Suitable Gauss and Filon-type methods for oscillatory integrals with an algebraic singularity

Abstract

The standard classic integration rules give inaccurate results for ∫ 0 1 t α f ( t ) sin ( ω / t r ) d t and ∫ 0 1 f ( t ) t α cos ( ω / t r ) d t where ω , r > 0 , α + r > − 1 are real numbers and f is any sufficiently smooth function on [ 0 , 1 ] . These integrals have been investigated for the special case α = 0 in Hascelik [A.I. Hascelik, On numerical computation of integrals with integrands of the form f ( x ) sin ( 1 / x r ) on [ 0 , 1 ] (2007), in press] and for the case ( r = 1 , α = 0 ) in Gautschi [W. Gautschi, Computing polynomials orthogonal with respect to densely oscillating and exponentially decaying weight functions and related integrals, J. Comput. Appl. Math. 184 (2005) 493–504]. In this work we construct suitable Gauss quadrature rules for approximating these integrals in high accuracy. The required three-term recurrence coefficients are computed by the Chebyshev algorithm using arbitrary precision arithmetic. We also give appropriate Filon-type methods for these integrals, with related error bounds. Some numerical examples are given to test the new methods.