A fully discrete local discontinuous Galerkin method for variable-order fourth-order equation with Caputo-Fabrizio derivative based on generalized numerical fluxes

In this paper, an effective numerical method for the variable-order(VO) fourth-order problem with Caputo-Fabrizio derivative will be constructed and analyzed. Based on generalized alternating numerical flux, appropriate spatial and temporal discretization, we get a fully discrete local discontinuous Galerkin(LDG) scheme. The theoretic properties of the fully discrete LDG scheme are proved in detail by mathematical induction, and the method is proved to be unconditionally stable and convergent with $\mathrm{O}(\tau +h^{k+1})$, where $h$ is the spatial step, $\tau$ is the temporal step and $k$ is the degree of the piecewise $P^k$ polynomial. In order to show the efficiency of our method, some numerical examples are carried out by Matlab.