Solutions of hypersingular integral equations over circular domains by a spectral method. (English) Zbl 1340.65316

Farina, Leandro; Ziebell, Juliana S.

Brandts, J. (ed.) et al., Proceedings of the international conference ‘Applications of mathematics’, Prague, Czech Republic, May 15–17, 2013. In honor of the 70th birthday of Karel Segeth. Prague: Academy of Sciences of the Czech Republic, Institute of Mathematics (ISBN 978-80-85823-61-5). 52-66 (2013).

This article deals with a system of hypersingular integral equations that can arise in the study of the interaction of water waves with submerged thin plates. The numerical solution is constructed for 2D problem using the expansion-collocation method.
Presented numerical method has base in works of the first author and Martin. They used semi-analytical expansion-collocation method for the solution of hypersingular integral equations on a disc. Martin’s solution of the problem of a wrinkled disc in an unbounded fluid became a basis for the boundary perturbation method. In this manuscript, the numerical solution for 2D problem using the expansion-collocation method in combination with Gegenbauer polynomials in the radial variable is constructed. This application is new.
Text of the manuscript is understandable. The manuscript is carefully processed with logical links. The article is supplemented with plenty of references to the literature.
For the entire collection see [Zbl 1277.00032].

Reviewer: Vratislava Mošová (Olomouc)

MSC:

65R20	Numerical methods for integral equations
76B15	Water waves, gravity waves; dispersion and scattering, nonlinear interaction
45E10	Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)

Keywords:

hypersingular integral equations; boundary perturbation method; expansion-collocation method; Chebyshev polynomials; Gegenbauer polynomials; water waves