Approximation to Logarithmic-Cauchy Type Singular Integrals with Highly Oscillatory Kernels
In this paper, a fast and accurate numerical Clenshaw-Curtis quadrature is proposed for the approximation of highly oscillatory integrals with Cauchy and logarithmic singularities, <inline-formula> <math display="inline"> <semantics> <mrow> <msubsup> <mi>...
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MDPI AG
2019-05-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/11/6/728 |
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doaj-658e4f05f4354ea5b3a3d1dd8e47d1332020-11-25T02:31:27ZengMDPI AGSymmetry2073-89942019-05-0111672810.3390/sym11060728sym11060728Approximation to Logarithmic-Cauchy Type Singular Integrals with Highly Oscillatory KernelsSAIRA0Shuhuang Xiang1School of Mathematics and Statistics, Central South University, Changsha 410083, ChinaSchool of Mathematics and Statistics, Central South University, Changsha 410083, ChinaIn this paper, a fast and accurate numerical Clenshaw-Curtis quadrature is proposed for the approximation of highly oscillatory integrals with Cauchy and logarithmic singularities, <inline-formula> <math display="inline"> <semantics> <mrow> <msubsup> <mi>⨍</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> <mfrac> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mi>log</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>−</mo> <mi>α</mi> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mi>i</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> </mrow> <mrow> <mi>x</mi> <mo>−</mo> <mi>t</mi> </mrow> </mfrac> <mi>d</mi> <mi>x</mi> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>∉</mo> <mo>(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> <mo>,</mo> <mi>α</mi> <mo>∈</mo> <mo>[</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>]</mo> </mrow> </semantics> </math> </inline-formula> for a smooth function <inline-formula> <math display="inline"> <semantics> <mrow> <mi>f</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. This method consists of evaluation of the modified moments by stable recurrence relation and Cauchy kernel is solved by steepest descent method that transforms the oscillatory integral into the sum of line integrals. Later theoretical analysis and high accuracy of the method is illustrated by some examples.https://www.mdpi.com/2073-8994/11/6/728Clenshaw-Curtis quadraturesteepest descent methodlogarithmic singularitiesCauchy singularityhighly oscillatory integrals |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
SAIRA Shuhuang Xiang |
| spellingShingle |
SAIRA Shuhuang Xiang Approximation to Logarithmic-Cauchy Type Singular Integrals with Highly Oscillatory Kernels Symmetry Clenshaw-Curtis quadrature steepest descent method logarithmic singularities Cauchy singularity highly oscillatory integrals |
| author_facet |
SAIRA Shuhuang Xiang |
| author_sort |
SAIRA |
| title |
Approximation to Logarithmic-Cauchy Type Singular Integrals with Highly Oscillatory Kernels |
| title_short |
Approximation to Logarithmic-Cauchy Type Singular Integrals with Highly Oscillatory Kernels |
| title_full |
Approximation to Logarithmic-Cauchy Type Singular Integrals with Highly Oscillatory Kernels |
| title_fullStr |
Approximation to Logarithmic-Cauchy Type Singular Integrals with Highly Oscillatory Kernels |
| title_full_unstemmed |
Approximation to Logarithmic-Cauchy Type Singular Integrals with Highly Oscillatory Kernels |
| title_sort |
approximation to logarithmic-cauchy type singular integrals with highly oscillatory kernels |
| publisher |
MDPI AG |
| series |
Symmetry |
| issn |
2073-8994 |
| publishDate |
2019-05-01 |
| description |
In this paper, a fast and accurate numerical Clenshaw-Curtis quadrature is proposed for the approximation of highly oscillatory integrals with Cauchy and logarithmic singularities, <inline-formula> <math display="inline"> <semantics> <mrow> <msubsup> <mi>⨍</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> <mfrac> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mi>log</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>−</mo> <mi>α</mi> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mi>i</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> </mrow> <mrow> <mi>x</mi> <mo>−</mo> <mi>t</mi> </mrow> </mfrac> <mi>d</mi> <mi>x</mi> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>∉</mo> <mo>(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> <mo>,</mo> <mi>α</mi> <mo>∈</mo> <mo>[</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>]</mo> </mrow> </semantics> </math> </inline-formula> for a smooth function <inline-formula> <math display="inline"> <semantics> <mrow> <mi>f</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. This method consists of evaluation of the modified moments by stable recurrence relation and Cauchy kernel is solved by steepest descent method that transforms the oscillatory integral into the sum of line integrals. Later theoretical analysis and high accuracy of the method is illustrated by some examples. |
| topic |
Clenshaw-Curtis quadrature steepest descent method logarithmic singularities Cauchy singularity highly oscillatory integrals |
| url |
https://www.mdpi.com/2073-8994/11/6/728 |
| work_keys_str_mv |
AT saira approximationtologarithmiccauchytypesingularintegralswithhighlyoscillatorykernels AT shuhuangxiang approximationtologarithmiccauchytypesingularintegralswithhighlyoscillatorykernels |
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