Analysis for one‐dimensional time‐fractional Tricomi‐type equations by LDG methods

In this paper, we consider the local discontinuous Galerkin (LDG) finite element method for one‐dimensional linear time‐fractional Tricomi‐type equation (TFTTE), which is obtained from the standard one‐dimensional linear Tricomi‐type equation by replacing the first‐order time derivative with a fractional derivative (of order α, with 1 < α ≤ 2). The proposed LDG is based on LDG finite element method for space and finite difference method for time. We prove that the method is unconditionally stable, and the numerical solution converges to the exact one with order O(h ^{k + 1} + τ ²), where h, τ and k are the space step size, time step size, polynomial degree, respectively. The comparison of the LDG results with the exact solutions is made, numerical experiments reveal that the LDG is very effective.