Browse > Article
http://dx.doi.org/10.4134/CKMS.c210010

HERMITE-TYPE EXPONENTIALLY FITTED INTERPOLATION FORMULAS USING THREE UNEQUALLY SPACED NODES  

Kim, Kyung Joong (School of Liberal Arts and Sciences Korea Aerospace University)
Publication Information
Communications of the Korean Mathematical Society / v.37, no.1, 2022 , pp. 303-326 More about this Journal
Abstract
Our aim is to construct Hermite-type exponentially fitted interpolation formulas that use not only the pointwise values of an 𝜔-dependent function f but also the values of its first derivative at three unequally spaced nodes. The function f is of the form, f(x) = g1(x) cos(𝜔x) + g2(x) sin(𝜔x), x ∈ [a, b], where g1 and g2 are smooth enough to be well approximated by polynomials. To achieve such an aim, we first present Hermite-type exponentially fitted interpolation formulas IN built on the foundation using N unequally spaced nodes. Then the coefficients of IN are determined by solving a linear system, and some of the properties of these coefficients are obtained. When N is 2 or 3, some results are obtained with respect to the determinant of the coefficient matrix of the linear system which is associated with IN. For N = 3, the errors for IN are approached theoretically and they are compared numerically with the errors for other interpolation formulas.
Keywords
Exponentially fitted; interpolation;
Citations & Related Records
Times Cited By KSCI : 2  (Citation Analysis)
연도 인용수 순위
1 K. J. Kim and R. Cools, Extended exponentially fitted interpolation formulas for oscillatory functions, Appl. Math. Comput. 224 (2013), 178-195. https://doi.org/10.1016/j.amc.2013.08.039   DOI
2 K. J. Kim, Exponentially fitted interpolation formulas involving first and higher-order derivatives, J. Appl. Math. Inform. 31 (2013), no. 5-6, 677-693. https://doi.org/10.14317/jami.2013.677   DOI
3 K. J. Kim, Exponentially fitted interpolation formulas depending on two frequencies, J. Appl. Math. Inform. 34 (2016), no. 3-4, 207-220. https://doi.org/10.14317/jami.2016.207   DOI
4 K. Kim, R. Cools, and L. Gr. Ixaru, Extended quadrature rules for oscillatory integrands, Appl. Numer. Math. 46 (2003), no. 1, 59-73. https://doi.org/10.1016/S0168-9274(03)00009-6   DOI
5 Matlab, Language of Technical Computing, Mathworks, Inc.
6 R. L. Burden and J. D. Faires, Numerical Analysis, Brooks/Cole, 2001.
7 D. Hollevoet, M. Van Daele, and G. Vanden Berghe, Exponentially fitted methods applied to fourth-order boundary value problems, J. Comput. Appl. Math. 235 (2011), no. 18, 5380-5393. https://doi.org/10.1016/j.cam.2011.05.049   DOI
8 K. J. Kim, Two-frequency-dependent Gauss quadrature rules, J. Comput. Appl. Math. 174 (2005), no. 1, 43-55. https://doi.org/10.1016/j.cam.2004.03.020   DOI
9 J. P. Coleman and L. Gr. Ixaru, Truncation errors in exponential fitting for oscillatory problems, SIAM J. Numer. Anal. 44 (2006), no. 4, 1441-1465. https://doi.org/10.1137/050641752   DOI
10 A. Ghizzetti and A. Ossicini, Quadrature Formulae, Academic Press, New York, 1970.
11 A. Cardone, R. D'Ambrosio, and B. Paternoster, High order exponentially fitted methods for Volterra integral equations with periodic solution, Appl. Numer. Math. 114 (2017), 18-29. https://doi.org/10.1016/j.apnum.2016.05.003   DOI
12 L. Gr. Ixaru and B. Paternoster, A Gauss quadrature rule for oscillatory integrands, Comput. Phys. Comm. 133 (2001), no. 2-3, 177-188. https://doi.org/10.1016/S0010- 4655(00)00173-9   DOI
13 L. Gr. Ixaru and G. Vanden Berghe, Exponential Fitting, Mathematics and its Applications, 568, Kluwer Academic Publishers, Dordrecht, 2004. https://doi.org/10.1007/978-1-4020-2100-8   DOI
14 K. J. Kim, Error analysis for frequency-dependent interpolation formulas using first derivatives, Appl. Math. Comput. 217 (2011), no. 19, 7703-7717. https://doi.org/10.1016/j.amc.2011.02.073   DOI
15 L. Gr. Ixaru, G. Vanden Berghe, and H. De Meyer, Frequency evaluation in exponential fitting multistep algorithms for ODEs, J. Comput. Appl. Math. 140 (2002), no. 1-2, 423-434. https://doi.org/10.1016/S0377-0427(01)00474-5   DOI
16 L. Gr. Ixaru, Operations on oscillatory functions, Comput. Phys. Comm. 105 (1997), no. 1, 1-19. https://doi.org/10.1016/S0010-4655(97)00067-2   DOI