Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

In this paper some useful formulas are developed to evaluate integrals having a singularity of the form (t-x) sup-m, m or = 1 Interpreting the integrals with strong singularities in Hadamard sense, the results are used to obtain approximate solutions of singular integral equations A mixed boundary value problem from the theory of elasticity is considered as an example Particularly for integral equations where the kernel contains, in addition to the dominant term (t,x) sup-m, terms which become unbounded at the end points, the present technique appears to be extremely effective to obtain rapidly converging numerical results

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On the solution of integral equations with strong ly singular kernels

Numerical solution of some classes of integral equations using Bernstein polynomials

The problem of oblique water wave diffraction by two equal thin, parallel, fixed vertical barriers with gaps present in uniform finite-depth water is investigated here. Three types of barrier configurations are considered. A one-term Galerkin approximation is used to evaluate upper and lower bounds for reflection and transmission coefficients for each configuration. These bounds are seen to be very close numerically for all wave numbers and as such their averages produce good numerical estimates for these coefficients. Only the bounds for the reflection coefficient are numerically computed. These are also numerically compared with the results obtained by using multiterm Galerkin approximations involving Chebyshev polynomials for a wide range of parameters. Numerical results for the reflection coefficients for the three barrier configurations are presented graphically. It is seen that total reflection occurs only for the surface-piercing barriers while total transmission occurs for all the three configurations considered here. It is also observed that the introduction of an equal second barrier to a submerged barrier increases the reflection coefficient considerably in some frequency bands and as such submerged double barrier configurations are preferable to a submerged single barrier for the purpose of reflecting more wave energy into the open sea.

Oblique Wave Diffraction by Parallel Thin Vertical Barriers with Gaps

This paper is concerned with two-dimensional scattering of a normally incident surface wave train on an obstacle in the form of a thick vertical barrier of rectangular cross section in water of uniform finite depth. Four different geometrical configurations of the barrier are considered. The barrier may be surface-piercing and partially immersed, or bottom-standing and submerged, or in the form of a submerged rectangular block not extending down to the bottom, or in the form of a thick vertical wall with a submerged gap. Appropriate multi-term Galerkin approximations involving ultraspherical Gegenbauer polynomials are used for solving the integral equations arising in the mathematical analysis. Very accurate numerical estimates for the reflection coefficient for each configuration of the barrier are then obtained. The reflection coefficient is depicted graphically against the wave number for each configuration. It is observed that the reflection coefficient depends significantly on the thickness for a wide range of values of the wave number, and as such, thickness plays a significant role in the modelling of efficient breakwaters.

Water-wave scattering by thick vertical barriers

Water wave scattering by a finite dock over a step-type bottom topography

Oblique wave scattering by submerged thin wall with gap in finite-depth water

A straightforward analysis involving the complex function-theoretic method is employed to determine the closed-form solution of a special hypersingular integral equation of the second kind, and its known solution is recovered.

Solution of a hypersingular integral equation of the second kind

In the present article, we have studied the problem of scattering of water waves by a porous plate submerged in the ocean and inclined at an angle to the vertical. The problem is formulated in terms of a hypersingular integral equation of the second kind. The hypersingular integral equation is then solved by a collocation method by representing the unknown function using Tchebychev polynomials. The reflection and transmission coefficients are then obtained and presented graphically. It is observed that porosity of the plate dissipates the wave energy and thereby reduces the reflection and transmission of wave energy. However, it is found that the presence of ice cover reduces the dissipation of energy and also the reflection of wave energy and thereby increases the transmission of wave energy. Also, the increase in angle of inclination of porous plate reduces the energy dissipation as well as the reflection of waves.

Sudeshna Banerjea

Papers

Water wave scattering by a finite dock over a step-type bottom topography

Oblique wave scattering by submerged thin wall with gap in finite-depth water

Solution of a hypersingular integral equation of the second kind

Scattering of water waves by an inclined porous plate submerged in ocean with ice cover

Solution of a singular integral equation in a double interval arising in the theory of water waves