High-Precision Matrix Computational Technique for Fractional Differential Equations using Generalized Symmetric Polynomials

In this work, a generalized main class of symmetric orthogonal polynomials is proposed and utilized within a spectral frame-work for solving fractional differential equations (FDEs) using tau method. Emphasis is placed on the monic symmetric and shifted monic symmetric representations of several kinds Chebyshev olynomials. These polynomials are employed to develop general unified operational differentiation matrices, which convert the original differential equations into equivalent systems of algebraic equations. The
symmetry and orthogonality properties inherent to these polynomial families enhance both the precision and stability of the numerical schemes. A comparative study is conducted to assess the performance of each type in terms of accuracy, convergence behavior, and computational efficiency. The results confirm that the proposed method provides a reliable and adaptable framework for effectively solving a wide range of linear and nonlinear differential equations within the context of numerical analysis and applied mathematics.