Direct methods for solving singular integral equations
with discontinuous coefficients
|
StatusThe thesis was presented on the 9 April, 2008Approved by NCAA on the 19 June, 2008 Abstract – 0.26 Mb / in romanianThesisCZU 519.8:519.6
|
Keywords
Singular integral equations with Cauchy kernel, direct methods, discontinuous coefficients, convergence manifold of an operator, local principles, nöetherian properties, complexity theory, fast Fourier transform.Summary
The main scope of the PhD thesis consists in the elaboration and the theoretical justification of direct methods for the approximate solving of one-dimensional singular integral equations (SIE) and their systems with coefficients which have discontinuities of the first kind on the contour of integration.
The computational schemes of the collocation method, the reduction method and the method of mechanic quadratures are proposed for solving SIE defined on the unit circumference of the complex plane. These schemes are obtained for the equations, kernels in regular part of which do not have discontinuity, as well as for the equations, regular kernels of which have weak singularities. The theorems regarding the solvability of the computational schemes of these methods and the theorems regarding the convergence of the corresponding approximate solutions to the exact solution in the norm of the Lebesgue space L2 are proved. The cases of finite number of points of discontinuity and countable set of points of discontinuity are considered. The obtained results are generalized for systems of SIE and the used methods for this case appreciably differ from the methods for a single equation. It is proved that the proposed computational schemes are stable relative to small perturbations of coefficients, a kernel and a right-hand member, and the condition numbers of the computational schemes are close to the condition numbers of the considered equations.
The effective method for the approximate solving of SIE, which combines the algorithms of the method of mechanic quadratures and the fast Fourier transform, is proposed. It is shown that the corresponding algorithm is substantially more economical from the point of view of calculations in comparison with the collocation method or the method of mechanic quadratures. The optimal asymptotical estimate of the rate of convergence in the scale of Sobolev spaces is justified. It is also marked out the class of iterative algorithms for solving of systems of equations which are obtained when discretizing SIE with discontinuous coefficients. These algorithms permit essential reducing of computational cost for finding an approximate solution, at that not losing in the quality of other numerical characteristics (they are numerically stable and allow us to check precision in the course of iterations without calculation of approximation of a solution).
For the concrete problem from the elasticity theory it is obtained the modeling SIE which was numerically solved by using the proposed algorithms. By means of the programming language C++ and the system MATLAB the graphical representations of the considered process are obtained.
Oficial Reviewers
- Gheorghe Moraru
doctor habilitat, professor, Technical University of Moldova - Vladislav Seiciuc
doctor, associate professor (docent), Moldova State University
Theses
