Abstract
The method of reduction of a Fredholm integral equation to the linear system is generalized to construction of a complex potential โ an analytic function in an unbounded multiply connected domain with a simple pole at infinity which maps the domain onto a plane with horizontal slits. We consider a locally sourceless, locally irrotational flow on an arbitrary given ๐-connected unbounded domain with impermeable boundary. The complex potential has the form of a Cauchy integral with one linear and ๐ logarithmic summands. The method is easily computable.
References
[1] D. F. Abzalilov and E. A. Shirokova, The approximate conformal mapping onto simply and doubly connected domains, Complex Var. Elliptic Equ. 62 (2017), no. 4, 554โ565. 10.1080/17476933.2016.1227978Search in Google Scholar
[2] N. I. Achieser, Theory of Approximation, Frederick Ungar, New York, 1956. Search in Google Scholar
[3] U. Bรถttger, B. Plรผmper and R. Rupp, Complex potentials, J. Math. Anal. Appl. 234 (1999), no. 1, 55โ66. 10.1006/jmaa.1999.6317Search in Google Scholar
[4] D. Crowdy and J. Marshall, Conformal mappings between canonical multiply connected domains, Comput. Methods Funct. Theory 6 (2006), no. 1, 59โ76. 10.1007/BF03321118Search in Google Scholar
[5] T. K. DeLillo, On some relations among numerical conformal mapping methods, J. Comput. Appl. Math. 19 (1987), no. 3, 363โ377. 10.1016/0377-0427(87)90205-6Search in Google Scholar
[6] T. K. DeLillo, The accuracy of numerical conformal mapping methods: a survey of examples and results, SIAM J. Numer. Anal. 31 (1994), no. 3, 788โ812. 10.1137/0731043Search in Google Scholar
[7] F. D. Gakhov, Boundary Value Problems, Pergamon Press, Oxford, 1966. 10.1016/B978-0-08-010067-8.50007-4Search in Google Scholar
[8] P. N. Ivanshin and E. A. Shirokova, Approximate conformal mappings and elasticity theory, J. Complex Anal. 2016 (2016), Article ID 4367205. 10.1155/2016/4367205Search in Google Scholar
[9] L. A. Lyusternik and V. I. Sobolev, The Elements of Functional Analysis, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1951. Search in Google Scholar
[10] A. H. M. Murid and L.-N. Hu, Numerical conformal mapping of bounded multiply connected regions by an integral equation method, Int. J. Contemp. Math. Sci. 4 (2009), no. 21โ24, 1121โ1147. Search in Google Scholar
[11] M. M. S. Nasser, A. H. M. Murid and A. W. K. Sangawi, Numerical conformal mapping via a boundary integral equation with the adjoint generalized Neumann kernel, TWMS J. Pure Appl. Math. 5 (2014), no. 1, 96โ117. Search in Google Scholar
[12] A. W. K. Sangawi, A. H. M. Murid and M. M. S. Nasser, Annulus with circular slit map of bounded multiply connected regions via integral equation method, Bull. Malays. Math. Sci. Soc. (2) 35 (2012), no. 4, 945โ959. Search in Google Scholar
[13] R. Schinzinger and P. A. A. Laura, Conformal Mapping, Dover, Mineola, 2003. Search in Google Scholar
[14] E. A. Shirokova, On the approximate conformal mapping of the unit disk on a simply connected domain, Russian Math. (Iz. VUZ) 58 (2014), no. 3, 47โ56. 10.3103/S1066369X14030050Search in Google Scholar
[15] R. Wegmann, Fast conformal mapping of multiply connected regions, J. Comput. Appl. Math. 130 (2001), no. 1โ2, 119โ138. 10.1016/S0377-0427(99)00387-8Search in Google Scholar
[16] R. Wegmann, Methods for numerical conformal mapping, Handbook of Complex Analysis: Geometric Function Theory. Vol. 2, Elsevier Science, Amsterdam (2005), 351โ477. 10.1016/S1874-5709(05)80013-7Search in Google Scholar
[17] R. Wegmann and M. M. S. Nasser, The RiemannโHilbert problem and the generalized Neumann kernel on multiply connected regions, J. Comput. Appl. Math. 214 (2008), no. 1, 36โ57. 10.1016/j.cam.2007.01.021Search in Google Scholar
[18] A. A. M. Yunus, A. H. M. Murid and M. M. S. Nasser, Numerical conformal mapping and its inverse of unbounded multiply connected regions onto logarithmic spiral slit regions and straight slit regions, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470 (2014), no. 2162, Article ID 20130514. 10.1098/rspa.2013.0514Search in Google Scholar PubMed PubMed Central
[19] A. A. M. Yunus, A. H. M. Murid and M. M. S. Nasser, Numerical evaluation of conformal mapping and its inverse for unbounded multiply connected regions, Bull. Malays. Math. Sci. Soc. (2) 37 (2014), no. 1, 1โ24. Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Lagrangian multipliers for generalized affine and generalized convex vector optimization problems of set-valued maps
- Perturbations on K-fusion frames
- Integral solutions of nondense impulsive conformable-fractional differential equations with nonlocal condition
- Fixed point theorems for a new generalization of contractive maps in incomplete metric spaces and its application in boundary value problems
- Construction of complex potentials for multiply connected domain
- Solution of a transport equation with discontinuous coefficients
- Solving systems of fractional two-dimensional nonlinear partial Volterra integral equations by using Haar wavelets
- Translation uniqueness of phase retrieval and magnitude retrieval of band-limited signals
- Robust synchronization of chaotic fractional-order systems with shifted Chebyshev spectral collocation method
- M-projective curvature tensor on an (LCS)2n+1-manifold
- An adapted integration method for Volterra integral equation of the second kind with weakly singular kernel
- z-arcs in the thirty degrees sector
Articles in the same Issue
- Frontmatter
- Lagrangian multipliers for generalized affine and generalized convex vector optimization problems of set-valued maps
- Perturbations on K-fusion frames
- Integral solutions of nondense impulsive conformable-fractional differential equations with nonlocal condition
- Fixed point theorems for a new generalization of contractive maps in incomplete metric spaces and its application in boundary value problems
- Construction of complex potentials for multiply connected domain
- Solution of a transport equation with discontinuous coefficients
- Solving systems of fractional two-dimensional nonlinear partial Volterra integral equations by using Haar wavelets
- Translation uniqueness of phase retrieval and magnitude retrieval of band-limited signals
- Robust synchronization of chaotic fractional-order systems with shifted Chebyshev spectral collocation method
- M-projective curvature tensor on an (LCS)2n+1-manifold
- An adapted integration method for Volterra integral equation of the second kind with weakly singular kernel
- z-arcs in the thirty degrees sector