{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:44:06Z","timestamp":1760060646902,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,9,9]],"date-time":"2025-09-09T00:00:00Z","timestamp":1757376000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ajman University, United Arab Emirates"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>This study presents the Fourier\u2013Gegenbauer integral Galerkin (FGIG) method, a new numerical framework that uniquely integrates Fourier series and Gegenbauer polynomials to solve the one-dimensional advection\u2013diffusion (AD) equation with spatially symmetric periodic boundary conditions, achieving exponential convergence and reduced computational cost compared to traditional methods. The FGIG method uniquely combines Fourier series for spatial periodicity and Gegenbauer polynomials for temporal integration within a Galerkin framework, resulting in highly accurate numerical and semi-analytical solutions. Unlike traditional approaches, this method eliminates the need for time-stepping procedures by reformulating the problem as a system of integral equations, reducing error accumulation over long-time simulations and improving computational efficiency. Key contributions include exponential convergence rates for smooth solutions, robustness under oscillatory conditions, and an inherently parallelizable structure, enabling scalable computation for large-scale problems. Additionally, the method introduces a barycentric formulation of the shifted Gegenbauer\u2013Gauss (SGG) quadrature to ensure high accuracy and stability for relatively low P\u00e9clet numbers. This approach simplifies calculations of integrals, making the method faster and more reliable for diverse problems. Numerical experiments presented validate the method\u2019s superior performance over traditional techniques, such as finite difference, finite element, and spline-based methods, achieving near-machine precision with significantly fewer mesh points. These results demonstrate its potential for extending to higher-dimensional problems and diverse applications in computational mathematics and engineering. The method\u2019s fusion of spectral precision and integral reformulation marks a significant advancement in numerical PDE solvers, offering a scalable, high-fidelity alternative to conventional time-stepping techniques.<\/jats:p>","DOI":"10.3390\/computation13090219","type":"journal-article","created":{"date-parts":[[2025,9,9]],"date-time":"2025-09-09T19:16:17Z","timestamp":1757445377000},"page":"219","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Fourier\u2013Gegenbauer Integral Galerkin Method for Solving the Advection\u2013Diffusion Equation with Periodic Boundary Conditions"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7369-4988","authenticated-orcid":false,"given":"Kareem T.","family":"Elgindy","sequence":"first","affiliation":[{"name":"Department of Mathematics and Sciences, College of Humanities and Sciences, Ajman University, Ajman P.O. Box 346, United Arab Emirates"},{"name":"Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman P.O. Box 346, United Arab Emirates"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1504\/PCFD.2017.082520","article-title":"Computational study of three numerical methods for some linear and nonlinear advection-diffusion-reaction problems","volume":"17","author":"Appadu","year":"2017","journal-title":"Prog. Comput. Fluid Dyn. Int. J."},{"key":"ref_2","first-page":"259","article-title":"Efficient explicit nonstandard finite difference scheme with positivity-preserving property","volume":"30","author":"Khalsaraei","year":"2017","journal-title":"Gazi Univ. J. 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