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This article deals with an establishment and sharp theoretical analysis of a numerical scheme devised for solving the multi-dimensional multi-term time fractional mixed diffusion and wave equations. The governing equation contains both fractional diffusion term and fractional wave term which make the numerical analysis challenging. With the help of the method of order reduction, we convert the time multi-term fractional diffusion and wave terms into the time multi-term fractional integral and diffusion terms respectively, and then develop L2-1σ formula for solving the latter problem. In addition, the formula is used to numerically solve the time distributed-order diffusion and wave equations. The stability and convergence of these numerical schemes are rigorously analyzed by the energy method. The convergence rates are of order two in both time and space. A difference scheme on nonuniform time grids is also constructed for solving the problem with weak regularity at the initial time. Finally, we illustrate our results with some examples.

A new high-order finite difference scheme to approximate the Caputo fractional derivative 1 2 ( D t α 0 C f ( t k ) + D t α 0 C f ( t k − 1 ) ) , k = 1 , 2 , … , N , with the convergence order O ( Δ t 4 − α ) , α ∈ ( 1 , 2 ) is obtained when f ‴ ( t 0 ) = 0 , where Δt denotes the time step size. Based on this scheme we introduce a finite difference method for solving fractional diffusion wave equation with the convergence order O ( Δ t 4 − α + h 2 ) , where h denotes the space step size. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

A time fractional mixed sub-diffusion and diffusion-wave equation involving at least two Caputo time derivatives of order γ ∈ ( 1 , 2 ) and α ∈ ( 0 , 1 ) is considered. A new analytical technique is introduced to analyze the standard finite difference method based on L 1 scheme. Both stability and convergence of the scheme are proved rigorously. The final convergence result shows clearly that the temporal accuracy arrives at the order of O ( τ min { 2 − α , 3 − γ } ) in a discrete H 1 -norm. This novel analytical technique can provide new insights in analyzing other multi-term mixed time fractional differential equations.