Numerical analysis for Klein-Gordon equation with time-space fractional derivatives

We present and analyze two numerical schemes for solving a nonlinear Klein-Gordon equation with time-space fractional derivatives. Numerical methods are base on finite difference scheme in fractional derivative and Fourier-spectral method in spatial variable. It is proved that the linearized method is conditionally stable while the nonlinearized one is unconditionally stable. In addition, the error estimate shows that the linearized method is in the order of O(Delta t+N beta-r), and the nonlinearized method converge with the order O(Delta t3-alpha+N beta-r), where Delta t, N, beta, and r are, respectively, step of time, polynomial degree, the fractional derivative in space, and regularity of u. Some numerical experiments are performed to demonstrate the theoretical results.