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A family of novel time-stepping methods for the fractional calculus operators is presented with a shifted parameter. The truncation error with second-order accuracy is proved under the framework of the shifted convolution quadrature. To improve the efficiency, two aspects are considered, that i) a fast algorithm is developed to reduce the computation complexity from O ( N t 2 ) to O ( N t log ⁡ N t ) and the memory requirement from O ( N t ) to O ( log ⁡ N t ) , where N t denotes the number of successive time steps, and ii) correction terms are added to deal with the initial singularity of the solution. The stability analysis and error estimates are provided in detail where in temporal direction the novel time-stepping methods are applied and the spatial variable is discretized by the finite element method. Numerical results for d-dimensional examples ( d = 1 , 2 , 3 ) confirm our theoretical conclusions and the efficiency of the fast algorithm.

For the Maxwell’s equations in a Havriliak–Negami (H-N) dispersive medium, the associated energy dissipation law has not been settled at both continuous level and discrete level. In this paper, we rigorously show that the energy of the H-N model can be bounded by the initial energy and the model is well-posed. We analyse a backward Euler-type semi-discrete scheme, and prove that the modified discrete energy decays monotonically in time. Such a strong stability ensures that the scheme is unconditionally stable. We also introduce a fast temporal convolution algorithm to alleviate the burden of the history dependence in the polarisation relation involving the singular kernel with the Mittag-Leffler function with three parameters. We provide ample numerical results to demonstrate the efficiency and accuracy of a full-discrete scheme via a spectra-Galerkin method in two dimensions. Finally, we consider an interesting application in the recovery of complex relative permittivity and some related physical quantities.

A large amount of shale gas fracturing flowback fluid (FFBF) from the process of shale gas exploitation causes obvious ecological harm to health of soil and water. However, biological hazard of soil microbial populations by fracturing flowback fluid remains rarely reported. In this study, the microbiological compositions were assessed via analyzing diversity of microbial populations. The results showed significant differences between polluted soil by fracturing flowback fluid and unpolluted soil in different pH and temperature conditions. And then, the microbe-index of biological integrity (M-IBI) was used to evaluate the toxicity of the fracturing flowback fluid based on analysis of microbial integrity. The results showed that polluted soil lacks key microbial species known to be beneficial to soil health, including denitrifying bacteria and cellulose-decomposing bacteria, and 35 °C is a critical value for estimating poor and sub-healthy level of damage to microbial integrity by fracturing flowback fluid. Our results provide a valuable reference for the evaluation of soil damage by fracturing flowback fluid.

In this paper, a new Crank–Nicolson finite element method for the time-fractional subdiffusion equation is developed, in which a novel time discretization called the modified L1 method is used to discretize the Riemann–Liouville fractional derivative. The present method is unconditionally stable and convergent of order O ( τ 1 + β + h r + 1 ) , where β ∈ ( 0 , 1 ) , τ and h are the step sizes in time and space, respectively, and r is the degree of the piecewise polynomial space. The derived method is reduced to the classical Crank–Nicolson method when β → 1 . The new time discretization is also used to solve the fractional cable equation. And the unconditional stability and convergence are given. Numerical examples are provided which support the theoretical analysis. The comparison with the existing methods are also given, which shows good performances of the present methods.

In this work, a finite difference method of tunable accuracy for fractional differential equations (FDEs) with end-point singularities is developed. Modified weighted shifted Grunwald–Letnikov (WSGL) formulas are proposed to approximate the left and right Riemann–Liouville fractional operators, which show better accuracy than the original WSGL formulas, due to the use of the correction terms. Finite difference schemes are constructed to solve two fractional boundary value problems and a space-fractional Allen–Cahn equation. Even if the singularity of the considered FDEs is unknown, satisfactory numerical solutions can still be obtained by suitably tuning the correction terms. Various numerical examples are presented to verify the effectiveness of the present method, and comparisons with other known methods are also made that demonstrate higher accuracy of the current method.

We develop spectral collocation methods for fractional differential equations with variable order with two end-point singularities. Specifically, we derive three-term recurrence relations for both integrals and derivatives of the weighted Jacobi polynomials of the form $(1+x)^{\mu_1}(1-x)^{\mu_2}P_{j}^{a,b}(x) \,({a,b,\mu_1,\mu_2>-1})$, which leads to the desired differentiation matrices. We apply the new differentiation matrices to construct collocation methods to solve fractional boundary value problems and fractional partial differential equations with two end-point singularities. We demonstrate that the singular basis enhances greatly the accuracy of the numerical solutions by properly tuning the parameters $\mu_1$ and $\mu_2$, even for cases for which we do not know explicitly the form of singularities in the solution at the boundaries. Comparison with other existing methods shows the superior accuracy of the proposed spectral collocation method.

In this paper, a new type of the discrete fractional Gr{\"o}nwall inequality is developed, which is applied to analyze the stability and convergence of a Galerkin spectral method for a linear time-fractional subdiffusion equation. Based on the temporal-spatial error splitting argument technique, the discrete fractional Gr{\"o}nwall inequality is also applied to prove the unconditional convergence of a semi-implicit Galerkin spectral method for a nonlinear time-fractional subdiffusion equation., Comment: 16 pages

In this paper, we consider the initial boundary value problem of the two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations. An alternating direction implicit (ADI) spectral method is developed based on Legendre spectral approximation in space and finite difference discretization in time. Numerical stability and convergence of the schemes are proved, the optimal error is O ( N − r + τ 2 ) , where N , τ , r are the polynomial degree, time step size and the regularity of the exact solution, respectively. We also consider the non-smooth solution case by adding some correction terms. Numerical experiments are presented to confirm our theoretical analysis. These techniques can be used to model diffusion and transport of viscoelastic non-Newtonian fluids.

In this work, we extend the fractional linear multistep methods in [C. Lubich, SIAM J. Math. Anal., 17 (1986), pp.704--719] to the tempered fractional integral and derivative operators in the sense that the tempered fractional derivative operator is interpreted in terms of the Hadamard finite-part integral. We develop two fast methods, Fast Method I and Fast Method II, with linear complexity to calculate the discrete convolution for the approximation of the (tempered) fractional operator. Fast Method I is based on a local approximation for the contour integral that represents the convolution weight. Fast Method II is based on a globally uniform approximation of the trapezoidal rule for the integral on the real line. Both methods are efficient, but numerical experimentation reveals that Fast Method II outperforms Fast Method I in terms of accuracy, efficiency, and coding simplicity. The memory requirement and computational cost of Fast Method II are $O(Q)$ and $O(Qn_T)$, respectively, where $n_T$ is the number of the final time steps and $Q$ is the number of quadrature points used in the trapezoidal rule. The effectiveness of the fast methods is verified through a series of numerical examples for long-time integration, including a numerical study of a fractional reaction-diffusion model.

We propose a new class of semi-implicit methods for solving nonlinear fractional differential equations and study their stability. Several versions of our new schemes are proved to be unconditionally stable by choosing suitable parameters. Subsequently, we develop an efficient strategy to calculate the discrete convolution for the approximation of the fractional operator in the semi-implicit method and we derive an error bound of the fast convolution. The memory requirement and computational cost of the present semi-implicit methods with a fast convolution are about $O(N\log n_T)$ and $O(Nn_T\log n_T)$, respectively, where $N$ is a suitable positive integer and $n_T$ is the final number of time steps. Numerical simulations, including the solution of a system of two nonlinear fractional diffusion equations with different fractional orders in two-dimensions, are presented to verify the effectiveness of the semi-implicit methods., Comment: 25 pages, 10 figures

This article aims to fill in the gap of the second-order accurate schemes for the time-fractional subdiffusion equation with unconditional stability. Two fully discrete schemes are first proposed f...

If a function can be explicitly expressed, then one can easily compute its Caputo derivative by the known methods. If a function cannot be explicitly expressed but it satisfies a differential equation, how to seek Caputo derivative of such a function has not yet been investigated. In this paper, we propose a numerical algorithm for computing the Caputo derivative of a function defined by a classical (integer-order) differential equation. By the properties of Caputo derivative derived in this paper, we can change the original typical differential system into an equivalent Caputo-type differential system. Numerical examples are given to support the derived numerical method.

In this paper, the high-order finite difference/element methods for the nonlinear anomalous diffusion equations of subdiffusion and superdiffusion are developed, where the high-order finite difference methods are used to approximate the time-fractional derivatives and the finite element methods are used in the spatial domain. The stability and error estimates are proved for both cases of superdiffusion and subdiffusion. Numerical examples are provided to confirm the theoretical analysis.

Starting with the asymptotic expansion of the error equation of the shifted Gr\"{u}nwald--Letnikov formula, we derive a new modified weighted shifted Gr\"{u}nwald--Letnikov (WSGL) formula by introducing appropriate correction terms. We then apply one special case of the modified WSGL formula to solve multi-term fractional ordinary and partial differential equations, and we prove the linear stability and second-order convergence for both smooth and non-smooth solutions. We show theoretically and numerically that numerical solutions up to certain accuracy can be obtained with only a few correction terms. Moreover, the correction terms can be tuned according to the fractional derivative orders without explicitly knowing the analytical solutions. Numerical simulations verify the theoretical results and demonstrate that the new formula leads to better performance compared to other known numerical approximations with similar resolution., Comment: 31 pages, 24 figures