A Cauchy integral method to solve the 2D Dirichlet and Neumann problems for irregular simply-connected domains
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 diffic...
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Kazan Federal University
2018-12-01
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| Series: | Учёные записки Казанского университета: Серия Физико-математические науки |
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| Online Access: | https://kpfu.ru/a-cauchy-integral-method-to-solve-the-2d-403971.html |
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| _version_ | 1846145417992994816 |
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| author | A. El-Shenawy E.A. Shirokova |
| author_facet | A. El-Shenawy E.A. Shirokova |
| author_sort | A. El-Shenawy |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-0b4a30fdb7df4301ad6d55b57bb34966 |
| institution | Kabale University |
| issn | 2541-7746 2500-2198 |
| language | English |
| publishDate | 2018-12-01 |
| publisher | Kazan Federal University |
| record_format | Article |
| series | Учёные записки Казанского университета: Серия Физико-математические науки |
| spelling | doaj-art-0b4a30fdb7df4301ad6d55b57bb349662024-12-02T05:56:54ZengKazan Federal UniversityУчёные записки Казанского университета: Серия Физико-математические науки2541-77462500-21982018-12-011604778787A Cauchy integral method to solve the 2D Dirichlet and Neumann problems for irregular simply-connected domainsA. El-Shenawy0E.A. Shirokova1Kazan Federal University, Kazan, 420008 RussiaKazan Federal University, Kazan, 420008 RussiaA 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.https://kpfu.ru/a-cauchy-integral-method-to-solve-the-2d-403971.htmlcauchy integralfourier polynomialdirichlet problemneumann problemfredholm integral equationsimply-connected domain |
| spellingShingle | A. El-Shenawy E.A. Shirokova A Cauchy integral method to solve the 2D Dirichlet and Neumann problems for irregular simply-connected domains Учёные записки Казанского университета: Серия Физико-математические науки cauchy integral fourier polynomial dirichlet problem neumann problem fredholm integral equation simply-connected domain |
| title | A Cauchy integral method to solve the 2D Dirichlet and Neumann problems for irregular simply-connected domains |
| title_full | A Cauchy integral method to solve the 2D Dirichlet and Neumann problems for irregular simply-connected domains |
| title_fullStr | A Cauchy integral method to solve the 2D Dirichlet and Neumann problems for irregular simply-connected domains |
| title_full_unstemmed | A Cauchy integral method to solve the 2D Dirichlet and Neumann problems for irregular simply-connected domains |
| title_short | A Cauchy integral method to solve the 2D Dirichlet and Neumann problems for irregular simply-connected domains |
| title_sort | cauchy integral method to solve the 2d dirichlet and neumann problems for irregular simply connected domains |
| topic | cauchy integral fourier polynomial dirichlet problem neumann problem fredholm integral equation simply-connected domain |
| url | https://kpfu.ru/a-cauchy-integral-method-to-solve-the-2d-403971.html |
| work_keys_str_mv | AT aelshenawy acauchyintegralmethodtosolvethe2ddirichletandneumannproblemsforirregularsimplyconnecteddomains AT eashirokova acauchyintegralmethodtosolvethe2ddirichletandneumannproblemsforirregularsimplyconnecteddomains AT aelshenawy cauchyintegralmethodtosolvethe2ddirichletandneumannproblemsforirregularsimplyconnecteddomains AT eashirokova cauchyintegralmethodtosolvethe2ddirichletandneumannproblemsforirregularsimplyconnecteddomains |