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A superlinear convergence scheme for the multi‐term and distribution‐order fractional wave equation with initial singularity

In this paper, a superlinear convergence scheme for the multi‐term and distribution‐order fractional wave equation with initial singularity is proposed. The initial singularity of the solution of the multi‐term time fractional partial differential equation often generate a singular source, it increases the difficulty to numerically solve the equation. Thus, after discretizing the spatial distribution‐order derivative by the midpoint quadrature, an integral transformation is applied to deal with the temporal direction for obtaining a temporal superlinear convergence scheme based on the uniform mesh. Then, the fully discrete scheme is constructed by using Crank–Nicolson technique and L1 approximation in time, and fractional centered difference approximation in space. The convergence and stability of the proposed scheme are rigorously analyzed. Finally, numerical experiments are presented to support the theoretical results of our scheme.