Spectral matrix computational Tau approach for fractional differential equations via fifth-kind chebyshev polynomials

This paper presents Tau approach for solving fractional differential equations (FDEs) via shifted Chebyshev polynomials of the fifth kind. By leveraging the unique properties of these polynomials, we develop operational matrices that facilitate the approximation of solutions to both linear and nonlinear FDEs. The proposed method employs a tau technique in the matrix form to transform the problem into a solvable algebraic system, ensuring computational efficiency and accuracy. This work presents a rigorous convergence analysis and demonstrates the efficacy of the proposed approach through a series of illustrative examples, showcasing a marked improvement in solution precision relative to conventional methodologies. This research contributes to the growing of work in fractional calculus and offers a robust tool for researchers and practitioners in applied mathematics and engineering.