995).Google Scholar
  8. 8.
    Z. Wu, “Multivariate compactly supported positive definite radial functions,” Adv. Comp. Math., No. 4, 283–292 (1995).Google Scholar
  9. 9.
    V. N. Malozemov, “An estimate of the exactness of a quadrature formula for periodic functions,” Vestn. Leningrad. Un-ta, 1, No. 1, 52–59 (1967).MathSciNetGoogle Scholar
  10. 10.
    A. A. Ligun, “Best quadrature formulas for some classes of periodic functions,” Mat. Zametki, 24, No. 5, 661–669 (1978).MATHMathSciNetGoogle Scholar
  11. 11.
    V. P. Motornyi, “On the best quadrature formula of the form {ie614-01} for some classes of differentiable periodic functions,” Izv. Akad. Nauk SSSR, Ser. Mat., 38, No. 3, 583–614 (1974).MathSciNetGoogle Scholar
  12. 12.
    A. A. Karatsuba, Fundamentals of Analytical Number Theory [in Russian], Nauka, Moscow (1983).Google Scholar
  13. 13.
    A. F. Timman, Theory of Approximation of Functions of a Real Variable [Russian translation], Fizmatgiz, Moscow (1960).Google Scholar
  14. 14.
    R. Kurant, Partial Differential Equations [Russian translation], Mir, Moscow (1964).Google Scholar
  15. 15.
    L. V. Kantorovich and V. I. Krylov, Approximate Methods of Higher Analysis [in Russian], Gos. Izd-vo Fiz.-Mat. Lit, Moscow-Leningrad (1962).Google Scholar
  16. 16.
    V. M. Kolodyazhny and V. O. Rvachov, “Some properties of atomic functions of many variables,” Dop. NAN Ukr., No. 1, 12–20 (2005).Google Scholar
  17. 17.
    V. M. Kolodyazhnyi and V. A. Rvachev, “Atomic functions of three variables invariant with respect to a rotation group,” Cybernetics and Systems Analysis, No. 6, 118–130 (2004).Google Scholar
  18. 18.
    V. M. Kolodyazhny and V. O. Rvachov, “Compactly supported functions generated by a biharmonic operator,” Dop. NAN Ukr., No. 2, 22–30 (2006).Google Scholar
  19. 19.
    V. M. Kolodyazhny, “Compactly supported functions generated by a polyharmonic operator,” Cybernetics and Systems Analysis, No. 5, 141–156 (2006).Google Scholar
  20. 20.
    T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, and P. Krysl, “Meshless methods: An overview and recent developments,” Comput. Methods Appl. Mech. Eng., 139, 3–47 (1996).MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  • V. M. Kolodyazhny
    • 1
  • V. A. Rvachov
    • 2
  1. 1.A. N. Podgornyi Institute of Problems of Mechanical EngineeringNational Academy of Sciences of UkraineKharkovUkraine
  2. 2.N. E. Zhukovskii National Aerospace University “Kharkov Aviation Institute”KharkovUkraine

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