Table of Contents
ISRN Applied Mathematics
Volume 2011, Article ID 341564, 16 pages
Research Article

Solution of Singular Integral Equations Involving Logarithmically Singular Kernel with an Application in a Water Wave Problem

1Department of Mathematics, Jadavpur University, Kolkata 700032, India
2Department of Mathematics, Netajinagar Day College, Regent Estate, NSC Bose Road, Kolkata 700023, India
3Department of Mathematics, Indian Institute of Science, Bangalore 560012, India

Received 10 March 2011; Accepted 1 April 2011

Academic Editors: A. Bellouquid and T. Y. Kam

Copyright Β© 2011 Sudeshna Banerjea et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A direct function theoretic method is employed to solve certain weakly singular integral equations arising in the study of scattering of surface water waves by vertical barriers with gaps. Such integral equations possess logarithmically singular kernel, and a direct function theoretic method is shown to produce their solutions involving singular integrals of similar types instead of the stronger Cauchy-type singular integrals used by previous workers. Two specific ranges of integration are examined in detail, which involve the following: Case(i) two disjoint finite intervals (0,π‘Ž)βˆͺ(𝑏,𝑐) and (π‘Ž,𝑏,𝑐beingfinite) and Case(ii) a finite union of 𝑛 disjoint intervals. The connection of such integral equations for Case(i), with a particular water wave scattering problem, is explained clearly, and the important quantities of practical interest (the reflection and transmission coefficients) are determined numerically by using the solution of the associated weakly singular integral equation.