Second‐order difference scheme for the time fractional mixed diffusion‐wave equation with initial weak regularity

An L1‐type difference scheme on graded meshes for time fractional mixed diffusion‐wave equation with two Caputo derivatives of fractional order and is established and analyzed. In particular, the solution of governing equation typically exhibits a weak regular behavior at an initial time. A fully discrete numerical scheme based on finite difference method is proposed and detailed theoretical analysis for the scheme is also investigated. The novelty of this paper is to show that the L1‐type difference scheme for the model problem under consideration is temporal second‐order accurate over graded meshes. Numerical tests are presented to support our theoretical results.