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.