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Volume 18, Issue 4
Fast Gauss-Related Quadrature for Highly Oscillatory Integrals with Logarithm and Cauchy-Logarithmic Type Singularities

​Idrissa Kayijuka, Serife Muge Ege, Fatma Serap Topal & Ali Konuralp

Int. J. Numer. Anal. Mod., 18 (2021), pp. 442-457.

Published online: 2021-05

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  • Abstract

This paper presents an efficient method for the computation of two highly oscillatory integrals having logarithmic and Cauchy-logarithmic singularities. This approach first requires the transformation of the original oscillatory integrals into a sum of line integrals with semi-infinite intervals. Afterwards, the coefficients of the three-term recurrence relation that satisfy the orthogonal polynomial are obtained by using the method based on moments, where classical Laguerre and Gautschi's logarithmic weight functions are employed. The algorithm reveals that with fixed $n$, the method is capable of achieving significant figures within a short time. Furthermore, the approach yields higher accuracy as the frequency increases. The results of numerical experiments are given to substantiate our theoretical analysis.

  • Keywords

Highly oscillatory integrals, modified Chebyshev algorithm, steepest descent method, Cauchy principal value integrals, logarithmic weight function, algebraic and logarithm singular integrals.

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  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-18-442, author = {Kayijuka , ​IdrissaEge , Serife MugeTopal , Fatma Serap and Konuralp , Ali}, title = {Fast Gauss-Related Quadrature for Highly Oscillatory Integrals with Logarithm and Cauchy-Logarithmic Type Singularities}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2021}, volume = {18}, number = {4}, pages = {442--457}, abstract = {

This paper presents an efficient method for the computation of two highly oscillatory integrals having logarithmic and Cauchy-logarithmic singularities. This approach first requires the transformation of the original oscillatory integrals into a sum of line integrals with semi-infinite intervals. Afterwards, the coefficients of the three-term recurrence relation that satisfy the orthogonal polynomial are obtained by using the method based on moments, where classical Laguerre and Gautschi's logarithmic weight functions are employed. The algorithm reveals that with fixed $n$, the method is capable of achieving significant figures within a short time. Furthermore, the approach yields higher accuracy as the frequency increases. The results of numerical experiments are given to substantiate our theoretical analysis.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/19115.html} }
TY - JOUR T1 - Fast Gauss-Related Quadrature for Highly Oscillatory Integrals with Logarithm and Cauchy-Logarithmic Type Singularities AU - Kayijuka , ​Idrissa AU - Ege , Serife Muge AU - Topal , Fatma Serap AU - Konuralp , Ali JO - International Journal of Numerical Analysis and Modeling VL - 4 SP - 442 EP - 457 PY - 2021 DA - 2021/05 SN - 18 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/19115.html KW - Highly oscillatory integrals, modified Chebyshev algorithm, steepest descent method, Cauchy principal value integrals, logarithmic weight function, algebraic and logarithm singular integrals. AB -

This paper presents an efficient method for the computation of two highly oscillatory integrals having logarithmic and Cauchy-logarithmic singularities. This approach first requires the transformation of the original oscillatory integrals into a sum of line integrals with semi-infinite intervals. Afterwards, the coefficients of the three-term recurrence relation that satisfy the orthogonal polynomial are obtained by using the method based on moments, where classical Laguerre and Gautschi's logarithmic weight functions are employed. The algorithm reveals that with fixed $n$, the method is capable of achieving significant figures within a short time. Furthermore, the approach yields higher accuracy as the frequency increases. The results of numerical experiments are given to substantiate our theoretical analysis.

Kayijuka , ​IdrissaEge , Serife MugeTopal , Fatma Serap and Konuralp , Ali. (2021). Fast Gauss-Related Quadrature for Highly Oscillatory Integrals with Logarithm and Cauchy-Logarithmic Type Singularities. International Journal of Numerical Analysis and Modeling. 18 (4). 442-457. doi:
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