A Refined Spectral Galerkin Approach Leveraging Romanovski–Jacobi Polynomials for Differential Equations

Ramy M. Hafez, Mohamed A. Abdelkawy () and Hany M. Ahmed
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Ramy M. Hafez: Department of Mathematics, Faculty of Education, Matrouh University, Marsa Matrouh 51511, Egypt
Mohamed A. Abdelkawy: Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11623, Saudi Arabia
Hany M. Ahmed: Department of Mathematics, Faculty of Technology and Education, Helwan University, Cairo 11281, Egypt

Mathematics, 2025, vol. 13, issue 9, 1-28

Abstract: This study explores the application of Romanovski–Jacobi polynomials (RJPs) in spectral Galerkin methods (SGMs) for solving differential equations (DEs). It uses a suitable class of modified RJPs as basis functions that meet the homogeneous initial conditions (ICs) given. We derive spectral Galerkin schemes based on modified RJP expansions to solve three models of high-order ordinary differential equations (ODEs) and partial differential equations (PDEs) of first and second orders with ICs. We provide theoretical assurances of the treatment’s efficacy by validating its convergent and error investigations. The method achieves enhanced accuracy, spectral convergence, and computational efficiency. Numerical experiments demonstrate the robustness of this approach in addressing complex physical and engineering problems, highlighting its potential as a powerful tool to obtain accurate numerical solutions for various types of DEs. The findings are compared to those of preceding studies, verifying that our treatment is more effective and precise than that of its competitors.

Keywords: Romanovski–Jacobi polynomials; spectral methods; partial differential equations; numerical methods; orthogonal polynomials (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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