A quick method of solution for multiple integral equations which are defined over a partition consisting of three intervals of the positive axis and whose kernel is the combination of trigonometric functions has been explained. The solution procedure can be extended to deal with similar integral equations defined over any finite partition of the positive axis. To represent the solution uniquely, certain solvability criteria are obtained in terms of the forcing functions involved. Limiting cases of dual integral equations over two disjoint intervals are discussed. © 2011 Elsevier Ltd. All rights reserved.

Srinivasa Rao Manam

Department of Mathematics

DUAL INTEGRAL EQUATIONS

FLEXURAL-GRAVITY WAVES

Forcing function

LIMITING CASE

Logarithmic singularity

SINGULAR INTEGRAL EQUATIONS

SOLUTION PROCEDURE

THREE INTERVAL

Trigonometric functions

TRIGONOMETRIC KERNELS

Function evaluation

Gravitational effects

Gravity Waves

Hydrodynamics

Surface Properties

Surface tension

Waves

Integral equations

Fulltext

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IIT Madras has been ranked as the top engineering institute in India for four years in a row by the National Institutional Ranking Framework of the MHRD

It currently offers undergraduate, postgraduate and research degrees across 16 disciplines in Engineering, Sciences, Humanities and Management. About 596 faculty belonging to science and engineering departments and centres of the Institute are engaged in teaching, research and industrial consultancy.

IIT Madras

Applied Mathematics Letters

Scattering of membrane coupled gravity waves in deep water by partial vertical barriers is investigated by the recently developed expansion formulae for wave structure interaction problems. The horizontal thin membrane is considered to be under uniform tension and is covering the free surface. The analysis is based on the linearized theory of water waves, and by combining the kinematic and dynamic conditions at the membrane covered surface, one may derive a not so well-posed mixed boundary value problem for Laplaces equation with third-order boundary condition. The flexible membrane is attached by a spring to the surface piercing barrier, giving suitable edge conditions for the unique solution. The boundary value problem has been converted into dual integral equations with kernels composed of trigonometric functions, which are then solved analytically. The important physical quantities such as reflection and transmission coefficients for both cases of submerged and surface piercing barriers are obtained analytically in terms of modified Bessel functions. It is found that complete reflection or transmission is possible at certain resonant frequencies for the incident membrane coupled waves. Numerical results are plotted and discussed for different values of the nondimensional membrane tension parameter. © 2010 Australian Mathematical Society.

Journal	Applied Mathematics Letters
ISSN	08939659
Open Access	Yes