DOI
10.34229/KCA2522-9664.25.1.13
UDC 519.6
OPTIMAL INTEGRATION OF RAPIDLY OSCILLATING FUNCTIONS
FOR ONE CLASS OF DIFFERENTIAL FUNCTIONS UNDER APPROXIMATE
A PRIORI INFORMATION
Abstract. The problem of calculating integrals of rapidly oscillating functions for a class of functions with continuous second and partially continuous
third derivatives limited by Lipschitz conditions with a constant Lipschitz L is considered.
The a priori information about the integrand function contains fixed values of the function and its first and second derivatives,
which are given at N fixed nodes of an arbitrary grid approximately, with a specific error.
This method of specifying a priori information narrows down the class of integrable functions to the so-called interpolation class of functions and allows generating a quadrature formula optimal in terms of accuracy for it and obtaining an optimal estimate of its error by applying the method of boundary functions.
Keywords: integrals of rapidly oscillating functions, interpolation classes of functions, approximate a priori information, accuracy-optimal quadrature formulas, method of boundary functions.
full text
REFERENCES
- 1. Davis P.J., Rabinowitz P. Methods of Numerical Integration. Academic Press, 1984. 626 p.
- 2. Clenshaw C.W., Curtis A.R. A method for numerical integration on an automatic computer. Numer. Math. 1960. Vol. 2. P. 197–205.
- 3. Haider Q., Liu L.C. Fourier and Bessel transformations of highly oscillatory functions. Journal of Physics A: Mathematical and General. 1992. N 24. P. 6755–6760.
- 4. Filon L.N. On a quadrature formula for trigonometric integrals. Proc. Roy. Soc. Edinburg. 1928. Vol. 49. P. 38–47.
- 5. Flinn E.A. A modification of Filon’s method of numerical integration. J. Assoc. Comput. Mach. 1960. N 7. P. 181–184.
- 6. Iserles A., Nrsett S.P. Efficient quadrature of highly oscillatory integrals using derivatives. Proceedings Royal Soc. 2005. Vol. 461, Iss. 2057. P. 1383–1399. https://doi.org/10.1098/ .
- 7. Ivanov V.V. Questions of accuracy and efficiency of computational algorithms: Review of achievements in the field of cybernetics and computer engineering [in Russian]. Kyiv: Institute of Cybernetics of the Academy of Sciences of the Ukrainian SSR, 1969. Iss. 2. 135 p.
- 8. Zhileikin Ya.M. On the error of approximate calculation of integrals of rapidly oscillating functions. JVM and MF. 1971. Vol. 11, N 1. P. 263–266.
- 9. Zhileikin Ya.M., Kukarkin A.B. Approximate calculation of integrals of rapidly oscillating functions [in Russian]. Moscow: Publishing house of Moscow University, 1987. 248 p.
- 10. Zadiraka V.K., Luts L.V., Shvidchenko I.V. Theory of calculations of integrals of fast-oscillating functions [in Ukrainian]. Kyiv: Nauk. Dumka, 2023. 472 с. https://doi.org/10.15407/ .
- 11. Luts L.V. Optimal calculation of integrals of rapidly oscillating functions for some classes of differential functions. Cybernetics and Systems Anaysysl. 2024. Vol. 60, N 2. Р. 276–284. https://doi.org/10.1007/ .
- 12. Traub J.F., Wozniakowski H. A General theory of optimal algorithms. Academic Press, 1980. 341 p.
