Abstract
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.
Highlights
The singular integral equation t−x φB t ln t x dt f x, x ∈ B, 1.1 where B is a single or a finite union of finite intervals is encountered in many branches of mathematical physics cf. Lewin 1, Ursell 2, and Mandal and Kundu 3 .This typeISRN Applied Mathematics of integral equation is usually solved in the literature by converting the weak singularity logarithmic type in the kernel to a Cauchy-type singularity which is strong in nature
For an integral equation with weakly singular kernel, the associated integral is defined in the standard Riemann sense, while for an integral equation with strongly singular kernel, the singular integral has to be defined appropriately so as to give a consistent mathematical sense
From these 2n conditions, we find the arbitrary constant Π2i n2 1Ai, and from the order condition, the arbitrary constant A1 can be determined, and the solution of the singular integral equation 2.26 can be obtained from 2.38
Summary
B t ln t x dt f x, x ∈ B, 1.1 where B is a single or a finite union of finite intervals is encountered in many branches of mathematical physics cf. Lewin 1 , Ursell 2 , and Mandal and Kundu 3 .This type. It is observed that the solution as obtained by the relations 2.22 and 2.23 of 2.2 neither involves any strong singularity nor any arbitrary constant. We observe that the original integral equation 2.26 can be solved by using the above relations if and only if the following consistency conditions are satisfied: ψ1 ai ψ1 bi 0, i 1, . We give the genesis of the integral equation 2.2 with B 0, a ∪ b, c and use its solution to the problem of scattering of water waves by a partially immersed vertical barrier with gap
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