مشخصات پژوهش

Shifted Chebyshev polynomials are well-suited for problems defined on finite intervals, providing orthogonality, fast convergence, and spectral accuracy. Incorporating them within a Petrov–Galerkin framework ensures better stability and accuracy compared to standard Galerkin or collocation approaches, especially for fractional differential equations.‎ ‎ Spectral methods are known for exponential convergence for smooth problems. By employing shifted Chebyshev polynomials, the proposed Petrov–Galerkin spectral method can deliver highly accurate approximations with fewer basis functions, reducing computational cost while maintaining precision.‎ ‎ The fractional-order operator introduces memory effects and non-locality, which are difficult to approximate using traditional methods. The Petrov–Galerkin spectral framework provides a stable and reliable way to approximate fractional derivatives and capture long-range interactions in time-dependent processes.‎ ‎ This research contributes to the ongoing development of spectral methods for fractional PDEs, offering new insights into basis selection, stability analysis, and numerical efficiency. It strengthens the toolbox available to applied mathematicians and computational scientists working on fractional dynamics.