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A three-point formula for numerical quadrature of oscillatory integrals with variable frequency
Abstract A simple three-point formula is constructed for the evaluation of general oscillatory integrals. A rigorous derivation of the local error term is presented, and the implications to highExpand
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The Initial‐Value Problem on a Plane Beach
The two-dimensional initial-value problem is examined for an inviscid incompressible fluid on a plane beach within the framework of a linearized but non-hydrostatic theory. Solutions generated byExpand
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An alternative dispersion equation for water waves over an inclined bed
A classical approach to extending the validity of Airy's dispersion relation for surface gravity waves by Friedrichs (1948) to gentle slopes (of special inclinations) is here re-examined withExpand
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Oblique wave incidence on a plane beach: the classical problem revisited
The non-hydrostatic description of three-dimensional waves incident over a plane beach at a long straight coastline is considered in terms of the inverse Kontorovich-Lebedev integral transform. ThisExpand
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Far field asymptotic of the two-dimensional linearised sloping beach problem
The inverse Mellin transform solution for the two-dimensional linearised inviscid beach problem is used to derive far field asymptotic expansions. It is shown that a Poincare expansion exists whichExpand
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The numerical inversion of two classes of Kontorovich-Lebedev transform by direct quadrature
Abstract The inversion of the conventional Kontorovich-Lebedev transform with the Macdonald function as kernel and also with a Kelvin function as kernel is considered. Strategies are presented firstExpand
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On the error and its control in a two-parameter generalised Newton-Cotes rule
A quadrature formula containing two free (phase) parameters k, k', and recently written by Van Daele et al. [13], is rederived using an extension of Lagrange's identity. By using this method, aExpand
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A note on a recent study of oscillatory integration rules
Abstract In a recent comparison of methods of highly oscillatory numerical integration the authors appear to have overlooked some works which seem to compete effectively with several of thoseExpand
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On the normal incidence of linear waves over a plane incline partially covered by a rigid lid
The effect is examined on infinitesimal standing waves over a plane beach when restricted by the arbitrary placing of a finite rigid (or permeable) lid of length l on the undisturbed surface. AExpand
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