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Introduction

  • Liviu Gr. Ixaru
  • Guido Vanden Berghe
Chapter
  • 378 Downloads
Part of the Mathematics and Its Applications book series (MAIA, volume 568)

Abstract

The simple approximate formula for the computation of the first derivative of a function y(x),
$$ y'\left( x \right) \approx \frac{1}{{2h}}\left[ {y\left( {x + h} \right) - y\left( {x - h} \right)} \right], $$
(1.1)
is known to work well when y(x) is smooth enough. However, if y(x) is an oscillatory function of the form
$$ y'\left( x \right) = {f_1}\left( x \right)\sin \left( {\omega x} \right) + {f_2}\left( x \right)\cos \left( {\omega x} \right) $$
(1.2)
with smooth f 1(x) and f 2(x), the slightly modified formula
$$ y'\left( x \right) \approx \frac{1}{{2h}} \cdot \frac{\theta }{{\sin \theta }} \cdot \left[ {y\left( {x + h} \right) - y\left( {x - h} \right)} \right], $$
(1.3)
whereθ=ωh, becomes appropriate.

Keywords

Multistep Method Oscillatory Integral Oscillatory Function Ofthe Form Order ODEs 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References

  1. [1]
    Alaylioglu, G., Evans, G. A. and Hyslop, J. (1976). The use of Chebyshev series for the evaluation of oscillatory integrals. Comput. J., 19: 258–267.CrossRefzbMATHMathSciNetGoogle Scholar
  2. [2]
    Andrew, A. L. and Paine, J. W. (1985). Correction of Numerov’s eigenvalue estimates. Numer. Math„ 47: 289–300.CrossRefzbMATHMathSciNetGoogle Scholar
  3. [3]
    Bakhvalov, N. S. and Vasil’eva, L. G. (1969). Evaluation of the integrals of oscillating functions by interpolation at nodes of Gaussian quadratures. USSR Comput. Math and Math. Phys., 8: 241–249.CrossRefGoogle Scholar
  4. [4]
    Bocher, P., De Meyer H. and Vanden Berghe G. (1994). Modified Gregory formulae based on mixed interpolation. Intern. Computer Math., 52: 109–122.CrossRefGoogle Scholar
  5. [5]
    Bocher, P., De Meyer H. and Vanden Berghe G. (1994). Numerical solution of Volterra equations based on mixed interpolation. Computers Math. Applic., 27: 1–11.CrossRefzbMATHGoogle Scholar
  6. [6]
    Bock, P. and Murray, F. J. (1952). The use of exponential sums in step by step integration. Mathematical Tables and other Aids to Computation, 6: 63–78 and 138–150.CrossRefMathSciNetGoogle Scholar
  7. [7]
    Brunner, H., Makroglou, A. and Miller, R. K. (1997). Mixed interpolation collocation methods for first and second order Volterra integro-differential equations with periodic solution. Appl. Num. Math. 23: 381–402.CrossRefzbMATHMathSciNetGoogle Scholar
  8. [8]
    Coleman, J. P. (1989). Numerical methods for y“ = f(x, y) via rational approximations for the cosine. IMA J. Numer. Anal., 9: 145–165.CrossRefzbMATHMathSciNetGoogle Scholar
  9. [9]
    Coleman, J. P. and Ixaru, L. Gr. (1996). P-stability and exponential-fitting methods for y“ = f (x, y). IMA J. Numer. Anal., 16: 179–199.CrossRefzbMATHMathSciNetGoogle Scholar
  10. [10]
    Davis, P. J. and Rabinowitz, P. (1984). Methods of Numerical Integration. Academic Press, New York.zbMATHGoogle Scholar
  11. [11]
    De Meyer, H., Vanthournout, J. and Vanden Berghe, G. (1990). On a new type of mixed interpolation. J. Comput. Appl. Math., 30: 55–69.CrossRefzbMATHMathSciNetGoogle Scholar
  12. [12]
    Denk, G. (1993). A new numerical method for the integration of highly oscillatory second-order ordinary differential equations. Appl. Numer. Math., 13: 57–67.CrossRefzbMATHMathSciNetGoogle Scholar
  13. [13]
    Dennis, S. C. R. (1960). The numerical integration of ordinary differential equations pos-sesing exponential type solutions. Proc. Cambridge PhiL Soc., 65: 240–246.CrossRefMathSciNetGoogle Scholar
  14. [14]
    Ehrenmark, U. T. (1988). A three-point formula for numerical quadrature of oscillatory integrals with variable frequency. J. Comput. AppL Math., 21: 87–99.CrossRefzbMATHMathSciNetGoogle Scholar
  15. [15]
    Evans, G. (1993). Practical Numerical Integration. John Wiley & Sons Ltd., Chichester.zbMATHGoogle Scholar
  16. [16]
    Evans, G. A. and Webster, J. R. (1997). A high order, progressive method for the evaluation of irregular oscillatory integrals. AppL Num. Math., 23: 205–218.CrossRefzbMATHMathSciNetGoogle Scholar
  17. [17]
    Gautschi, W. (1961). Numerical integration of ordinary differential equations based on trigonometric polynomials. Numer Math., 3: 381–397.CrossRefzbMATHMathSciNetGoogle Scholar
  18. [18]
    Gautschi, W. (1970). Tables of Gaussian quadrature rules for the calculation of Fourier coefficients. Math. Comp., 24: microfiche.Google Scholar
  19. [19]
    Greenwood, R. E. (1949). Numerical integration of linear sums of exponential functions. Ann. Math. Stat., 20: 608–611.CrossRefzbMATHMathSciNetGoogle Scholar
  20. [20]
    Henrici, P. (1962). Discrete Variable Methods in Ordinary Differential Equations. Wiley, New York.zbMATHGoogle Scholar
  21. [21]
    Ixaru, L. Gr. (1984). Numerical Methods for Differential Equations and Applications. Reidel, Dordrecht - Boston - Lancaster.zbMATHGoogle Scholar
  22. [22]
    Ixaru, L. Gr. (1997). Operations on oscillatory functions. Comput. Phys. Commun., 105: 1–19.CrossRefzbMATHMathSciNetGoogle Scholar
  23. [23]
    Levin, D. (1982). Procedures for computing one and two dimensional integrals of functions with rapid irregular oscillations. Math. Comp, 38: 531–538.CrossRefzbMATHMathSciNetGoogle Scholar
  24. [24]
    Liniger, W., Willoughby, R. A. (1970). Efficient integration methods for stiff systems of ordinary differential equations. SIAM J. Numer AnaL, 7: 47–66.CrossRefzbMATHMathSciNetGoogle Scholar
  25. [25]
    Lyche, T. (1974). Chebyshevian multistep methods for ordinary differential equations. Numer Math., 19: 65–75.CrossRefMathSciNetGoogle Scholar
  26. [26]
    Patterson, T. N. L. (1976). On high precision methods for the evaluation of Fourier integrals with finite and infinite limits. Numer Math., 24: 41–52.CrossRefMathSciNetGoogle Scholar
  27. [27]
    Piessens, R. (1970). Gaussian quadrature formulae for the integration of oscillating func-tions. ZAMM, 50: 698–700.CrossRefzbMATHGoogle Scholar
  28. [28]
    Salzer, H. E. (1962). Trigonometric interpolation and predictor-corrector formulas for numerical integration. ZAMM, 9: 403–412.CrossRefGoogle Scholar
  29. [29]
    Sheffield, C. (1969). Generalized multi-step methods with an application to orbit compu-tation. Celestial Mech., 1: 46–58.CrossRefzbMATHMathSciNetGoogle Scholar
  30. [30]
    Stiefel, E. and Bettis, D. G. (1969). Stabilization of Cowell’s method. Numer Math., 13: 154–175.CrossRefzbMATHMathSciNetGoogle Scholar
  31. [31]
    Van Daele, M., De Meyer, H. and Vanden Berghe, G. (1992). Modified Newton—Cotes formulae for numerical quadrature of oscillatory integrals with two independent frequencies. Intern. J. Comput. Math., 42: 83–97.CrossRefzbMATHGoogle Scholar
  32. [32]
    Vanden Berghe, G., De Meyer, H. and Vanthournout, J. (1990). On a class of modified Newton—Cotes quadrature formulae based upon mixed-type interpolation. J. Comput. Appl. Math., 31: 331–349.CrossRefzbMATHMathSciNetGoogle Scholar
  33. [33]
    Vanden Berghe, G., De Meyer, H. and Vanthournout, J. (1990). A modified Numerov integration method for second order periodic initial—value problems. Intern. J. Computer Math., 32: 233–242.CrossRefzbMATHGoogle Scholar
  34. [34]
    Vanden Berghe, G. and De Meyer, H. (1991). Accurate computation of higher Sturm—Liouville eigenvalues. Numer. Math., 59: 243–254.CrossRefMathSciNetGoogle Scholar
  35. [35]
    Vanthournout, J., Vanden Berghe, G. and De Meyer, H. (1990). Families of backward differentiation methods based on a new type of mixed interpolation. Computers Math. Applic., 11: 19–30.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2004

Authors and Affiliations

  • Liviu Gr. Ixaru
    • 1
  • Guido Vanden Berghe
    • 2
  1. 1.“Horia Hulubei”, Department of Theoretical PhysicsNational Institute for Research and Development for Physics and Nuclear EngineeringBucharestRomania
  2. 2.Department of Applied Mathematics and Computer ScienceUniversity of GentGentBelgium

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