A Comprehensive Study On Solving Multitype Differential Equations Using Romanovski–Jacobi Matrix Methods

This research presents advanced spectral algorithms based on Romanovski–Jacobi polynomials for solving various types of differential equations. The study explores the integration of these polynomials into pseudospectral methods, enabling efficient handling of higher-order ordinary differential equations (ODEs) and second-order linear partial differential equations (PDEs). The proposed methods leverage the operational matrices of derivatives to transform differential equations into systems of algebraic equations, ensuring high accuracy and rapid convergence. Numerical experiments validate the effectiveness of the algorithms in solving problems with complex geometries and boundary conditions. The results demonstrate the superiority of Romanovski–Jacobi-based techniques compared to traditional spectral methods, offering a robust and versatile framework for scientific and engineering applications.