A Cauchy integral method to solve the 2D Dirichlet and Neumann problems for irregular simply-connected domains

A. El-Shenawy*, E.A. Shirokova**

Kazan Federal University, Kazan, 420008 Russia

E-mail: *atallahtm@yahoo.com, **Elena.Shirokova@kpfu.ru

Received December 29, 2017

Full text PDF

Abstract

A method for construction of solutions to the continuous approximate 2D Dirichlet and Neumann problems in the arbitrary simply-connected domain with a smooth boundary has been discussed. The numerical finite difference method for solving the Dirichlet problem for an irregular domain meets the difficulties connected with construction of an adequate difference scheme for this domain and its discretization. We have reduced the solving of the Dirichlet problem to the solving of a linear integral equation. Unlike in the case of the Fredholm’s solution to the problem, we have applied the properties of Cauchy integral boundary values rather than the logarithmic potential of a double layer. We have searched the solution to the integral equation in the form of a Fourier polynomial with the coefficients being the solution of a linear equation system. The continuous solution to the Dirichlet problem has the form of the Cauchy integral real part. The values near the boundary of the domain have been obtained with the help of analytic continuation of the Cauchy integral over an inner curve. Comparison of the Dirichlet problem exact solution and the continuous approximate solution has shown an error less than 10⁻⁵. The Neumann problem solution has been reduced to the Dirichlet problem solution for the conjugate harmonic function. Comparison of the Neumann problem exact solution and the continuous approximate solution has shown an error less than 10⁻⁴.

Keywords: Cauchy integral, Fourier polynomial, Dirichlet problem, Neumann problem, Fredholm integral equation, simply-connected domain

References

1. Fredholm E.I. Sur une classe d’equations fonctionnelles. Acta Math., 1903, vol. 27, pp. 365– 390. doi:10.1007/BF02421317. (In French)
2. Hess J.L., Smith A.M.O. Calculation of non-lifting potential flow about arbitrary threedimensional bodies. J. Ship Res., 1964, vol. 8, pp.22–44.
3. Jaswon M. Integral equation methods in potential theory, I. Proc. R. Soc. London, Ser. A, 1963, vol. 275, no. 1360, pp. 23–32. doi: 10.1098/rspa.1963.0152.
4. Symm G. Integral equation methods in potential theory, II. Proc. R. Soc. Ser. A, 1963, vol. 275, no. 1360, 1963, pp. 33–46. doi: 10.1098/rspa.1963.0153.
5. Tikhonov A.N., Samarskii A.A. Equations of Mathematical Physics. New York, Dover Publ., 1963. 800 p.
6. Tricomi F.G. Integral Equations. New York, Dover Publ., 1982. 238 p.
7. Samarskii A.A., Gulin A.V. Chislennye metody [Numerical Methods]. Moscow, Nauka, 1989. 432 p. (In Russian)
8. Gakhov F.D. Boundary Value Problems. Oxford, Pergamon Press, 1966. 584 p.
9. Shirokova E.A. On the approximate conformal mapping of the unit disk on simply connected domain. Russ. Math., 2014, vol. 58, no. 3, pp. 47–56. doi: 10.3103/S1066369X14030050.
10. Ivanshin P.N., Shirokova E.A. Approximate conformal mappings and elasticity theory. J. Complex Anal., 2016, vol. 2016, art. 4367205, pp. 1–8. doi: 10.1155/2016/4367205.
11. Abzalilov D.F., Shirokova E.A. The approximate conformal mappings onto simply and doubly connected domains. Complex Var. Elliptic Equations, 2017, vol. 62, no. 4, pp. 554– 565. doi: 10.1080/17476933.2016.1227978.
12. Krantz S.G. Handbook of Complex Variables. New York, Springer, 1999. 290 p.

For citation: El-Shenawy A., Shirokova E.A. A Cauchy integral method to solve the 2D Dirichlet and Neumann problems for irregular simply-connected domains. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, vol. 160, no. 4, pp. 778–787.

The content is available under the license Creative Commons Attribution 4.0 License.