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In this paper, we construct two high order difference schemes for semilinear fractional diffusion-wave equations by a symmetric order reduction method and standard order reduction method at first. To improve the computational efficiency, an efficient time two-grid algorithm is then proposed. The global convergence order of the two-grid schemes reaches $ O(\tau_F^2+\tau_C^4+h^2) $, where $ \tau_F $ and $ \tau_C $ represent the time-step sizes on the fine and coarse grids respectively, while $ h $ is the space-step size. Furthermore, stability and convergence analysis of the derived schemes are carefully verified by the energy method. Finally, a numerical experiment is carried out to show the effectiveness of theoretical statements. [ABSTRACT FROM AUTHOR]

In this paper, we study the two-dimensional Volterra integro-differential equations for viscoelastic rods and membranes in a bounded smooth domain. The memory kernel of this equation makes it not easy to construct an efficient numerical scheme and perform theoretical analysis. The numerical method is considered by the finite difference approach for spatial discretization and Crank–Nicolson (CN) alternating direction implicit (ADI) scheme in the time direction. The integral terms are approximated by the fractional convolution quadrature. We prove that the proposed method is unconditional stable and derive error estimates in L 2 norm. The literature reported on the ADI finite difference method of this model is extremely sparse. Numerical results support the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we study the finite difference/iterative method for the fractional telegraph equation with Hadamard derivatives. The model is first transformed into an equivalent fractional integro‐differential equation, then the technic of exponential type meshes is adopted. The resulting discrete coefficients of Hadamard fractional integral own several graceful properties, which are crucial in numerical analysis. To conquer the difficulty caused by weak regularity, we also propose iterative schemes. The error estimate is equal to O(logt)n+γ,1