Article
Mathematics, Applied
Jianqiang Xie, Muhammad Aamir Ali, Zhiyue Zhang
Summary: This paper focuses on the error estimation of a novel time second-order splitting conservative finite difference method for high-dimensional nonlinear fractional Schrodinger equation. The paper demonstrates the discrete preservation property and shows the accuracy of the method in terms of L2-norm. Numerical experiments are conducted to validate the accuracy and conservation property of the proposed method.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Samira Labidi, Khaled Omrani
Summary: This paper investigates a high-order finite difference scheme for the nonlinear Schrodinger equation with wave operator, showing the existence of solutions using a variant of Brouwer fixed point theorem, discussing the stability and uniqueness of the difference scheme, and ultimately proving the convergence of the scheme through the energy method.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Mathematics, Interdisciplinary Applications
Lewa Alzaleq, Valipuram Manoranjan
Summary: In this paper, a numerical scheme is developed to solve the Klein-Gordon equation with cubic nonlinearity while conserving the discrete energy. The theoretical proof demonstrates that the scheme also conserves other energy-like discrete quantities. Furthermore, the convergence and stability of the scheme are proven. Numerical simulations are presented to showcase the performance of the energy-conserving scheme.
FRACTAL AND FRACTIONAL
(2022)
Article
Mathematics, Applied
Xiujun Cheng, Xiaoqiang Yan, Hongyu Qin, Huiru Wang
Summary: The numerical computation for the two-dimensional generalized nonlinear Schrodinger equations with wave operator is considered in this work. A three-level scheme for the equivalent system is proposed which conserves energy and is linearly implicit. The energy-conserving property, boundedness of the numerical solution and convergence analysis are derived, with numerical experiments confirming the theoretical results.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Anatoly A. Alikhanov, Chengming Huang
Summary: This paper focuses on constructing L2 type difference analog of the Caputo fractional derivative, studying its fundamental features, and using it to generate difference schemes with different orders in space and time for time fractional diffusion equations. The stability and convergence of the schemes are proven, and numerical computations support the obtained results.
APPLIED MATHEMATICS AND COMPUTATION
(2021)
Article
Mathematics, Applied
Yuyu He, Hongtao Chen
Summary: In this paper, an efficient and conservative compact difference scheme based on the SAV approach for the BP equation is constructed. The scheme preserves mass and discrete modified energy, with uniquely solvable properties and bounded estimates for the numerical solution. Convergence rates of second-order in temporal direction and fourth-order in spatial direction are detailed, with theoretical analysis verified through numerical experiments.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Interdisciplinary Applications
Zeting Liu, Baoli Yin, Yang Liu
Summary: In this paper, an explicit-implicit spectral element scheme is developed to solve the space fractional nonlinear Schrodinger equation (SFNSE). The scheme is formulated based on the Legendre spectral element approximation in space and the Crank-Nicolson leap frog difference discretization in time. Both mass and energy conservative properties are discussed and numerical stability and convergence of the scheme are proved. Numerical experiments are performed to confirm the high accuracy and efficiency of the proposed numerical scheme.
FRACTAL AND FRACTIONAL
(2023)
Article
Mathematics, Applied
Zhuoran Li, Ying Wang
Summary: In this paper, an elementary counterexample is given to show that the global L-3 Schrodinger maximal estimate fails if s < 1/3. The argument also applies to the case of 2D fractional Schrodinger operators and does not rely on any facts from number theory.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2023)
Article
Mathematics, Applied
Shuguang Li, Oleg V. Kravchenko, Kai Qu
Summary: In this work, a novel compact difference scheme for the generalized Rosenau-KdVRLW equation is investigated, which preserves some conservative properties of the original equation. The proposed numerical scheme is three-level and nonlinear implicit, and it adopts a novel approximation in the temporal direction and a direct compact discretization method for the higher-order derivatives in the spatial direction. The convergence, stability, and efficiency of the scheme are proven through theoretical analysis and numerical experiments.
NUMERICAL ALGORITHMS
(2023)
Article
Computer Science, Interdisciplinary Applications
Xiaofeng Wang, Hong Cheng, Weizhong Dai
Summary: This article presents two conservative and fourth-order compact finite-difference schemes for solving the generalized Rosenau-Kawahara-RLW equation. The proposed schemes are energy-conserved, convergent, and unconditionally stable, and the numerical convergence orders in both l(2)-norm and l(infinity)-norm are of O(tau(2) + h(4)). Numerical experiments demonstrate the efficiency and reliability of the proposed schemes.
ENGINEERING WITH COMPUTERS
(2022)
Article
Mathematics, Applied
Kanyuta Poochinapan, Ben Wongsaijai
Summary: This paper presents a fourth-order difference scheme for solving the Allen-Cahn equation in both 1D and 2D. The scheme utilizes compact difference operators and an additional stabilized term, and applies the Crank-Nicolson/Adams-Bashforth method to handle the nonlinearity. The proposed scheme maintains the energy-decaying property of the equation and achieves high accuracy. Numerical simulations confirm the reliability of the method.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Mathematics, Applied
Hongtao Chen, Yuyu He
Summary: In this paper, a conservative compact difference scheme for the generalized Kawahara equation is constructed based on the scalar auxiliary variable (SAV) approach. The discrete conservative laws of mass and Hamiltonian energy and boundedness estimates are studied in detail. The error estimates in discrete L-infinity norm and L-2 norm of the presented scheme are analyzed using mathematical induction and discrete energy method. An efficient algorithm, which only requires solving two decoupled equations, is proposed for the presented scheme.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Zhiyong Xing, Liping Wen, Hanyu Xiao
Summary: This paper investigates the Sine-Gordon Equations with the Riesz space fractional derivative. A fourth-order conservative difference scheme is proposed for the one-dimensional problem and extended to the two-dimensional problem. The efficiency of the proposed fast algorithm is verified through several numerical examples.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Computer Science, Interdisciplinary Applications
Jianqiang Xie, Quanxiang Wang, Zhiyue Zhang
Summary: This paper focuses on developing and analyzing two types of finite difference schemes with energy conservation properties for solving the Boussinesq paradigm equation. Firstly, two schemes are proposed for solving the resulting system using auxiliary potential function and the limit as |x| goes to infinity, including a two-level nonlinear difference scheme and a three-level linearized difference scheme. The theoretical analysis of the proposed energy conservative finite difference schemes is then presented, which includes the discrete energy conservation properties, unique solvability, and optimal error estimates. Numerical experiments demonstrate the physical behaviors and efficiency of the proposed schemes.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2023)
Editorial Material
Mathematics, Applied
Asma Rouatbi, Khaled Omrani
Summary: The paper proves a second-order convergence in the maximum norm of the two-dimensional regularized long-wave equation, but due to the use of a wrong inequality in the citation, fundamental errors were found, particularly in the proof of Theorem 3.3 and the convergence Theorem. Clarifying comments are provided in this brief paper.
APPLIED MATHEMATICS AND COMPUTATION
(2021)
Article
Mathematics, Applied
Yanping Chen, Qingfeng Li, Yang Wang, Yunqing Huang
NUMERICAL ALGORITHMS
(2020)
Article
Mathematics, Applied
Hanzhang Hu, Yanping Chen
NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS
(2020)
Article
Mathematics, Applied
Huasheng Wang, Yanping Chen, Yunqing Huang, Wenting Mao
Summary: This paper investigates a boundary value problem of a fractional convection-diffusion equation with general two-sided fractional derivative, studying the well-posedness of the variation formulation under certain assumptions. A Petrov-Galerkin method is developed using Jacobi poly-fractonomials for the trial and test space, allowing for optimal error estimates in properly weighted Sobolev space with a diagonal matrix of the leading term. It is shown that even for smooth data, only algebraic convergence is obtained due to the regularity of the solution, with numerical examples presented to demonstrate the validity of the theoretical results.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2021)
Article
Mathematics, Applied
Yang Wang, Yanping Chen, Yunqing Huang
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2020)
Article
Mathematics, Applied
Jiaoyan Zeng, Yanping Chen, Guichang Liu
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2020)
Article
Mathematics, Applied
Yanping Chen, Xiuxiu Lin, Yunqing Huang, Qian Lin
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2020)
Article
Mathematics, Applied
Yonghui Qin, Yanping Chen, Yunqing Huang, Heping Ma
NUMERICAL MATHEMATICS-THEORY METHODS AND APPLICATIONS
(2020)
Article
Mathematics, Applied
Zhaojie Zhou, Jiabin Song, Yanping Chen
NUMERICAL MATHEMATICS-THEORY METHODS AND APPLICATIONS
(2020)
Article
Mathematics, Applied
Jun Liu, Chen Zhu, Yanping Chen, Hongfei Fu
Summary: In this paper, a novel Crank-Nicolson ADI quadratic spline collocation method is developed for the approximation of two-dimensional two-sided Riemann-Liouville space-fractional diffusion equation. The method is unconditionally stable for certain values of alpha and beta, and it is shown to be convergent with second order in time and min{3 - alpha, 3 - beta} order in space. Numerical examples are included to confirm the theoretical results.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Mathematics, Applied
Ying Liu, Yanping Chen, Yunqing Huang, Qingfeng Li
Summary: This article presents an efficient two-grid method for solving the mathematical model of semiconductor devices, approximating the electric potential and concentration equations with mixed finite element and standard Galerkin methods. By linearizing the full discrete scheme using Newton iteration, small scaled nonlinear equations are solved on the coarse grid followed by linear equations on the fine grid. Detailed analysis of error estimation for two-grid solutions is provided, demonstrating asymptotically optimal approximations when the mesh size satisfies H = O(h(1/2)). Numerical experiments illustrate the efficiency of the two-grid method.
APPLIED MATHEMATICS AND MECHANICS-ENGLISH EDITION
(2021)
Article
Mathematics, Applied
Wenting Mao, Yanping Chen, Huasheng Wang
Summary: This paper presents an efficient spectral algorithm and method to solve fractional initial value problems, using a special set of general Jacobi functions to form trial and test spaces. Rigorous error analysis is conducted in non-uniformly weighted Sobolev spaces, with optimal error estimates obtained. The postprocessing technique is used and superconvergence estimates are derived, along with asymptotically exact a-posteriori error estimators. Numerical experiments are included for theoretical support.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Li-Bin Liu, Yanping Chen
Summary: This study focuses on a nonlinear fractional differential equation with Caputo fractional derivative and introduces a new monitor function for designing an adaptive grid algorithm to solve this type of equation. Numerical results demonstrate that the presented adaptive method outperforms other monitor functions in solving this kind of nonlinear fractional differential equation.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Qingfeng Li, Yanping Chen, Yunqing Huang, Yang Wang
Summary: This paper presents two efficient two-grid algorithms with L1 scheme for solving two-dimensional nonlinear time fractional diffusion equations, which can reduce computational cost and maintain asymptotically optimal accuracy.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2021)
Article
Mathematics, Applied
Bo Tang, Yanping Chen, Xiuxiu Lin
Summary: In this paper, a space-time spectral Galerkin method is re-examined for solving multi-term time fractional diffusion equations, with improved a posteriori error estimates proposed and validated through numerical examples to confirm the effectiveness of the theoretical claims.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Mathematics, Applied
Yanping Chen, Hanzhang Hu
Summary: This study proposes a combined method of mixed finite element method for the pressure equation and finite element method with characteristics for the concentration equation to solve the coupled system of incompressible two-phase flow in porous media. A two-grid algorithm based on the Newton iteration method is developed and analyzed for the nonlinear coupled system. Theoretical and numerical results demonstrate the efficiency and asymptotically optimal accuracy of the proposed schemes as long as the mesh sizes satisfy a specific condition.
ZAMM-ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND MECHANIK
(2021)
