Fast Approximation of Algebraic and Logarithmic Hypersingular Type Singular Integrals with Highly Oscillatory Kernel

Idrissa Kayijuka ,Ali Konuralp ,Şerife Müge Ege ,Fatma Serap Topal

doi:10.28924/2291-8639-18-2020-965

Idrissa Kayijuka , Ali Konuralp + Show 2 more

Open Access

https://doi.org/10.28924/2291-8639-18-2020-965

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Abstract

Herein, highly oscillatory integrals with hypersingular type singularities are studied. After transforming the original integral into a sum of line integrals over a positive semi-infinite interval, a Gauss-related quadrature rule is constructed. The vehicle utilized is the moment’s information. The comparison of two algorithms (Chebyshev and its modified one) to produce the recursion coefficients that satisfy orthogonal polynomial with respect to Gautschi logarithmic weight function, is investigated. Lastly, numerical examples are given to substantiate the effectiveness of the proposed method.

Highlights

The main aspect of our interest in this paper is the computational efficiency of hypersingular integral of the form (1.1) I[s] [f ; μ] =b (x − a)α (b − x)β [ln (x − a) ln (b − x)]s f (x) eiωx a (x − μ)k dx, (k ∈ N ), Received July 20th, 2020; accepted August 10th, 2020; published September 23rd, 2020.2010 Mathematics Subject Classification. 65D30, 65D32
When k = 1 the integral is considered as the Cauchy Principal Value (CPV) integral with a highly oscillatory kernel, 3
When k = 2 the integral is understood in the Hadamard Finite Part (HFP) sense integral with a highly oscillatory kernel, 4

Summary

Introduction

The main aspect of our interest in this paper is the computational efficiency of hypersingular integral of the form (1.1). Algorithms to produce the recursion coefficients of the three-term recurrence relation that satisfy an orthogonal polynomial with respect to Gautschi logarithmic weight function over positive half-infinite interval are compared by employing the moment’s information approach. Automatic computation of the above coefficients plays a significant role in the construction of the numerical quadrature rule (3.6) Their information will facilitate the roots of orthogonal polynomials to be quickly evaluated as characteristic values (eigenvalues) of the Jn (Jacobi) matrix. For constructing recursive coefficients that satisfy three-term recurrence relation for orthogonal polynomials with respect to the given weight functions w (y; m) = yme−y and w (y; m) = (y − 1 − ln y) yme−y for m = α, β > −1, the moments obtained in (3.11)

Algorithm 1

Algorithm 2

Numerical Experiments

Conclusions