应用数学中线性 PDE 的 Jacobi-Galerkin 光谱法综述

R. Hafez, Y. H. Youssri
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引用次数: 0

摘要

本研究探讨了解决时空薛定谔方程、波方程、Airy方程和光束方程的谱Galerkin方法。为了便于创建半解析近似解,它使用了由空间和时间维度的雅可比多项式(JPs)线性组合形成的多项式基。通过使用这些多项式来扩展精确解,论文希望能满足同质起始和 Dirichlet 边界的要求。值得注意的是,如果解足够平滑,雅各比-格勒金 (JG) 方法会表现出指数收敛率。这一结果强调了 JG 方法作为一种有效数值求解方法的潜力,它有望在出现这些方程的其他领域,如量子力学、声学、光学和结构力学中得到广泛应用。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Review on Jacobi-Galerkin Spectral Method for Linear PDEs in Applied Mathematics
This study explores the spectral Galerkin approach to solving the space-time Schrödinger, wave, Airy, and beam equations. In order to facilitate the creation of a semi-analytical approximation solution, it uses polynomial bases that are formed from a linear combination of Jacobi polynomials (JPs) in both spatial and temporal dimensions. By using these polynomials to expand the exact solution, the paper hopes to satisfy the homogeneous starting and Dirichlet boundary requirements. Notably, the Jacobi Galerkin (JG) method exhibits exponential convergence rates if the solution is sufficiently smooth. This result emphasizes the JG approach' s potential as an effective numerical solution method, which has promise for a variety of applications in other domains where these equations occur, such as quantum mechanics, acoustics, optics, and structural mechanics.
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