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A High-Order Scheme for Fractional Ordinary Differential Equations with the Caputo-Fabrizio Derivative

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Communications on Applied Mathematics and Computation 2020年第2期2卷 179-199页

作者： Junying Cao Ziqiang Wang Chuanju Xu School of Data Science and Information Engineering Guizhou Minzu UniversityGuiyang 550025China School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing Xiamen UniversityXiamen 361005China

In this paper, we consider numerical solutions of fractional ordinary diferential equations with the Caputo-Fabrizio derivative, and construct and analyze a high-order time-stepping scheme for this equation. The proposed method makes use of quadratic interpolation function in sub-intervals, which allows to produce fourth-order convergence. A rigorous stability and convergence analysis of the proposed scheme is given. A series of numerical examples are presented to validate the theoretical claims. Traditionally a scheme having fourth-order convergence could only be obtained by using block-by-block technique. The advantage of our scheme is that the solution can be obtained step by step, which is cheaper than a block-by-block-based approach.

关键词： Caputo-Fabrizio derivative Fractional diferential equations High-order numerical scheme

OBSERVABILITY INEQUALITY, LOG-TYPE HAUSDORFF CONTENT AND HEAT EQUATIONS

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arXiv 2024年

作者： Huang, Shanlin Wang, Gengsheng Wang, Ming School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China Center for Applied Mathematics Tianjin University Tianjin300072 China School of Mathematics and Statistics HNP-LAMA Central South University Hunan Changsha410083 China

This paper studies observability inequalities for heat equations on both bounded domains and the whole space Rd. The observation sets are measured by log-type Hausdorff contents, which are induced by certain log-type gauge functions closely related to the heat kernel. On a bounded domain, we derive the observability inequality for observation sets of positive log-type Hausdorff content. Notably, the aforementioned inequality holds not only for all sets with Hausdorff dimension s for any s ∈ (d−1, d], but also for certain sets of Hausdorff dimension d − 1. On the whole space Rd, we establish the observability inequality for observation sets that are thick at the scale of the log-type Hausdorff content. Furthermore, we prove that for the 1-dimensional heat equation on an interval, the Hausdorff content we have chosen is an optimal scale for the observability inequality. To obtain these observability inequalities, we use the adapted Lebeau-Robiano strategy from [14]. For this purpose, we prove the following results at scale of the log-type Hausdorff content, the former being derived from the latter: We establish a spectral inequality/a Logvinenko-Sereda uncertainty principle;we set up a quantitative propagation of smallness of analytic functions;we build up a Remez’ inequality;and more fundamentally, we provide an upper bound for the log-type Hausdorff content of a set where a monic polynomial is small, based on an estimate in Lubinsky [35], which is ultimately traced back to the classical Cartan Lemma. In addition, we set up a capacity-based slicing lemma (related to the log-type gauge functions) and establish a quantitative relationship between Hausdorff contents and capacities. These tools are crucial in the studies of the aforementioned propagation of smallness in high-dimensional situations. Copyright © 2024, The Authors. All rights reserved.

关键词： Set theory

Operator Learning Enhanced Physics-informed Neural Networks for Solving Partial Differential Equations Characterized by Sharp Solutions

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arXiv 2023年

作者： Lin, Bin Mao, Zhiping Wang, Zhicheng Karniadakis, George Em School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University Xiamen361005 China Laboratory of Ocean Energy Utilization Ministry of Education School of Energy and Power Engineering Dalian University of Technology Dalian China Division of Applied Mathematics Brown University ProvidenceRI02906 United States

Physics-informed Neural Networks (PINNs) have been shown as a promising approach for solving both forward and inverse problems of partial differential equations (PDEs). Meanwhile, the neural operator approach, including methods such as Deep Operator Network (DeepONet) and Fourier neural operator (FNO), has been introduced and extensively employed in approximating solution of PDEs. Nevertheless, to solve problems consisting of sharp solutions poses a significant challenge when employing these two approaches. To address this issue, we propose in this work a novel framework termed Operator Learning Enhanced Physics-informed Neural Networks (OL-PINN). Initially, we utilize DeepONet to learn the solution operator for a set of smooth problems relevant to the PDEs characterized by sharp solutions. Subsequently, we integrate the pre-trained DeepONet with PINN to resolve the target sharp solution problem. We showcase the efficacy of OL-PINN by successfully addressing various problems, such as the nonlinear diffusion-reaction equation, the Burgers equation and the incompressible Navier-Stokes equation at high Reynolds number. Compared with the vanilla PINN, the proposed method requires only a small number of residual points to achieve a strong generalization capability. Moreover, it substantially enhances accuracy, while also ensuring a robust training process. Furthermore, OL-PINN inherits the advantage of PINN for solving inverse problems. To this end, we apply the OL-PINN approach for solving problems with only partial boundary conditions, which usually cannot be solved by the classical numerical methods, showing its capacity in solving ill-posed problems and consequently more complex inverse problems. Copyright © 2023, The Authors. All rights reserved.

关键词： Partial differential equations

Bound/Positivity Preserving Sav Schemes for the Patlak-Keller-Segel-Navier-Stokes System

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SSRN

SSRN 2022年

作者： Huang, Xueling Shen, Jie School of Mathematical Sciences Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing Xiamen University Fujian Xiamen361005 China Department of Mathematics Purdue University West LafayetteIN47907 United States

The Patlak-Keller-Segel-Navier-Stokes system describes the biological chemotaxis phenomenon in the fluid environment. It is a coupled nonlinear system with unknowns being the cell density, the concentration of chemoattractants, the fluid velocity and the pressure, and it satisfies an energy dissipation law, preserves the bound/positivity and mass of the cell density. We develop in this paper a class of scalar auxiliary variable (SAV) schemes with relaxation which preserve these properties unconditionally at the discrete level, and only require solving decoupled linear systems with constant coefficients at each time step. We present ample numerical results to validate these schemes, and investigate the blow-up phenomenon. In particular, it is found that the fluid flow could delay the blow-up time when compared with the PatlakKeller-Segel system without flow, but weather it may surpress the blow-up requires further investigation. © 2022, The Authors. All rights reserved.

关键词： Energy dissipation

A GENERALIZED SAV APPROACH WITH RELAXATION FOR DISSIPATIVE SYSTEMS

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arXiv 2022年

作者： Zhang, Yanrong Shen, Jie School of Mathematical Sciences Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing Xiamen University Fujian Xiamen361005 China Department of Mathematics Purdue University West LafayetteIN47907 United States

The scalar auxiliary variable (SAV) approach [30] and its generalized version GSAV proposed in [19] are very popular methods to construct efficient and accurate energy stable schemes for nonlinear dissipative systems. However, the discrete value of the SAV is not directly linked to the free energy of the dissipative system, and may lead to inaccurate solutions if the time step is not sufficiently small. Inspired by the relaxed SAV method proposed in [20] for gradient flows, we propose in this paper a generalized SAV approach with relaxation (R-GSAV) for general dissipative systems. The R-GSAV approach preserves all the advantages of the GSAV appraoch, in addition, it dissipates a modified energy that is directly linked to the original free energy. We prove that the k-th order implicit-explicit (IMEX) schemes based on R-GSAV are unconditionally energy stable, and we carry out a rigorous error analysis for k = 1, 2, 3, 4, 5. We present ample numerical results to demonstrate the improved accuracy and effectiveness of the R-GSAV approach. © 2022, CC BY.

关键词： Energy dissipation

O(t^(-β))-SYNCHRONIZATION AND ASYMPTOTIC SYNCHRONIZATION OF DELAYED FRACTIONAL ORDER NEURAL NETWORKS

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Acta Mathematica Scientia 2022年第4期42卷 1273-1292页

作者： Anbalagan PRATAP Ramachandran RAJA Jinde CAO Chuangxia HUANG Jehad ALZABUT Ovidiu BAGDASAR Department of Mathematics Alagappa UniversityKaraikudi 630004India Ramanujan Centre for Higher Mathematics Alagappa UniversityKaraikudi 630004India School of Mathematics Southeast UniversityNanjing 211189China Yonsei Frontier Lab Yonsei UniversitySeoul03722South Korea Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering Changsha University of Science and TechnologyChangsha410114China Department of Mathematics and General Sciences Prince Sultan UniversityRiyadh 12435Saudi Arabia Department of Electronics Computing and MathematicsUniversity of DerbyUnited Kingdom

This article explores the O(t^(-β))synchronization and asymptotic synchronization for fractional order BAM neural networks(FBAMNNs)with discrete delays,distributed delays and non-identical *** designing a state feedback control law and a new kind of fractional order Lyapunov functional,a new set of algebraic sufficient conditions are derived to guarantee the O(t^(-β))Synchronization and asymptotic synchronization of the considered FBAMNNs model;this can easily be evaluated without using a MATLAB LMI control ***,two numerical examples,along with the simulation results,illustrate the correctness and viability of the exhibited synchronization results.

关键词： O(t^(-β))-synchronization asymptotic synchronization BAM neural networks fractional order state feedback control law

Analysis and Hermite spectral approximation of diffusive-viscous wave equations in unbounded domains arising in geophysics

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arXiv 2022年

作者： Ling, Dan Mao, Zhiping School of Mathematics and Statistics Xi’an Jiaotong University Shaanxi Xi’an710049 China School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University Fujian Xiamen361005 China

The diffusive-viscous wave equation (DVWE) is widely used in seismic exploration since it can explain frequency-dependent seismic reflections in a reservoir with hydrocarbons. Most of the existing numerical approximations for the DVWE are based on domain truncation with ad hoc boundary conditions. However, this would generate artificial reflections as well as truncation errors. To this end, we directly consider the DVWE in unbounded domains. We first show the existence, uniqueness, and regularity of the solution of the DVWE. We then develop a Hermite spectral Galerkin scheme and derive the corresponding error estimate showing that the Hermite spectral Galerkin approximation delivers a spectral rate of convergence provided sufficiently smooth solutions. Several numerical experiments with constant and discontinuous coefficients are provided to verify the theoretical result and to demonstrate the effectiveness of the proposed method. In particular, We verify the error estimate for both smooth and non-smooth source terms and initial conditions. In view of the error estimate and the regularity result, we show the sharpness of the convergence rate in terms of the regularity of the source term. We also show that the artificial reflection does not occur by using the present *** Codes 65M70, 35A01, 35Q86, 35B65 Copyright © 2022, The Authors. All rights reserved.

关键词： Seismology

HIGH ORDER ASYMPTOTIC PRESERVING FINITE DIFFERENCE WENO SCHEMES WITH CONSTRAINED TRANSPORT FOR MHD EQUATIONS IN ALL SONIC MACH NUMBERS

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arXiv 2022年

作者： Chen, Wei Wu, Kailiang Xiong, Tao School of Mathematical Sciences Xiamen University Xiamen Fujian361005 China Guangdong Shenzhen518055 China School of Mathematical Sciences Xiamen University Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen Fujian361005 China

In this paper, a high-order semi-implicit (SI) asymptotic preserving (AP) and divergence-free finite difference weighted essentially nonoscillatory (WENO) scheme is proposed for magnetohydrodynamic (MHD) equations. We consider the sonic Mach number Ε ranging from 0 to O(1). High-order accuracy in time is obtained by SI implicit-explicit Runge–Kutta (IMEX-RK) time discretization. High-order accuracy in space is achieved by finite difference WENO schemes with characteristic-wise reconstructions. A constrained transport method is applied to maintain a discrete divergence-free condition. We formally prove that the scheme is AP. Asymptotic accuracy (AA) in the incompressible MHD limit is obtained if the implicit part of the SI IMEX-RK scheme is stiffly accurate. Numerical experiments are provided to validate the AP, AA, and divergence-free properties of our proposed approach. Besides, the scheme can well capture discontinuities such as shocks in an essentially nonoscillatory fashion in the compressible regime, while it is also a good incompressible solver with uniform large-time step conditions in the low sonic Mach limit. © 2022, CC BY.

关键词： Mach number

Improved phase-field-based lattice Boltzmann method for thermocapillary flow

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Physical Review E 2022年第1期105卷 015314-015314页

作者： Liqing Yue Zhenhua Chai Huili Wang Baochang Shi School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan 430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan 430074 China School of Mathematical and Computer Sciences Wuhan Textile University Wuhan 430074 China

In this paper, we present an improved phase-field-based lattice Boltzmann (LB) method for thermocapillary flows with large density, viscosity, and thermal conductivity ratios. The present method uses three LB models to solve the conservative Allen-Cahn equation, the incompressible Navier-Stokes equations, and the temperature equation. To overcome the difficulty caused by the convection term in solving the convection-diffusion equation for the temperature field, we first rewrite the temperature equation as a diffuse equation where the convection term is regarded as the source term and then construct an improved LB model for the diffusion equation. The macroscopic governing equations can be recovered correctly from the present LB method; moreover, the present LB method is much simpler and more efficient. In order to test the accuracy of this LB method, several numerical examples are considered, including the planar thermal Poiseuille flow of two immiscible fluids, the two-phase thermocapillary flow in a nonuniformly heated channel, and the thermocapillary Marangoni flow of a deformable bubble. It is found that the numerical results obtained from the present LB method are consistent with the theoretical prediction and available numerical data, which indicates that the present LB method is an effective approach for the thermocapillary flows.

关键词： Drops & bubbles Multiphase flows Lattice-Boltzmann methods Navier-Stokes equation