Abstract

We consider a mathematical model of scattering of \(H \)-polarized electromagnetic waves by a screen with a curvilinear boundary based on a singular integro-differential equation with a Cauchy kernel and a logarithmic singularity. The integrands contain both the unknown function and its first derivative. For the numerical analysis of this model, two computational schemes are constructed based on the representation of the unknown function in the form of a linear combination of orthogonal Chebyshev polynomials and spectral relations, which permit one to obtain simple analytical expressions for the singular component of the equation. The expansion coefficients of the solution in terms of the basis of Chebyshev polynomials are calculated as a solution of the corresponding system of linear algebraic equations. The results of numerical experiments show that the error in the approximate solution on a grid of 20–30 nodes does not exceed the roundoff error.

中文翻译：

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奇异积分微分方程近似解的正交多项式方法应用于二维衍射问题

摘要

我们考虑\(H\)散射的数学模型-基于具有柯西核和对数奇点的奇异积分微分方程的曲线边界的屏幕极化电磁波。被积函数包含未知函数及其一阶导数。对于该模型的数值分析，基于未知函数以正交切比雪夫多项式和谱关系的线性组合的形式表示，构建了两种计算方案，这使得人们可以得到奇异分量的简单解析表达式。方程。以切比雪夫多项式为基础的解的展开系数被计算为相应线性代数方程组的解。