A Boundary Integral Equation Related to the Ahlfors Map
DOI:
https://doi.org/10.11113/mjfas.v17n2.1911Keywords:
Ahlfors map, Multiply connected region, Kerzman-Stein kernel, Neumann kernelAbstract
The Ahlfors map is a mapping function that maps a multiply connected region onto a unit disk. This paper presents a new boundary integral equation related to the Ahlfors map of a bounded multiply connected region. The boundary integral equation is constructed from a boundary relationship satisfied by the Ahlfors map of a multiply connected region.References
Bell S. R., 1986, Numerical computation of the Ahlfors map of a multiply connected planar domain, Journal of Mathematical Analysis and Applications, 120, 211-217.
Tegtmeyer T. J., 1998, The Ahlfors Map and Szegӧ Kernel in Multiply Connected Domains, [Ph.D. thesis], Purdue University.
Tegtmeyer T. J. and Thomas A. D., 1999, The Ahlfors map and Szegӧ kernel for an annulus, Rocky Mountain Journal of Mathematics, 29(2), 709–723.
Kerzman N. and Stein E. M., 1978, The Cauchy kernel, the Szegӧ kernel, and the Riemann mapping function, Mathematische Annalen, 236(1), 85-93.
Murid A. H. M. and Razali M. R. M., 1999, An integral equation method for conformal mapping of doubly connected regions, Matematika, 15(2), 79-93.
Nasser M. M. S. and Murid A. H. M., 2013, A boundary integral equation with the generalized Neumann kernel for the Ahlfors map, Clifford Analysis, Clifford Algebras and their Applications, 2(4), 307-312.
Nazar K., Murid A. H. M. and Sangawi A. W. K., 2015, Integral equation for the Ahlfors map on multiply connected regions, Jurnal Teknologi, 73(1), 1-9.
Nazar K., Murid A. H. M and Sangawi A. W. K., 2015, Some integral equations related to the Ahlfors map for multiply connected regions, AIP Conference Proceedings, 1691, 1-8.
Bell S. R., 1992, The Cauchy Transform, Potential Theory, and Conformal Mapping, CRC Press, Boca Raton.
Kerzman N. and Trummer M. R., 1986, Numerical conformal mapping via the Szegӧ kernel, Journal of Computational and Applied Mathematics, 14(1-2), 111–123.
Helsing J. and Ojala R., 2008, On the evaluation of layer potentials close to their sources, Journal of Computational Physics, 227(5), 2899–2921.
Nazar K., Sangawi A. W. K., Murid A. H. M. and Hoe Y. S., 2011, The computation of zeros of Ahlfors map for doubly connected regions, AIP Conference Proceedings, 1750, 1-9.
Nazar K., Murid A. H. M. and Sangawi A. W. K., 2017, The computation of zeros of Ahlfors map for multiply connected regions, AIP Conference Proceedings, 1755, 1-8.
Sangawi A. W. K., 2012, Boundary Integral Equations for Conformal Mappings of Bounded Multiply Connected Regions, [Ph.D. thesis], Universiti Teknologi Malaysia.
Henrici P., 1986, Applied and Computational Complex Analysis Vol. 3, New York: John Wiley and Sons.
Wegmann R., Murid A. H. M. and Nasser M. M. S., 2005, The Riemann–Hilbert problem and the generalized Neumann kernel, Journal of Computational and Applied Mathematics, 182(2), 388–415.
Wegmann R. and Nasser M. M. S., 2008, The Riemann–Hilbert problem and the generalized Neumann kernel on multiply connected regions, Journal of Computational and Applied Mathematics, 214, 36–57.
Nasser M. M. S., Murid A. H. M., Ismail M. and Alejaily E. M. A., 2011, Boundary integral equations with the generalized Neumann kernel for Laplace’s equation in multiply connected regions, Applied Mathematics and Computation, 217(9), 4710–4727.
Wahid N. H. A. A., Murid A. H. M. and Muminov, M. I., 2019, Analytical solution for finding the second zero of the Ahlfors map for an annulus region, Journal of Mathematics, 2019, 1-11.
