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The airfoil integral equation over two disjoint intervals, separated by a distance κ, is considered. An analytic solution is obtained in terms of a polynomial series that satisfies a generalized three-term recurrence relation, with the solution being particularly tailored for cases where the input function is a Chebyshev polynomial of the first kind. A new class of polynomials is defined on two disjoint intervals, demonstrating a generalization of a classical integral relationship previously established for Chebyshev polynomials on a single interval. The solutions are efficiently calculated, and comparisons with the original problem, defined on a continuous domain, are presented. Additionally, as κ→0, we derive approximate solutions for the airfoil equation and exhibit the first terms of an asymptotic expansion of them in power series of κwithin weighted Sobolev spaces. The paper further elaborates on the implementation of spectral methods, specifically the collocation and Galerkin versions, for solving the general airfoil integral equation over two disjoint intervals.

The numerical ensemble prediction is a well accepted method for improving the performance of atmospheric models. In the context of ocean wave modeling little has been researched or documented about this technique. An essential study of the method of ensemble prediction applied to deep water waves has been carried out. A framework is defined for obtaining perturbations of the directional wave spectra and for employing an ensemble of wind fields generated by an atmospheric model. The third‐generation global wave model WAM is used with real atmospheric conditions to investigate the effect on wave predictions of perturbed initial conditions and atmospheric forcing. Due to spectral shape stabilisation, perturbing wave initial conditions has limited utility in ensemble prediction. However, the members could be used in wave data assimilation schemes in an interactive way. Using ensembles of the atmospheric condition can generate diverging solutions, justifying the ensemble procedure by itself. In the cases studied, it is observed that the ensemble mean outperformed the other members. The solution behaviour suggests using a lower‐order approximation of the model to generate ensemble members with less computational cost.

The numerical ensemble prediction is a well accepted method for improving the performanceof atmospheric models. In the context of ocean wave modeling little has been researched ordocumented about this technique. An essential study of the method of ensemble predictionapplied to deep water waves has been carried out. A framework is defined for obtaining perturbationsof the directional wave spectra and for employing an ensemble of wind fields generatedby an atmospheric model. The third-generation global wave model WAM is used with realatmospheric conditions to investigate the effect on wave predictions of perturbed initial conditionsand atmospheric forcing. Due to spectral shape stabilisation, perturbing wave initialconditions has limited utility in ensemble prediction. However, the members could be used inwave data assimilation schemes in an interactive way. Using ensembles of the atmosphericcondition can generate diverging solutions, justifying the ensemble procedure by itself. In thecases studied, it is observed that the ensemble mean outperformed the other members. Thesolution behaviour suggests using a lower-order approximation of the model to generateensemble members with less computational cost.

A novel application of the boundary integral method to horizontal well analysis in the field of petroleum engineering is presented. The transient pressure satisfies the heat equation, non-local and non-linear boundary conditions. The turbulent flow inside the well is modelled by considering a pressure gradient along the well. The heat potential is used and Chebyshev collocation along with a time discretization is employed. Some numerical results are presented to show the features of this new approach. Copyright © 2000 John Wiley & Sons, Ltd.

The article discusses a photography project by still life photographer Leandro Farina for "Port" magazine, which features black-and-white images of various desk tools.