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In the present paper, the regularity and finite difference methods for Caputo-Hadamard fractional differential equations with initial value singularity are taken into consideration. To overcome the weak singularity and enhance convergence precision, a fitted scheme on nonuniform meshes is applied to such problems. Firstly, based on L log , 2 - 1 σ approximation, the temporal convergence accuracy of fitted scheme for the sub-diffusion equations is O (N - min { 2 r α , 2 }) , where N denotes the number of time steps, α is the fractional order and r is the mesh grading parameter. It is indicated that the performance of the fitted scheme is better than that of the standard L log , 2 - 1 σ scheme on (i) exponent meshes (i.e., r = 1 ) and (ii) graded meshes with the optimal choice of the mesh grading. Secondly, a second-order fitted scheme on exponent meshes for the diffusion-wave equations is obtained. Furthermore, for the sake of improving the computational efficiency and demonstrating the effectiveness of the decomposition of the solution, the fast algorithm and further decomposition of the solution for the sub-diffusion equations are investigated. Ultimately, some examples are presented to verify the availability of our theoretical results. [ABSTRACT FROM AUTHOR]

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 derive the improved regularity for two-dimensional nonlinear time fractional telegraph equations by virtue of the technic of decomposition at first. Then, the famous L 2 - 1 σ formula is adopted to approximate the Caputo derivative and the central finite difference method is used for spatial discretization. The convergence accuracy of the proposed method is second order in both temporal and spatial direction. Meanwhile, for the sake of reducing the time of solving the high dimensional nonlinear problems, an efficient time two-grid algorithm is proposed. Furthermore, stability analysis of the proposed scheme is studied by the energy method. At last, numerical experiments are presented to verify the validity of the theoretical statements. [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

In this paper, we study the numerical method for time-fractional KdV–Burgers' equation with initial singularity. The famous L 2 - 1 σ formula on graded meshes is adopted to approximate the Caputo derivative. Meanwhile, a nonlinear finite difference method on uniform grids is deduced for spatial discretization. The proposed method is second order in time and first order in space. With the help of the fractional Grönwall inequality, the unconditional stability and convergence of the current scheme are analyzed based on some skills. To raise the accuracy in spatial direction, a second order method is then carefully deduced. At last, theoretical results are verified by numerical experiments. [ABSTRACT FROM AUTHOR]