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

Victor D. Didenko & Johan Helsing

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 cj , j = 0, 1,··· , m relies on the invertibility of certain operators Acj belonging to an algebra of Toeplitz operators. The operators Acj do not depend on the shape of the contour, but on the opening angle θj of the corresponding corner cj 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 θj . In the interval (0.1π, 1.9π), it is found that there are 8 values of θj where the invertibility of the operator Acj 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

  • AMS Subject Headings

65R20 45L05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{EAJAM-1-403, author = {Victor D. Didenko and Johan Helsing}, title = {Features of the Nyström Method for the ShermanLauricella 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 cj , j = 0, 1,··· , m relies on the invertibility of certain operators Acj belonging to an algebra of Toeplitz operators. The operators Acj do not depend on the shape of the contour, but on the opening angle θj of the corresponding corner cj 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 θj . In the interval (0.1π, 1.9π), it is found that there are 8 values of θj where the invertibility of the operator Acj 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 ShermanLauricella Equation on Piecewise Smooth Contours AU - Victor D. Didenko & Johan Helsing 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 KW - Nystrom method KW - stability AB -

The stability of the Nyström method for the Sherman-Lauricella equation on contours with corner points cj , j = 0, 1,··· , m relies on the invertibility of certain operators Acj belonging to an algebra of Toeplitz operators. The operators Acj do not depend on the shape of the contour, but on the opening angle θj of the corresponding corner cj 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 θj . In the interval (0.1π, 1.9π), it is found that there are 8 values of θj where the invertibility of the operator Acj 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 ShermanLauricella 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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