Spectral Collocation Methods And Convergence Analysis Of Two Classes Of Time-fractional Partial Differential Equations

Fractional integro-differential equations are the generalization of the tradition-al integro-differential equations. they mainly include fractional wave equations,fractional Fokker-Planck equations and so on. Because these partial differential e-quations are difficult to solve, or only need the approximate solutions of high order accuracy in a real application, so solving the numerical solutions of high accuracy becomes very important and practical value.In this paper,we use the Jacobi spectral collocation method to solve time fractional wave equation and the time fractional Fokker-Planck equation, because the two kinds of time fractional partial differential equations are equivalent to the Volterra integral equation of the second kind, so The original problem is consid-ered as the solution of the Volterra integral equation with weakly singular kernel by the definition and related properties of Caputio or Riemann-Liouvville fractional derivative at first, an appropriate linear variation is then performed on the result-ing equation, this new equation has better regularity, and then using the Jacobi spectral collocation method separately from the time and the space, namely, tak-ing Jacobi-Gauss points as the collocation points, approximating integral term by Gauss integral formula, so the full discrete scheme is obtained. In the end, the error between the exact solution and the approximate solution of the original equation has exponential convergence in L? -norm and weighted L2?-norm, which is proved strictly in theory. Meanwhile, we give the numerical examples, which proves the validity of the spectral collocation method for solving these two equations and the correctness of theoretical results.