On the approximate evaluation of oscillatory-singular integrals

M.K Hota,P Ojha,A.K Saha,P.K Mohanty

doi:10.1080/23311835.2017.1314066

Abstract

AbstractIn this paper an efficient numerical scheme is proposed for the numerical computation of the Cauchy type oscillatory integral ∫-11coswxxf(x)dx; where f(x) is a well-behaved function without having any kind of singularity in the range of integration [−1; 1]. The scheme is devised with the help of quadrature rule meant for the approximate evaluation of Cauchy principal value of integrals of the type ∫-11f(x)dx; and a quasi exact quadrature meant for the numerical integration of Filon-type integrals. The error bounds are determined and the scheme numerically verified by some standard test integrals.

Highlights

Singular integral of these two kinds:f (x) I(f ) = P ∫ x −1 and (1.1) f (x)I(f, w) = P ∫ x cos wxdx (1.2)−1 where, f(x) is analytic on [−1, 1] though look as similar but, the second differs from the former by its high oscillatory characteristics for large w ∈ R−{0}
It is observed that classical quadrature rules meant for the numerical approximation of the CPV integral (1.1) diverge significantly and lead to uncontrolled instabilities when these rules are employed for the numerical integration of the second integral (1.2) with increasing values of |w|
It is well-known that these two improper integrals exist, if f satisfies the Holder’s condition in [−1, 1]

Summary

Introduction

It is observed that classical quadrature rules meant for the numerical approximation of the CPV integral (1.1) diverge significantly and lead to uncontrolled instabilities when these rules are employed for the numerical integration of the second integral (1.2) with increasing values of |w|. It is well-known that these two improper integrals exist, if f satisfies the Holder’s condition in [−1, 1]. We suggest an algebraic method to determine the nodes and weights of the generalized quadrature rule Qn(x) meant for the numerical approximation of the CPV integral (1.1).

Scheme for the approximate evaluation of Cauchy type oscillatory integral

Conclusion