In this paper, we study a fast and linearized finite difference method to solve the nonlinear time-fractional wave equation with multi fractional orders. We first propose a discretization to the multi-term Caputo derivative based on the recently established fast L2-1σ formula and a weighted approach. Then we apply the discretization to construct a fully fast linearized discrete scheme for the nonlinear problem under consideration. The nonlinear term, which just fulfills the Lipschitz condition, will be evaluated on the previous time level. Therefore only linear systems are needed to be solved for obtaining numerical solutions. The proposed scheme is shown to have second-order unconditional convergence with respect to the discrete H1-norm. Numerical examples are provided to justify the efficiency.
In this paper, we develop a linearized fractional Crank-Nicolson-Galerkin FEM for Kirchhoff type quasilinear time-fractional integro-differential equation $\left(\mathcal{D}^α\right)$. In general, the solutions to the time-fractional problems exhibit a weak singularity at time $t=0$. This singular behavior of the solutions is taken into account while deriving the convergence estimates of the developed numerical scheme. We prove that the proposed numerical scheme has an accuracy rate of $O(M^{-1}+N^{-2})$ in $L^{\infty}(0,T;L^{2}(Ω))$ as well as in $L^{\infty}(0,T;H^{1}_{0}(Ω))$, where $M$ and …
We introduce a symmetric fractional-order reduction (SFOR) method to construct numerical algorithms on general nonuniform temporal meshes for semilinear fractional diffusion-wave equations. By using the novel order reduction method, the governing problem is transformed to an equivalent coupled system, where the explicit orders of time-fractional derivatives involved are all $α/2$ $(1<α<2)$. The linearized L1 scheme and Alikhanov scheme are then proposed on general time meshes. Under some reasonable regularity assumptions and weak restrictions on meshes, the optimal convergence is derived for t…
A linearized numerical scheme is proposed to solve the nonlinear time fractional parabolic problems with time delay. The scheme is based on the standard Galerkin finite element method in the spatial direction, the fractional Crank-Nicolson method and extrapolation methods in the temporal direction. A novel discrete fractional Grönwall inequality is established. Thanks to the inequality, the error estimate of fully discrete scheme is obtained. Several numerical examples are provided to verify the effectiveness of the fully discrete numerical method.
In this paper, we propose a fast multipole method (FMM) for 3-D linearized Poisson-Boltzmann (PB) equation in layered media. The main framework of the algorithm is analogous to the FMM for Helmholtz and Laplace equation in layered media [1,2], using an extension of the Funk-Hecke formula for pure imaginary wave number. Moreover, a recurrence formula is provided for the run-time computation of the Sommerfeld-type integrals used in the FMM algorithm. Due to the similarity between Helmholtz and linearized PB equation, the recurrence formula can also be used for the FMM of Helmholtz equation in la…
In this article, we propose a linearized fully-discrete scheme for solving a time fractional nonlocal diffusion-wave equation of Kirchhoff type. The scheme is established by using the finite element method in space and the $L1$ scheme in time. We derive the $α$-robust \textit{a priori} bound and \textit{a priori} error estimate for the fully-discrete solution in $L^{\infty}\big(H^1_0(Ω)\big)$ norm, where $α\in (1,2)$ is the order of time fractional derivative. Finally, we perform some numerical experiments to verify the theoretical results.
In this paper, we study a time-fractional initial-boundary value problem of Kirchhoff type involving memory term for non-homogeneous materials. The energy argument is applied to derive the a priori bounds on the solution of the considered problem. Consequently, we prove the existence and uniqueness of the weak solution to the problem under consideration. We keep the time variable continuous and discretize the space domain using a conforming FEM to obtain the semi discrete formulation of the problem. The semi discrete error analysis is carried out by modifying the standard Ritz-Volterra project…
An all-at-once linear system arising from the nonlinear tempered fractional diffusion equation with variable coefficients is studied. Firstly, the nonlinear and linearized implicit schemes are proposed to approximate such the nonlinear equation with continuous/discontinuous coefficients. The stabilities and convergences of the two schemes are proved under several suitable assumptions, and numerical examples show that the convergence orders of these two schemes are $1$ in both time and space. Secondly, a nonlinear all-at-once system is derived based on the nonlinear implicit scheme, which may s…
In this paper, we propose a fast second-order approximation to the variable-order (VO) Caputo fractional derivative, which is developed based on $L2$-$1_σ$ formula and the exponential-sum-approximation technique. The fast evaluation method can achieve the second-order accuracy and further reduce the computational cost and the acting memory for the VO Caputo fractional derivative. This fast algorithm is applied to construct a relevant fast temporal second-order and spatial fourth-order scheme ($FL2$-$1_σ$ scheme) for the multi-dimensional VO time-fractional sub-diffusion equations. Theoreticall…
The nonlinear space-fractional problems often allow multiple stationary solutions, which can be much more complicated than the corresponding integer-order problems. In this paper, we systematically compute the solution landscapes of nonlinear constant/variable-order space-fractional problems. A fast approximation algorithm is developed to deal with the variable-order spectral fractional Laplacian by approximating the variable-indexing Fourier modes, and then combined with saddle dynamics to construct the solution landscape of variable-order space-fractional phase field model. Numerical experim…
In this paper, a linearized semi-implicit finite difference scheme is proposed for solving the two-dimensional (2D) space fractional nonlinear Schrödinger equation (SFNSE).The scheme has the property of mass and energy conservation on the discrete level, with an unconditional stability and a second order accuracy for both time and spatial variables. The main contribution of this paper is an optimal pointwise error estimate for the 2D SFNSE, which is rigorously established and proved for the first time. Moreover, a novel technique is proposed for dealing with the nonlinear term in the equation,…
In this paper, we present linearized learning methods to accelerate the convergence of training for stationary nonlinear Navier-Stokes equations. To solve the stationary nonlinear Navier-Stokes (NS) equation, we integrate the procedure of linearization of the nonlinear convection term in the NS equation into the training process of multi-scale deep neural network approximation of the NS solution. Four forms of linearizations are considered. After a benchmark problem, we solve the highly oscillating stationary flows utilizing the proposed linearized learning with multi-scale neural network for …
In this paper, a modified Euler-Maruyama (EM) method is constructed for a kind of multi-term Riemann-Liouville stochastic fractional differential equations and the strong convergence order min{1-α_m, 0.5} of the proposed method is proved with Riemann-Liouville fractional derivatives' orders 0<α_1<α_2<...<α_m <1. Then, based on the sum-of-exponentials approximation, a fast implementation of the modified EM method which is called a fast EM method is derived to greatly improve the computational efficiency. Finally, some numerical examples are carried out to support the theoretical results and sho…
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…
We investigate the nonlinear vibration of a fractional viscoelastic cantilever beam, subject to base excitation, where the viscoelasticity takes the general form of a distributed-order fractional model, and the beam curvature introduces geometric nonlinearity into the governing equation. We utilize the extended Hamilton principle to derive the governing equation of motion for specific material distribution functions that lead to fractional Kelvin-Voigt viscoelastic model. By spectral decomposition in space, the resulting governing fractional PDE reduces to nonlinear time-fractional ODEs. We us…
Efficient and fast predictor-corrector methods are proposed to deal with nonlinear Caputo-Fabrizio fractional differential equations, where Caputo-Fabrizio operator is a new proposed fractional derivative with a smooth kernel. The proposed methods achieve a uniform accuracy order with the second-order scheme for linear interpolation and the third-order scheme for quadratic interpolation. The convergence analysis is proved by using the discrete Gronwall's inequality. Furthermore, applying the recurrence relation of the memory term, it reduces CPU time executed the proposed methods. The proposed…
In this paper, we develop two fast implicit difference schemes for solving a class of variable-coefficient time-space fractional diffusion equations with integral fractional Laplacian (IFL). The proposed schemes utilize the graded $L1$ formula for the Caputo fractional derivative and a special finite difference discretization for IFL, where the graded mesh can capture the model problem with a weak singularity at initial time. The stability and convergence are rigorously proved via the $M$-matrix analysis, which is from the spatial discretized matrix of IFL. Moreover, the proposed schemes use t…
In this paper, a high-order and fast numerical method is investigated for the time-fractional Black-Scholes equation. In order to deal with the typical weak initial singularities of the solution, we construct a finite difference scheme with variable time steps, where the fractional derivative is approximated by the nonuniform Alikhanov formula and the sum-of-exponentials (SOE) technique. In the spatial direction, an average approximation with fourth-order accuracy is employed. The stability and the convergence with second-order in time and fourth-order in space of the proposed scheme are relig…
In this article, we develop a new method to approximate numerically the fractional Laplacian of functions defined on $\mathbb R$, as well as some more general singular integrals. After mapping $\mathbb R$ into a finite interval, we discretize the integral operator using a modified midpoint rule. The result of this procedure can be cast as a discrete convolution, which can be evaluated efficiently using the Fast-Fourier Transform (FFT). The method provides an efficient, second order accurate, approximation to the fractional Laplacian, without the need to truncate the domain. We first prove that…
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…
Considering fractional fast diffusion equations on bounded open polyhedral domains in $\mathbb{R}^N$, we give a fully Galerkin approximation of the solutions by $C^0$-piecewise linear finite elements in space and backward Euler discretization in time, a priori estimates and the rates of convergence for the approximate solutions are proved, which extends the results of \emph{Carsten Ebmeyer and Wen Bin Liu, SIAM J. Numer. Anal., 46(2008), pp. 2393--2410}. We also generalize the a priori estimates and the rates of convergence to a parabolic integral equation under the framework of \emph{Qiang Du…
Fast Linearized Alternating Direction Minimization Algorithm with Adaptive Parameter Selection for Multiplicative Noise Removal
Department of Mathematics, Third Military Medical University, Chongqing 400038, Chongqing, People's Republic of China（数学系，第三军医大学）；Department of Mathematics and System, School of Sciences, National University of Defense Technology, Changsha 410073, Hunan, People's Republic of China（数学与系统系，国防科技大学）
We develop a fast divided-and-conquer indirect collocation method for the homogeneous Dirichlet boundary value problem of variable-order space-fractional diffusion equations. Due to the impact of the space-dependent variable order, the resulting stiffness matrix of the numerical approximation does not have a Toeplitz-like structure. In this paper we derive a fast approximation of the coefficient matrix by the means of a sum of Toeplitz matrices multiplied by diagonal matrices. We show that the approximation is asymptotically consistent with the original problem, which requires $O(kN\log^2 N)$ …
Fractional order models have proven to be a very useful tool for the modeling of the mechanical behaviour of viscoelastic materials. Traditional numerical solution methods exhibit various undesired properties due to the non-locality of the fractional differential operators, in particular regarding the high computational complexity and the high memory requirements. The infinite state representation is an approach on which one can base numerical methods that overcome these obstacles. Such algorithms contain a number of parameters that influence the final result in nontrivial ways. Based on numer…
Efficient long-time integration of nonlinear fractional differential equations is significantly challenging due to the integro-differential nature of the fractional operators. In addition, the inherent non-smoothness introduced by the inverse power-law kernels deteriorates the accuracy and efficiency of many existing numerical methods. We develop two efficient first- and second-order implicit-explicit (IMEX) methods for accurate time-integration of stiff/nonlinear fractional differential equations with fractional order $α\in (0,1]$ and prove their convergence and linear stability properties. T…
In this paper, we develop a robust fast method for mobile-immobile variable-order (VO) time-fractional diffusion equations (tFDEs), superiorly handling the cases of small or vanishing lower bound of the VO function. The valid fast approximation of the VO Caputo fractional derivative is obtained using integration by parts and the exponential-sum-approximation method. Compared with the general direct method, the proposed algorithm ($RF$-$L1$ formula) reduces the acting memory from $\mathcal{O}(n)$ to $\mathcal{O}(\log^2 n)$ and computational cost from $\mathcal{O}(n^2)$ to $\mathcal{O}(n \log^2 …
In this paper, we first establish a new fractional magnetohydrodynamic (MHD) coupled flow and heat transfer model for a generalized second-grade fluid. This coupled model consists of a fractional momentum equation and a heat conduction equation with a generalized form of Fourier law. The second-order fractional backward difference formula is applied to the temporal discretization and the Legendre spectral method is used for the spatial discretization. The fully discrete scheme is proved to be stable and convergent with an accuracy of $O(τ^2+N^{-r})$, where $τ$ is the time step size and $N$ is …
In this paper, we study the numerical solutions of the multi-dimensional spatial fractional Allen-Cahn equations. After semi-discretization for the spatial fractional Riesz derivative, a system of nonlinear ordinary differential equations with Toeplitz structure is obtained. For the sake of reducing the computational complexity, a two-level Strang splitting method is proposed, where the Toeplitz matrix in the system is split into the sum of a circulant matrix and a skew-circulant matrix. Therefore, the proposed method can be quickly implemented by the fast Fourier transform, substituting to ca…
In this paper, we first propose an unconditionally stable implicit difference scheme for solving generalized time-space fractional diffusion equations (GTSFDEs) with variable coefficients. The numerical scheme utilizes the $L1$-type formula for the generalized Caputo fractional derivative in time discretization and the second-order weighted and shifted Grünwald difference (WSGD) formula in spatial discretization, respectively. Theoretical results and numerical tests are conducted to verify the $(2 - γ)$-order and 2-order of temporal and spatial convergence with $γ\in(0,1)$ the order of Caputo …
Analysis of $L1$-Galerkin FEMs for time-fractional nonlinear parabolic problems
School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院）；Department of Mathematics, City University of Hong Kong（香港城市大学数学系）；Institute of Sciences, PLA University of Science and Technology（中国人民解放军理工大学理学院）；Department of Scientific Computing, Florida State University（佛罗里达州立大学科学计算系）；Beijing Computational Science Research Center（北京计算科学研究中心）
In this paper, we solve the linearized Poisson-Boltzmann equation, used to model the electric potential of macromolecules in a solvent. We derive a corrected trapezoidal rule with improved accuracy for a boundary integral formulation of the linearized Poisson-Boltzmann equation. More specifically, in contrast to the typical boundary integral formulations, the corrected trapezoidal rule is applied to integrate a system of compacted supported singular integrals using uniform Cartesian grids in $\mathbb{R}^3$, without explicit surface parameterization. A Krylov method, accelerated by a fast multi…
High order fast algorithm for the Caputo fractional derivative
Chongqing University（重庆大学）；Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所）
In this paper, fast numerical methods are established for solving a class of time distributed-order and Riesz space fractional diffusion-wave equations. We derive new difference schemes by the weighted and shifted Gr$\ddot{\rm{u}}$nwald formula in time and the fractional centered difference formula in space. The unconditional stability and second-order convergence in time, space and distributed-order of the difference schemes are analyzed. In the one-dimensional case, the Gohberg-Semencul formula utilizing the preconditioned Krylov subspace method is developed to solve the symmetric positive d…
