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

الملخص EN

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,a)∪(b,c) and (a,b,c being finite ) 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.