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P. A. Krutitskii 《Differential Equations》2010,46(9):1287-1298
We consider the Dirichlet problem for the Laplace equation in a plane domain with smooth cuts of arbitrary form for the case
in which the solution is not continuous at the endpoints of the cuts. We present a well-posed statement of the problem, prove
the existence and uniqueness theorems for the classical solution, obtain an integral representation of the solution, and use
it to analyze the properties of the solution. We show that, as a rule, the Dirichlet problem in this setting has no weak solutions,
even though there exists a classical solution. 相似文献
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We consider a boundary value problem for the stationary diffusion equation outside cuts on the plane. The Dirichlet condition is posed on one side of each cut, and an oblique derivative condition is posed on the other side. We prove existence and uniqueness theorems for the solution of the boundary value problem. We obtain an integral representation of a solution in the form of potentials. The densities of these potentials are found from a system of Fredholm integral equations of the second kind, which is uniquely solvable. We obtain closed asymptotic formulas for the gradient of the solution of the boundary value problem at the endpoints of the cuts. 相似文献
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P. A. Krutitskii 《Mathematical Methods in the Applied Sciences》2001,24(16):1247-1256
We suggest a new approach of reduction of the Neumann problem in acoustic scattering to a uniquely solvable Fredholm integral equation of the second kind with weakly singular kernel. To derive this equation we placed an additional boundary with an appropriate boundary condition inside the scatterer. The solution of the problem is obtained in the form of a single layer potential on the whole boundary. The density in the potential satisfies a uniquely solvable Fredholm integral equation of the second kind and can be computed by standard codes. Copyright © 2001 John Wiley & Sons, Ltd. 相似文献
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P.A. Krutitskii 《Journal of Differential Equations》2004,198(2):422-441
The mixed Dirichlet-Neumann problem for the Laplace equation in a bounded connected plane domain with cuts (cracks) is studied. The Neumann condition is given on closed curves making up the boundary of a domain, while the Dirichlet condition is specified on the cuts. The existence of a classical solution is proved by potential theory and boundary integral equation method. The integral representation for a solution is obtained in the form of potentials. The density in potentials satisfies the uniquely solvable Fredholm integral equation of the second kind and index zero. Singularities of the gradient of the solution at the tips of cuts are investigated. 相似文献
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We present a numerical method for solving the system of integral-algebraic equations arising in the study of the oblique derivative problem for the Laplace equation outside open curves on the plane. The problem describes the electric current in a semiconductor film with curvilinear electrodes in the presence of a magnetic field. The integral-algebraic system has singularities, and the kernel in the integral equation is represented in the form of a Cauchy integral. The numerical scheme is of the second approximation order despite the singularities. 相似文献
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P. A. Krutitskii 《Differential Equations》2009,45(1):86-101
We consider a boundary value problem for the Laplace equation outside cuts on a plane. Boundary conditions of the third kind, which are in general different on different sides of each cut, are posed on the cuts. We show that the classical solution of the problem exists and is unique. We obtain an integral representation for the solution of the problem in the form of potentials whose densities are found from a uniquely solvable system of Fredholm integral equations of the second kind. 相似文献
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