Classical spectral methods are confined to numerically solving Fredholm integro-differential equations in regular domains, such as rectangles and discs. This paper aims to numerically address two-dimensional Fredholm integro-differential equations in complex geometries by combining spectral methods with mapping techniques. Initially, we transform the computational domain into a rectangular one via coordinate mapping. Subsequently, classical spectral methods are applied within this rectangular domain for numerical simulations. Our analysis primarily discusses the existence, uniqueness and convergence of numerical solutions. Numerical results demonstrate that the proposed method achieves high-order accuracy.

中文翻译：

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一种用于复杂几何结构中二维 Fredholm 积分微分方程的高效谱方法


经典谱方法仅限于在常规域（如矩形和圆盘）中对 Fredholm 积分微分方程进行数值求解。本文旨在通过将光谱方法与映射技术相结合，以数值方式解决复杂几何结构中的二维 Fredholm 积分微分方程。最初，我们通过坐标映射将计算域转换为矩形域。随后，在这个矩形域内应用经典光谱方法进行数值模拟。我们的分析主要讨论了数值解的存在性、唯一性和收敛性。数值结果表明，所提方法实现了高阶精度。