Abstarct: This dissertation focuses on the study and development of spectral collocation-baxsed numerical methods using Jacobi polynomials for solving fractional partial differential equations. In this context, the Jacobi collocation method is employed to solve the fractional advection-dispersion equation in porous media, and through convergence analysis and numerical simulations, the high efficiency and accuracy of this method in solving such problems are demonstrated. Furthermore, the Jacobi spectral collocation method is extended to solve one- and two-dimensional nonlinear fractional sub-diffusion equations, and through error estimation and comparison with existing methods, its exponential convergence rate in both temporal and spatial discretizations is confirmed. Finally, a space-time collocation algorithm is proposed for solving the variable-order fractional wave equation, in which Jacobi-Gauss-Lobatto and Jacobi-Gauss-Radau schemes are used for spatial and temporal discretization, respectively, transforming the problem into a system of algebraic equations. The numerical results indicate the high accuracy and efficiency of the proposed methods in solving complex fractional problems with various boundary and initial conditions.