This thesis investigates a numerical method based on a Petrov--Galerkin spectral approach utilizing Lucas polynomials for the solution of time-fractional differential equations. The proposed method employs specially designed test functions that satisfy the boundary conditions of the time-fractional differential equation, leading to an efficient algebraic formulation. The fractional derivative with respect to time is treated analytically, while spatial discretization is performed via the Petrov-Galerkin framework, ultimately yielding a system of linear algebraic equations. To evaluate the accuracy and efficiency of the proposed scheme, the Tau-Legendre spectral method is selected as a comparative numerical approach. Several numerical examples are presented to compare the performance of both methods. The numerical results are reported in tables and figures. The outcomes confirm the effectiveness of both methods, demonstrating that the Lucas polynomial-based Petrov--Galerkin method delivers approximations with good accuracy and exhibits computational performance comparable to that of the Tau--Legendre method in terms of both accuracy and efficiency.
