COMPUTATIONAL EFFICIENCY OF SINGULAR AND OSCILLATORY INTEGRALS WITH ALGEBRAIC SINGULARITIES
IDRISSA KAYIJUKA *
Department of Mathematics, Ege University, Izmir, 35040, Turkey.
JEAN PIERRE NDABAKURANYE
School of Physics, University of Melbourne, Victoria, 3010, Australia.
A. IHSAN HASÇELIK
Department of Mathematics, Gaziantep University, Gaziantep, 27310, Turkey.
*Author to whom correspondence should be addressed.
Abstract
In this paper, we present two methods: Modified Clenshaw-Curtis and the Gauss-Jacobi methods. These methods are commonly used in the evaluation of the finite Fourier transforms of integrands with endpoint singularities. In the first method, the integrand is truncated by the Chebyshev series, term by term, and then its singularity types are evaluated using recurrence relations. This method is more efficient for low-frequency values. On the other hand, the Gauss Jacobi method is found to be accurate in the evaluation of integrals with fairly high-frequency values; such as 1000. MATHEMATICA codes, for both methods, are provided for the purpose of testing the efficiency of automatic computation. Lastly, the illustrative examples are considered with regards to reliability, accuracy, and comparison of the methods outlined.
Keywords: Singular oscillatory integrals, Clenshaw-Curtis quadrature, Recurrence Relations, Chebyshev polynomials.