Article
Mathematics, Applied
Yuyu He, Xiaofeng Wang, Ruihua Zhong
Summary: In this paper, a novel three-point fourth-order compact operator is used to construct a new linearized conservative compact finite difference scheme for the symmetric regularized long wave equations. The discrete conservative laws, boundedness, and unique solvability are studied, and the convergence order and stability of the scheme are proved by the discrete energy method. Numerical examples are provided to support the theoretical analysis.
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Mohamed Rahmeni, Khaled Omrani
Summary: This paper considers a high-order finite difference method for the two-dimensional coupled nonlinear Schrodinger equations. The method is proven to preserve the total mass and energy in a discrete sense and an optimal error estimate is established using the standard energy method. Numerical results are compared with exact solutions and other existing methods, showing improved accuracy in both spatial and temporal directions.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Yanyan Wang, Zhaopeng Hao, Rui Du
Summary: In this paper, a conservative three-layer linearized difference scheme for the two-dimensional nonlinear Schrodinger equation with fractional Laplacian is proposed. The scheme is proven to be uniquely solvable and it conserves mass and energy in the discrete sense. The scheme is also shown to be unconditionally convergent and stable under l(infinity)-norm, with a convergence order of O(tau(2) + h(2)).
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
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
Zhiyong Xing, Liping Wen
Summary: This paper proposes a linearized difference scheme for the Sine-Gordon equation with a Caputo time derivative, which is simpler and easier for theoretical analysis compared to existing schemes. The solvability, boundedness, and convergence of the difference scheme are rigorously established in the L-infinity norm. Several numerical experiments are provided to support the theoretical results.
ADVANCES IN APPLIED MATHEMATICS AND MECHANICS
(2021)
Article
Mathematics, Interdisciplinary Applications
Merfat Basha, Eyaya Fekadie Anley, Binxiang Dai
Summary: In this paper, the nonlinear Riesz space-fractional convection-diffusion equation with a reaction term in a finite domain in two dimensions is studied. The Crank-Nicolson difference method for temporal discretization and the weighted-shifted Grunwald-Letnikov difference method for spatial discretization are proposed to achieve second-order convergence in time and space. The D'Yakonov alternating-direction implicit technique is applied to find alternative solutions and reduce computational cost. Theoretical analyses prove unconditional stability and convergence. Numerical experiments with known exact solutions verify the effectiveness and computational accuracy of the proposed method.
FRACTAL AND FRACTIONAL
(2023)
Article
Mathematics, Applied
Xiaofeng Wang
Summary: This paper proposes and analyzes a compact finite difference scheme for the generalized dissipative symmetric regularized long-wave equations. The solvability and estimates of the proposed scheme are rigorously proved using mathematical induction. The convergence and stability of the scheme are demonstrated using the discrete energy method. Numerical experiments verify the theoretical analysis and reliability of the proposed scheme.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics
Victor N. Orlov, Asmaa M. Elsayed, Elsayed Mahmoud
Summary: This paper investigates the solution to one-dimensional fractional differential equations with two types of fractional derivative operators of orders in the range of (1,2). Two linearized schemes of the numerical method are constructed. The considered FDEs are equivalently transformed by the Riemann-Liouville integral into their integro-partial differential problems to reduce the requirement for smoothness in time. The analysis of stability and convergence is rigorously discussed. Finally, numerical experiments are described, which confirm the obtained theoretical analysis.
Article
Computer Science, Interdisciplinary Applications
Longbin Wu, Qiang Ma, Xiaohua Ding
Summary: This paper presents an energy-preserving scheme for the nonlinear fractional Klein-Gordon Schrodinger equation using the scalar auxiliary variable approach. By introducing a scalar variable, the system is transformed into a new equivalent system, and a linear implicit energy-preserving scheme is obtained by applying the extrapolated Crank-Nicolson method in the temporal direction and Fourier pseudospectral method in the spatial direction.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2021)
Article
Mathematics, Applied
Waleed Mohamed Abd-Elhameed, Youssri Hassan Youssri
Summary: The current paper proposes a numerical algorithm for treating the linearized time-fractional KdV equation based on selecting two different sets of basis functions. By using the spectral tau method to convert the equation and its underlying conditions into a linear system of algebraic equations, numerical treatment can be performed with suitable standard procedures. Numerical examples and comparisons with other methods demonstrate the applicability and accuracy of the presented algorithm.
Article
Mathematics, Applied
Jingna Zhang, Jianfei Huang, Temirkhan S. Aleroev, Yifa Tang
Summary: A nonlinear initial boundary value problem with time Caputo and space Riesz fractional-order derivatives is considered in this study. A linearized ADI scheme is constructed and proven to be unconditionally stable and convergent with high accuracy. Numerical experiments demonstrate the effectiveness of the proposed scheme and theoretical findings.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2021)
Article
Mathematics, Applied
Dingwen Deng, Qiang Wu
Summary: In this paper, a three-level finite difference method (FDM) that conserves energy and mass laws is derived for 1D and 2D nonlinear coupled Schrodinger-Boussinesq equations (NCSBEs). Error estimations are proven using the discrete energy analysis method, showing convergence to exact solutions. The efficiency and accuracy of the proposed algorithms are confirmed by numerical results.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2021)
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
Dongyang Shi, Houchao Zhang
Summary: A linearized mass and energy-conservative nonconforming finite element method is proposed for the nonlinear KGS equations. The error is separated into two parts, one from the temporal discretization and the other from the spatial discretization, and rigorous analysis is used to derive uniform bounds of the solution and error estimates. Superclose and superconvergence error estimates are obtained without restrictions on the time step-size, using L-p norms and the special characteristics of EQ(1)(rot) element.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Hanen Boujlida, Kaouther Ismail, Khaled Omrani
Summary: This study investigates a high-order accuracy finite difference scheme for solving the one-dimensional extended Fisher-Kolmogorov (EFK) equation. A new compact difference scheme is proposed and the a priori estimates and unique solvability are discussed using the discrete energy method. The unconditional stability and convergence of the difference solution are proved. Numerical experiments demonstrate the accuracy and efficiency of the proposed technique.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Meng Li, Chengming Huang, Wanyuan Ming
NUMERICAL ALGORITHMS
(2020)
Article
Mathematics, Applied
Mingfa Fei, Chengming Huang, Nan Wang, Guoyu Zhang
Summary: This paper presents a linearized Galerkin-Legendre spectral method for solving the one-dimensional nonlinear fractional Ginzburg-Landau equation, with a focus on its unique solvability and boundedness properties. The method is unconditionally convergent in the maximum norm with second-order accuracy in time and spectral accuracy in space. Additionally, a split-step alternating direction implicit Galerkin-Legendre spectral method for two-dimensional problems is introduced without theoretical analysis, and the effectiveness of both proposed schemes is demonstrated through numerical examples.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
Meng Li, Chengming Huang, Yongliang Zhao
NUMERICAL ALGORITHMS
(2020)
Article
Mathematics, Applied
Mingfa Fei, Chengming Huang, Pengde Wang
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2020)
Article
Mathematics, Applied
Min Li, Chengming Huang
APPLIED MATHEMATICS AND COMPUTATION
(2020)
Article
Mathematics, Applied
Mingfa Fei, Nan Wang, Chengming Huang, Xiaohua Ma
APPLIED NUMERICAL MATHEMATICS
(2020)
Article
Mathematics, Applied
Nan Wang, Mingfa Fei, Chengming Huang, Guoyu Zhang, Meng Li
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2020)
Article
Mathematics, Applied
Guoyu Zhang, Chengming Huang, Mingfa Fei, Nan Wang
Summary: In this study, a linearized finite element method for solving two-dimensional fractional Klein-Gordon equations was proposed, which demonstrated bounded numerical solution with second-order accuracy. The convergence of the numerical solution was proved using mathematical induction.
NUMERICAL ALGORITHMS
(2021)
Article
Mathematics, Applied
Peng Hu, Chengming Huang
Summary: This paper addresses the delay dependent stability of the stochastic exponential Euler method for stochastic delay differential equations and stochastic delay partial differential equations. By using root locus technique, the necessary and sufficient condition for the numerical delay dependent stability is derived, showing the method can preserve the underlying system's stability. The study also investigates the stability of semidiscrete and fully discrete systems for linear stochastic delay partial differential equations.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Min Li, Chengming Huang, Peng Hu, Jiao Wen
Summary: This paper introduces a split-step theta method for solving stochastic Volterra integral equations with general smooth kernels. The method shows superconvergence when the kernel function satisfies certain conditions, and it exhibits superior stability compared to traditional methods when the test equation degrades to the deterministic case. Numerical experiments are conducted to verify the theoretical results.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Min Li, Chengming Huang, Yaozhong Hu
Summary: This paper examines the exact asymptotic separation rate of doubly singular stochastic Volterra integral equations with two different initial values, using the Gronwall inequality with doubly singular kernel. A bound for the leading coefficient of the asymptotic separation rate for two distinct solutions is obtained for a special linear singular SVIEs, demonstrating the sharpness of the asymptotic results.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Mathematics, Applied
Zheng Ma, Chengming Huang
Summary: In this paper, we propose a method for solving Volterra integro-differential equations with weakly singular kernels. By increasing the degrees of piecewise fractional polynomials, exponential rates of convergence can be achieved for certain solutions. The method is easy to implement and has the same computational complexity as polynomial collocation methods.
NUMERICAL ALGORITHMS
(2023)
Article
Mathematics, Applied
Zheng Ma, Chengming Huang
Summary: In this paper, a collocation method is developed for solving third-kind Volterra integral equations. To achieve high-order convergence for problems with non-smooth solutions, a collocation scheme on a modified graded mesh is constructed using a basis of fractional polynomials, depending on a parameter lambda. The proposed method derives an error estimate in the L-infinity norm, showing that the optimal order of global convergence can be obtained by choosing the appropriate parameter lambda and modified mesh, even for solutions with low regularity. Numerical experiments confirm the theoretical results and demonstrate the performance of the method.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Zexiong Zhao, Chengming Huang
Summary: This paper focuses on the numerical solution of Volterra integro-differential equations with weakly singular kernels. A smoothing transformation is applied to improve the regularity of the original equation. The collocation method based on barycentric rational interpolation is introduced and the convergence and superconvergence of the numerical solution are analyzed. Numerical results are presented to validate the theoretical predictions of convergence and superconvergence.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Jiao Wen, Aiguo Xiao, Chengming Huang
COMPUTATIONAL & APPLIED MATHEMATICS
(2020)
