| 1. |
- Didenko, Victor, et al.
(författare)
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Approximate solution of boundary integral equations for biharmonic problems in non-smooth domains
- 2013
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Ingår i: Proceedings in Applied Mathematics and Mechanics. - : Wiley. - 1617-7061. ; 13:1, s. 435-438
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Konferensbidrag (refereegranskat)abstract
- This paper deals with approximate solutions to integral equations arising in boundary value problems for the biharmonic equation in simply connected piecewise smooth domains. The approximation method considered demonstrates excellent convergence even in the case of boundary conditions discontinuous at corner points. In an application we obtain very accurate approximations for some characteristics of two-dimensional Stokes flow in non-smooth domains.
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| 2. |
- Didenko, Victor D., et al.
(författare)
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On the Stability of the Nystrom Method for the Muskhelishvili Equation on Contours with Corners
- 2013
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Ingår i: SIAM Journal on Numerical Analysis. - : Society for Industrial & Applied Mathematics (SIAM). - 0036-1429 .- 1095-7170. ; 51:3, s. 1757-1776
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Tidskriftsartikel (refereegranskat)abstract
- The stability of the Nystrom method for the Muskhelishvili equation on piecewise smooth simple contours Gamma is studied. It is shown that in the space L-2 the method is stable if and only if certain operators A tau(j) from an algebra of Toeplitz operators are invertible. The operators A tau(j) depend on the parameters of the equation considered, on the opening angles theta(j) of the corner points t(j) is an element of Gamma, and on parameters of the approximation method mentioned. Numerical experiments show that there are opening angles where the operators A tau(j) are noninvertible. Therefore, for contours with such corners the method under consideration is not stable. Otherwise, the method is always stable. Numerical examples show an excellent convergence of the method.
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| 3. |
- Didenko, Victor, et al.
(författare)
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Features of the Nyström method for the Sherman-Lauricella equation on Piecewise Smooth Contours
- 2011
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Ingår i: East Asian Journal on Applied Mathematics. - : Global Science Press. - 2079-7370 .- 2079-7362. ; 1:4, s. 403-414
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Tidskriftsartikel (refereegranskat)abstract
- The stability of the Nyström method for the Sherman-Lauricella equation on contours with corner points $c_j$, $j=0,1,...,m$ relies on the invertibility of certain operators $A_{c_j}$ belonging to an algebra of Toeplitz operators. The operators $A_{c_j}$ do not depend on the shape of the contour, but on the opening angle $\theta_j$ of the corresponding corner $c_j$ and on parameters of the approximation method mentioned. They have a complicated structure and there is no analytic tool to verify their invertibility. To study this problem, the original Nyström method is applied to the Sherman-Lauricella equation on a special model contour that has only one corner point with varying opening angle $\theta_j$. In the interval $(0.1\pi,1.9\pi)$, it is found that there are $8$ values of $\theta_j$ where the invertibility of the operator $A_{c_j}$ may fail, so the corresponding original Nyström method on any contour with corner points of such magnitude cannot be stable and requires modification.
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| 4. |
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| 5. |
- Didenko, Victor, et al.
(författare)
-
Stability of the Nyström Method for the Sherman–Lauricella Equation
- 2011
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Ingår i: SIAM Journal on Numerical Analysis. - : Society for Industrial & Applied Mathematics (SIAM). - 0036-1429 .- 1095-7170. ; 49:3, s. 1127-1148
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Tidskriftsartikel (refereegranskat)abstract
- The stability of the Nyström method for the Sherman–Lauricella equation on piecewise smooth closed simple contour $\Gamma$ is studied. It is shown that in the space $L_2$ the method is stable if and only if certain operators associated with the corner points of $\Gamma$ are invertible. If $\Gamma$ does not have corner points, the method is always stable. Numerical experiments show the transformation of solutions when the unit circle is continuously transformed into the unit square, and then into various rhombuses. Examples also show an excellent convergence of the method.
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