Abstract Modeling an infinite domain arises while simulating physical applications frequently. It is like attempting to model the effects and properties of the ocean in a bucket. In this paper, we develop a spectral Galerkin method based on exponential Jacobi functions (EJF) in unbounded domains. We also establish some basic results on exponential Jacobi orthogonal approximations, which serve as the mathematical foundation of spectral methods for various partial differential equations in unbounded domains. We derive the exponential Lagrange interpolation formula and its related error estimates. As examples, hyperbolic partial differential equations in multidimensions are considered. Related spectral Galerkin schemes are proposed. The test and trial function spaces are carefully chosen to obtain desired convergence properties. Through three examples, we demonstrate that the proposed method yields highly accurate results.

中文翻译：

---

指数 Jacobi-Galerkin 方法及其在无界域多维问题中的应用

摘要 对无限域进行建模是在频繁模拟物理应用程序时出现的。这就像尝试对桶中海洋的影响和特性进行建模。在本文中，我们开发了一种基于无界域中指数雅可比函数 (EJF) 的谱伽辽金方法。我们还建立了指数雅可比正交逼近的一些基本结果，这些结果作为无界域中各种偏微分方程的谱方法的数学基础。我们推导出指数拉格朗日插值公式及其相关的误差估计。例如，考虑了多维中的双曲偏微分方程。提出了相关的谱伽辽金方案。仔细选择测试和试验函数空间以获得所需的收敛特性。