Optimal error estimates of spectral Galerkin method for mixed diffusion equations

Hao, Zhaopeng

doi:10.1007/s10092-023-00505-9

Cited by 1 publication

(1 citation statement)

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“…However, a variety of problems can be solved using the broader Jacobi technique; see [13][14][15][16]. The p-version of the finite element approach, the boundary element method, and a spectral method for axisymmetrical domains, on the other hand, were all examined using some Jacobi approximation results, and various rational spectral methods in [17][18][19][20][21][22][23], besides, in the study of quadratures involving the values of derivatives of functions at endpoints. It is crucial to understand that, in the context of the Galerkin technique, the basis functions (BFs) used to determine how well spectral methods approximate problems compared to finite difference and finite element methods [24].…”

Section: Introductionmentioning

confidence: 99%

Review on Jacobi-Galerkin Spectral Method for Linear PDEs in Applied Mathematics

Hafez,

Youssri

2024

Contemp. Math.

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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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