Physical laws governing population dynamics are generally expressed as differential equations. Research in recent decades has incorporated fractional-order (non-integer) derivatives into differential models of natural phenomena, such as reaction-diffusion systems. In this paper, we develop a method to numerically solve a multi-component and multi-dimensional space-fractional system. For space discretization, we apply a Fourier spectral method that is suited for multidimensional PDE systems. Efficient approximation of time-stepping is accomplished with a locally one dimensional exponential time differencing approach. We show the effect of different fractional parameters on growth models and consider the convergence, stability, and uniqueness of solutions, as well as the biological interpretation of parameters and boundary conditions.
In this paper, a stabilized second order in time accurate linear exponential time differencing (ETD) scheme for the no-slope-selection thin film growth model is presented. An artificial stabilizing term $Aτ^2\frac{\partialΔ^2 u}{\partial t}$ is added to the physical model to achieve energy stability, with ETD-based multi-step approximations and Fourier collocation spectral method applied in the time integral and spatial discretization of the evolution equation, respectively. Long time energy stability and detailed $\ell^{\infty}(0,T; \ell^2)$ error analysis are provided based on the energy met…
We explore the class of exponential integrators known as exponential time differencing (ETD) method in this letter to design low complexity nonlinear Fourier transform (NFT) algorithms that compute discrete approximations of the scattering coefficients in terms of the Chebyshev polynomials for real values of the spectral parameter. In particular, we discuss ETD Runge-Kutta methods which yield algorithms with complexity $O(N\log^2N)$ (where $N$ is the number of samples of the signal) and an order of convergence that matches the underlying one-step method.
Data-Driven Free-Energy Learning from Trajectories via Structure-Preserving Exponential Time Differencing
Laboratory of Mathematics and Complex Systems, Ministry of Education and School of Mathematical Sciences, Beijing Normal University（数学与复杂系统教育部重点实验室，北京师范大学数学科学学院）
Efficiently and accurately simulating multi-dimensional M-coupled nonlinear Schrödinger equations with fourth-order time integrators and Fourier spectral method
In this paper, we consider the error analysis of a conservative Fourier pseudo-spectral method that conserves mass and energy for the space fractional nonlinear Schrödinger equation. We give a new fractional Sobolev norm that can construct the discrete fractional Sobolev space, and we also can prove some important lemmas for the new fractional Sobolev norm. Based on these lemmas and energy method, a priori error estimate for the method can be established. Then, we are able to prove that the Fourier pseudo-spectral method is unconditionally convergent with order $O(τ^{2}+N^{α/2-r})$ in the disc…
To achieve efficient and accurate long-time integration, we propose a fast, accurate, and stable high-order numerical method for solving fractional-in-space reaction-diffusion equations. The proposed method is explicit in nature and utilizes the fourth-order compact finite difference scheme and matrix transfer technique (MTT) in space with FFT-based implementation. Time integration is done through the accurate fourth-order modified exponential time differencing Runge-Kutta scheme. The linear stability analysis and various numerical experiments including two-dimensional (2D) Fitzhugh-Nagumo, Gi…
We propose a Fourier pseudo-spectral scheme for the space-fractional nonlinear Schrödinger equation. The proposed scheme has the following features: it is linearly implicit, it preserves two invariants of the equation, its unique solvability is guaranteed without any restrictions on space and time step sizes. The scheme requires solving a complex symmetric linear system per time step. To solve the system efficiently, we also present a certain variable transformation and preconditioner.
A third order exponential time differencing numerical scheme for no-slope-selection epitaxial thin film model with energy stability
Southwest University of Science and Technology（西南科技大学）；The Hong Kong Polytechnic University（香港理工大学）；The University of Massachusetts, North Dartmouth（麻省大学北达特茅斯分校）
In this paper, we consider a second-order scalar auxiliary variable (SAV) Fourier spectral method to solve the nonlinear fractional generalized wave equation. Unconditional energy conservation or dissipation properties of the fully discrete scheme are first established. Next, we utilize the temporal-spatial error splitting argument to obtain unconditional optimal error estimate of the fully discrete scheme, which overcomes time-step restrictions caused by strongly nonlinear system, or the restrictions that the nonlinear term needs to satisfy the assumption of global Lipschitz condition in all …
Resolving the Gibbs Phenomenon in Fractional Fourier Series via Inverse Polynomial Reconstruction
School of Mathematics, China University of Mining and Technology（中国矿业大学数学学院）；Department of Mathematics, Riphah International University Faisalabad（菲萨尔巴德立夫海国际大学数学系）；School of Mathematics and Statistics, Guangxi Normal University（广西师范大学数学与统计学院）
In this paper, we propose a fast spectral-Galerkin method for solving PDEs involving integral fractional Laplacian in $\mathbb{R}^d$, which is built upon two essential components: (i) the Dunford-Taylor formulation of the fractional Laplacian; and (ii) Fourier-like bi-orthogonal mapped Chebyshev functions (MCFs) as basis functions. As a result, the fractional Laplacian can be fully diagonalised, and the complexity of solving an elliptic fractional PDE is quasi-optimal, i.e., $O((N\log_2N)^d)$ with $N$ being the number of modes in each spatial direction. Ample numerical tests for various decayi…
This paper is concerned with the inverse random source problem for a stochastic time fractional diffusion equation, where the source is assumed to be driven by a Gaussian random field. The direct problem is shown to be well-posed by examining the well-posedness and regularity of the solution for the equivalent stochastic two-point boundary value problem in the frequency domain. For the inverse problem, the Fourier modulus of the diffusion coefficient of the random source is proved to be uniquely determined by the variance of the Fourier transform of the boundary data. As a phase retrieval for …
Fast simulation of Volterra processes using random Fourier features with application to the log-stationary fractional Brownian motion
Ceremade, CNRS-UMR 7534, Université Paris-Dauphine PSL（巴黎第九大学PSL，CNRS UMR 7534 Cerede）；Guangdong Provincial/Zhuhai Key Laboratory of Interdisciplinary Research and Application for Data Science, Beijing Normal-Hong Kong Baptist University（北京师范大学-香港浸会大学联合国际学院数据科学跨学科研究与应用广东省/珠海市重点实验室）
