Volume 1, Issue 4
Features of the Nyström Method for the Sherman-Lauricella Equation on Piecewise Smooth Contours

East Asian J. Appl. Math., 1 (2011), pp. 403-414.

Published online: 2018-02

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• 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 $θ_j$ 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 $θ_j$ . In the interval ($0.1π$, $1.9π$), it is found that there are 8 values of $θ_j$ 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.

• Keywords

Sherman-Lauricella equation, Nystrom method, stability.

65R20, 45L05

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@Article{EAJAM-1-403, author = {}, title = {Features of the Nyström Method for the Sherman-Lauricella Equation on Piecewise Smooth Contours}, journal = {East Asian Journal on Applied Mathematics}, year = {2018}, volume = {1}, number = {4}, pages = {403--414}, 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 $θ_j$ 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 $θ_j$ . In the interval ($0.1π$, $1.9π$), it is found that there are 8 values of $θ_j$ 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.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.240611.070811a}, url = {http://global-sci.org/intro/article_detail/eajam/10912.html} }
TY - JOUR T1 - Features of the Nyström Method for the Sherman-Lauricella Equation on Piecewise Smooth Contours JO - East Asian Journal on Applied Mathematics VL - 4 SP - 403 EP - 414 PY - 2018 DA - 2018/02 SN - 1 DO - http://doi.org/10.4208/eajam.240611.070811a UR - https://global-sci.org/intro/article_detail/eajam/10912.html KW - Sherman-Lauricella equation, Nystrom method, stability. AB -

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 $θ_j$ 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 $θ_j$ . In the interval ($0.1π$, $1.9π$), it is found that there are 8 values of $θ_j$ 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.

Victor D. Didenko & Johan Helsing. (1970). Features of the Nyström Method for the Sherman-Lauricella Equation on Piecewise Smooth Contours. East Asian Journal on Applied Mathematics. 1 (4). 403-414. doi:10.4208/eajam.240611.070811a
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