Using modified moments for numerical solution of oscillatory integrals

Authors

  • Mina Shahbaznezhad MS.s student at mathematical faculty in islamic azad university of karaj of iran
  • H. Derili

DOI:

https://doi.org/10.14419/ijamr.v3i2.1819

Published:

2014-03-10

Abstract

Numerical methods for strongly oscillatory and singular functions are given in this paper. We present an alternative numerical solution for of oscillatory integrals of the general form

Where  and  are smooth functions in the interval [a, b]. Also  and   . In order to achieve this goal and avoid singularity, the mentioned integral is solved by the modified moments using interpolating at the Chebyshev points. Finally, we give some experiments for showing efficiency and validity of the method.

 

Keywords: Approximation, Highly Oscillatory Integrals, Hermite Interpolation, Orthogonal Polynomials, Moments, Uniform Approximation.

References

G.A. Evans, J.R. Webster, A high order, progressive method for the evaluation of irregular oscillatory integrals, Applied Numerical Mathematics 23 (1997) 205–218.

G.A. Evans, J.R. Webster, A comparison of some methods for the evaluation of highly oscillatory integrals, Journal of Computational and Applied Mathematics 112 (1999) 55–69.

M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC, 1964.

M.R. Capobianco and G. Criscuolo, On quadrature for Cauchy principal value integrals of oscillatory functions, J. Comput. Appl. Math. 156 (2003), pp. 471 486.

P.I. Davis and P. Rabinowitz, Methods of Numerical Integration, 2nd ed., Academic Press, New York, 1984.

W. Gautschi, Numerical Analysis – An Introduction, Birkhäuser, Basel, 1997.

[I.S. Gradshteyn and I.M. Ryzhik, Tables of Integrals, Series, and Products, 6th ed., Academic Press, San Diego, CA, 2000.

L.N.G. Filon, on a quadrature formula for trigonometric integrals, Proc. Roy. Soc. Edinburgh 49 (1928) 38–47.

P.C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion, SIAM, Philadelphia, 1998.

D. Huybrechs, on the evaluation of highly oscillatory integrals by analytic continuation, SIAM J. Numer. Anal. 44 (3) (2006) 1026–1048.

A. Iserles, S.P. Norsett, Efficient quadrature of highly oscillatory integrals using derivatives, Proc. Roy. Soc. 461 (2005) 1383–1399.

G.E. Okecha, Quadrature formulae for Cauchy principal value integrals of oscillatory kind, Math. Comput. 49 (1987), pp. 259–268.

G.E. Okecha, Hermite interpolation and a method for evaluating Cauchy principal value integrals of oscillatory kind, Kragujevac J. Math. 29 (2006), pp. 91–98.

J. Oliver, The numerical solution of linear recurrence relations, Numer. Math. 11 (1968), pp. 349–360.

H.Wang and S. Xiang, Uniform approximations to Cauchy principal value integrals of oscillatory functions, Appl. Math. Comput. 215 (2009), pp. 1886–1894.

S. Xiang, Efficient Filon-type methods for ∫_a^b▒〖e^iωg(x) f(x)dx〗, Numer. Math. 105 (2007), pp. 633–658.

W.C. Chew, Waves and Fields in Inhomogeneous Media, Van Nostrand Reinhold, New York, 1990.

View Full Article: