Fanhai Zeng

Fanhai Zeng, Zhongqiang Zhang, George Em Karniadakis

Starting with the asymptotic expansion of the error equation of the shifted Grünwald--Letnikov formula, we derive a new modified weighted shifted Grünwald--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.

Ling Guo, Fanhai Zeng, Ian Turner et al.

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.

Fanhai Zeng, Ian Turner, Kevin Burrage

A unified fast time-stepping method for both fractional integral and derivative operators is proposed. The fractional operator is decomposed into a local part with memory length $ΔT$ and a history part, where the local part is approximated by the direct convolution method and the history part is approximated by a fast memory-saving method. The fast method has $O(n_0+\sum_{\ell}^L{q}_α(N_{\ell}))$ active memory and $O(n_0n_T+ (n_T-n_0)\sum_{\ell}^L{q}_α(N_{\ell}))$ operations, where $L=\log(n_T-n_0)$, $n_0={ΔT}/τ,n_T=T/τ$, $τ$ is the stepsize, $T$ is the final time, and ${q}_α{(N_{\ell})}$ is the number of quadrature points used in the truncated Laguerre--Gauss (LG) quadrature. The error bound of the present fast method is analyzed. It is shown that the error from the truncated LG quadrature is independent of the stepsize, and can be made arbitrarily small by choosing suitable parameters that are given explicitly. Numerical examples are presented to verify the effectiveness of the current fast method.

Fanhai Zeng, Ian Turner, Kevin Burrage et al.

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.

Zeting Liu, Fawang Liu, Fanhai Zeng

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.

Fanhai Zeng, Changpin Li

In this paper, the fractional differential matrices based on the Jacobi-Gauss points are derived with respect to the Caputo and Riemann-Liouville fractional derivative operators. The spectral radii of the fractional differential matrices are investigated numerically. The spectral collocation schemes are illustrated to solve the fractional ordinary differential equations and fractional partial differential equations. Numerical examples are also presented to illustrate the effectiveness of the derived methods, which show better performances over some existing methods.

Simin Shekarpaz, Fanhai Zeng, George Karniadakis

We introduce a new approach for solving forward systems of differential equations using a combination of splitting methods and physics-informed neural networks (PINNs). The proposed method, splitting PINN, effectively addresses the challenge of applying PINNs to forward dynamical systems and demonstrates improved accuracy through its application to neuron models. Specifically, we apply operator splitting to decompose the original neuron model into sub-problems that are then solved using PINNs. Moreover, we develop an $L^1$ scheme for discretizing fractional derivatives in fractional neuron models, leading to improved accuracy and efficiency. The results of this study highlight the potential of splitting PINNs in solving both integer- and fractional-order neuron models, as well as other similar systems in computational science and engineering.

Xiangcheng Zheng, Fanhai Zeng, Hong Wang

We study a fully discrete finite element method for variable-order time-fractional diffusion equations with a time-dependent variable order. Optimal convergence estimates are proved with the first-order accuracy in time (and second order accuracy in space) under the uniform or graded temporal mesh without full regularity assumptions of the solutions. Numerical experiments are presented to substantiate the analysis.

Fanhai Zeng, Ian Turner, Kevin Burrage et al.

In this paper, we develop regularized discrete least squares collocation and finite volume methods for solving two-dimensional nonlinear time-dependent partial differential equations on irregular domains. The solution is approximated using tensor product cubic spline basis functions defined on a background rectangular (interpolation) mesh, which leads to high spatial accuracy and straightforward implementation, and establishes a solid base for extending the computational framework to three-dimensional problems. A semi-implicit time-stepping method is employed to transform the nonlinear partial differential equation into a linear boundary value problem. A key finding of our study is that the newly proposed mesh-free finite volume method based on circular control volumes reduces to the collocation method as the radius limits to zero. Both methods produce a large constrained least-squares problem that must be solved at each time step in the advancement of the solution. We have found that regularization yields a relatively well-conditioned system that can be solved accurately using QR factorization. An extensive numerical investigation is performed to illustrate the effectiveness of the present methods, including the application of the new method to a coupled system of time-fractional partial differential equations having different fractional indices in different (irregularly shaped) regions of the solution domain.

Hui Zhang, Xiaoyun Jiang, Fanhai Zeng et al.

The reaction-diffusion model can generate a wide variety of spatial patterns, which has been widely applied in chemistry, biology, and physics, even used to explain self-regulated pattern formation in the developing animal embryo. In this work, a second-order stabilized semi-implicit time-stepping Fourier spectral method is presented for the reaction-diffusion systems of equations with space described by the fractional Laplacian. We adopt the temporal-spatial error splitting argument to illustrate that the proposed method is stable without imposing the CFL condition, and we prove an optimal L2-error estimate. We also analyze the linear stability of the stabilized semi-implicit method and obtain a practical criterion to choose the time step size to guarantee the stability of the semi-implicit method. Our approach is illustrated by solving several problems of practical interest, including the fractional Allen-Cahn, Gray-Scott and FitzHugh-Nagumo models, together with an analysis of the properties of these systems in terms of the fractional power of the underlying Laplacian operator, which are quite different from the patterns of the corresponding integer-order model.

Kevin Burrage, Pamela M. Burrage, Ian W. Turner et al.

In this paper we study the class of mixed-index time fractional differential equations in which different components of the problem have different time fractional derivatives on the left hand side. We prove a theorem on the solution of the linear system of equations, which collapses to the well-known Mittag-Leffler solution in the case the indices are the same, and also generalises the solution of the so-called linear sequential class of time fractional problems. We also investigate the asymptotic stability properties of this class of problems using Laplace transforms and show how Laplace transforms can be used to write solutions as linear combinations of generalised Mittag-Leffler functions in some cases. Finally we illustrate our results with some numerical simulations.

Wanrong Cao, Fanhai Zeng, Zhongqiang Zhang et al.

We propose second-order implicit-explicit (IMEX) time-stepping schemes for nonlinear fractional differential equations with fractional order $0<β<1$. From the known structure of the non-smooth solution and by introducing corresponding correction terms, we can obtain uniformly second-order accuracy from these schemes. We prove the convergence and linear stability of the proposed schemes. Numerical examples illustrate the flexibility and efficiency of the IMEX schemes and show that they are effective for nonlinear and multi-rate fractional differential systems as well as multi-term fractional differential systems with non-smooth solutions.

Fanhai Zeng

We propose two stable and one conditionally stable finite difference schemes of second-order in both time and space for the time-fractional diffusion-wave equation. In the first scheme, we apply the fractional trapezoidal rule in time and the central difference in space. We use the generalized Newton-Gregory formula in time for the second scheme and its modification for the third scheme. While the second scheme is conditionally stable, the first and the third schemes are stable. We apply the methodology to the considered equation with also linear advection-reaction terms and also obtain second-order schemes both in time and space. Numerical examples with comparisons among the proposed schemes and the existing ones verify the theoretical analysis and show that the present schemes exhibit better performances than the known ones.