The tempered fractional calculus has been successfully applied for depicting the time evolution of a system describing non-Markovian diffusion particles.The related governing equations are a series of partial differential equations with tempered fractional derivatives.Using the polynomial interpolation technique,in this paper,we present three efficient numerical formulas,namely the tempered L1 formula,the tempered L1-2 formula,and the tempered L2-1σ formula,to approximate the Caputo-tempered fractional derivative of order α ∈ (0,1).The truncation error of the tempered L1 formula is of order 2-α,and the tempered L1-2 formula and L2-1σ formula are of order 3-α.As an application,we construct implicit schemes and implicit ADI schemes for one-dimensional and two-dimensional time-tempered fractional diffusion equations,respectively.Furthermore,the unconditional stability and convergence of two developed difference schemes with tempered L1 and L2-1σ formulas are proved by the Fourier analysis method.Finally,we provide several numerical examples to demonstrate the correctness and effectiveness of the theoretical analysis.
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