Computing the zeros of the Szegö kernel for doubly connected regions using conformal mapping

Nuraddeen S. Gafai; Ali H. M. Murid; Samir Naqos; Nur H. A. A. Wahid; Nuraddeen S. Gafai; Ali H. M. Murid; Samir Naqos; Nur H. A. A. Wahid

doi:10.3934/math.2023607

AIMS Mathematics

2023, Volume 8, Issue 5: 12040-12061. doi: 10.3934/math.2023607

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Research article

Computing the zeros of the Szegö kernel for doubly connected regions using conformal mapping

1.
Department of Mathematics and Statistics, Umaru Musa Yar'adua University Katsina, Nigeria
2.
Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, UTM Johor Bahru 81310, Johor, Malaysia
3.
School of Mathematical Sciences, College of Computing, Informatics and Media, Universiti Teknologi MARA, Shah Alam 40450, Selangor, Malaysia

Received: 02 January 2023 Revised: 13 March 2023 Accepted: 17 March 2023 Published: 21 March 2023
MSC : 30C40, 33C15, 65E05, 65E10, 65R20

An explicit formula for the zero of the Szegö kernel for an annulus region is well-known. There exists a transformation formula for the Szegö kernel from a doubly connected region onto an annulus. Based on conformal mapping, we derive an analytical formula for the zeros of the Szegö kernel for a general doubly connected region with smooth boundaries. Special cases are the explicit formulas for the zeros of the Szegö kernel for doubly connected regions bounded by circles, limacons, ellipses, and ovals of Cassini. For a general doubly connected region with smooth boundaries, the zero of the Szegö kernel must be computed numerically. This paper describes the application of conformal mapping via integral equation with the generalized Neumann kernel for computing the zeros of the Szegö kernel for smooth doubly connected regions. Some numerical examples and comparisons are also presented. It is shown that the conformal mapping approach also yields good accuracy for a narrow region or region with boundaries that are close to each other.
- Szegö kernel,
- conformal mapping,
- doubly connected regions,
- generalized Neumann kernel,
- integral equation
Citation: Nuraddeen S. Gafai, Ali H. M. Murid, Samir Naqos, Nur H. A. A. Wahid. Computing the zeros of the Szegö kernel for doubly connected regions using conformal mapping[J]. AIMS Mathematics, 2023, 8(5): 12040-12061. doi: 10.3934/math.2023607

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Abstract

An explicit formula for the zero of the Szegö kernel for an annulus region is well-known. There exists a transformation formula for the Szegö kernel from a doubly connected region onto an annulus. Based on conformal mapping, we derive an analytical formula for the zeros of the Szegö kernel for a general doubly connected region with smooth boundaries. Special cases are the explicit formulas for the zeros of the Szegö kernel for doubly connected regions bounded by circles, limacons, ellipses, and ovals of Cassini. For a general doubly connected region with smooth boundaries, the zero of the Szegö kernel must be computed numerically. This paper describes the application of conformal mapping via integral equation with the generalized Neumann kernel for computing the zeros of the Szegö kernel for smooth doubly connected regions. Some numerical examples and comparisons are also presented. It is shown that the conformal mapping approach also yields good accuracy for a narrow region or region with boundaries that are close to each other.

References

[1]	N. Kerzman, E. M. Stein, The Cauchy kernel, the Szegö kernel, and the Riemann mapping function, Math. Ann., 236 (1978), 85–93. https://doi.org/10.1007/BF01420257 doi: 10.1007/BF01420257
[2]	N. Kerzman, M. R. Trummer, Numerical conformal mapping via the Szegö kernel, J. Comput. Appl. Math., 14 (1986), 111–123. https://doi.org/10.1016/0377-0427(86)90133-0 doi: 10.1016/0377-0427(86)90133-0
[3]	T. J. Tegtmeyer, A. D. Thomas, The Ahlfors map and Szegö kernel for an annulus, Rocky Mt. J. Math., 29 (1999), 709–723.
[4]	N. H. A. A. Wahid, A. H. M. Murid, M. I. Muminov, Convergence of the series for the Szegö kernel for an annulus region, AIP Conf. Proc., 1974 (2018), 1–8. http://doi.org/10.1063/1.5041669 doi: 10.1063/1.5041669
[5]	N. S. Gafai, A. H. M. Murid, N. H. A. A. Wahid, Infinite product representation for the Szegö kernel for an annulus, J. Funct. Spaces, 2022 (2022), 1–9. http://doi.org/10.1155/2022/3763450 doi: 10.1155/2022/3763450
[6]	S. R. Bell, The Cauchy transform, potential theory, and conformal mapping, Boca Raton: CRC Press, 1992.
[7]	S. R. Bell, Numerical computation of the Ahlfors map of a multiply connected planar domain, J. Math. Anal. Appl., 120 (1986), 211–217. https://doi.org/10.1016/0022-247X(86)90211-8 doi: 10.1016/0022-247X(86)90211-8
[8]	T. J. Tegtmeyer, The Ahlfors map and Szegö kernel in multiply connected domains, PhD thesis, Purdue University, 1998.
[9]	K. Nazar, A. W. K. Sangawi, A. H. M. Murid, Y. S. Hoe, The computation of zeros of Ahlfors map for doubly connected regions, AIP Conf. Proc., 1750 (2016), 020007. https://doi.org/10.1063/1.4954520 doi: 10.1063/1.4954520
[10]	K. Nazar, A. H. M. Murid, A. W. K. Sangawi, Integral equation for the Ahlfors map on multiply connected regions, J. Teknol., 73 (2015), 1–9.
[11]	A. H. M. Murid, N. H. A. A. Wahid, M. I. Muminov, Methods and comparisons for computing the zeros of the Ahlfors map for doubly connected regions, AIP Conf. Proc., 2423 (2021), 020026. https://doi.org/10.1063/5.0075348 doi: 10.1063/5.0075348
[12]	P. Henrici, Applied and computational complex analysis, Vol. 3, New York: John Wiley, 1986.
[13]	P. K. Kythe, Computational conformal mapping, Boston: Birkhauser, 1998.
[14]	G. T. Symm, Conformal mapping of doubly-connected domains, Numer. Math., 13 (1969), 448–457. https://doi.org/10.1007/BF02163272 doi: 10.1007/BF02163272
[15]	M. M. S. Nasser, A boundary integral equation for conformal mapping of bounded multiply connected regions, Comput. Methods Funct. Theory, 9 (2009), 127–143. https://doi.org/10.1007/BF03321718 doi: 10.1007/BF03321718
[16]	M. M. S. Nasser, Numerical conformal mapping via a boundary integral equation with the generalized Neumann kernel, SIAM J. Sci. Comput., 31 (2009), 1695–1715. https://doi.org/10.1137/070711438 doi: 10.1137/070711438
[17]	M. M. S. Nasser, F. A. A. Al-Shihri, A fast boundary integral equation method for conformal mapping of multiply connected regions, SIAM J. Sci. Comput., 35 (2013), A1736–A1760. https://doi.org/10.1137/120901933 doi: 10.1137/120901933
[18]	M. M. S. Nasser, Fast solution of boundary integral equations with the generalized Neumann kernel, Electroin. Trans. Numer. Anal., 44 (2015), 189–229.
[19]	N. H. A. A. Wahid, A. H. M. Murid, M. I. Muminov, Analytical solution for finding the second zero of the Ahlfors map for an annulus region, J. Math., 2019 (2019), 1–11. https://doi.org/10.1155/2019/6961476 doi: 10.1155/2019/6961476
[20]	M. M. S. Nasser, M. Vuorinen, Computation of conformal invariants, Appl. Math. Comput., 389 (2021), 125617. https://doi.org/10.1016/j.amc.2020.125617 doi: 10.1016/j.amc.2020.125617
[21]	R. Wegmann, M. M. S. Nasser, The Riemann-Hilbert problem and the generalized Neumann kernel on multiply connected regions, J. Comput. Appl. Math., 214 (2008), 36–57. https://doi.org/10.1016/j.cam.2007.01.021 doi: 10.1016/j.cam.2007.01.021
[22]	M. M. S. Nasser, Fast computation of the circular map, Comput. Methods Funct. Theory, 15 (2014) 189–223. https://doi.org/10.1007/s40315-014-0098-3 doi: 10.1007/s40315-014-0098-3
[23]	K. E. Atkinson, The Numerical solution of integral equations of the second kind, Cambridge: Cambridge University Press, 1997.
[24]	L. N. Trefethen, J. A. C. Weideman, The exponentially convergent trapezoidal rule, SIAM Rev., 56 (2014), 385–458. https://doi.org/10.1137/130932132 doi: 10.1137/130932132
[25]	L. Greengard, Z. Gimbutas, FMMLIB2D: A MATLAB toolbox for fast multipole method in two dimensions, Version 1.2. Edition, 2012. Available from: http://www.cims.nyu.edu/cmcl/fmm2dlib/fmm2dlib.html.
[26]	E. B. Saff, A. D. Snider, Fundamentals of complex analysis with applications to engineering and science, 3 Eds., Pearson, 2014.
[27]	J. Helsing, R. Ojala, On the evaluation of layer potentials close to their sources, J. Comput. Phys., 227 (2008), 2899–2921. https://doi.org/10.1016/j.jcp.2007.11.024 doi: 10.1016/j.jcp.2007.11.024
[28]	M. M. S. Nasser, A. H. M. Murid, Z. Zamzamir, A boundary integral method for the Riemann-Hilbert problem in domains with corners, Complex Var. Elliptic Equ., 53 (2008), 989–1008. https://doi.org/10.1080/17476930802335080 doi: 10.1080/17476930802335080
[29]	A. Rathsfeld, Iterative solution of linear systems arising from the Nyström method for doubly-layer potential equation over curves with corners, Math. Methods Appl. Sci., 16 (1993), 443–455. https://doi.org/10.1002/mma.1670160604 doi: 10.1002/mma.1670160604
[30]	R. Kress, A Nyström method for boundary integral equations in domains with corners, Numer. Math., 58 (1990), 145–161. https://doi.org/10.1007/BF01385616 doi: 10.1007/BF01385616
[31]	H. Hakula, T. Quash, A. Rasila, Conjugate function method for numerical conformal mapping, J. Comput. Appl. Math., 237 (2013), 340–353. https://doi.org/10.1016/j.cam.2012.06.003 doi: 10.1016/j.cam.2012.06.003
[32]	H. Hakula, A. Rasila, M. Vuorinen, On moduli of rings and quadrilateral algorithms and experiment, SIAM J. Sci. Comput., 33 (2011), 279–302. https://doi.org/10.1137/090763603 doi: 10.1137/090763603

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