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
Chen Zhu, Bingyin Zhang, Hongfei Fu, Jun Liu
Summary: This paper considers a three-dimensional time-dependent Riesz space-fractional diffusion equation and proposes an ADI difference scheme, which is proven to be unconditionally stable and with second-order accuracy. Numerical experiments demonstrate the effectiveness and efficiency of the method for large-scale modeling and simulations. Additionally, a linearized ADI scheme for the nonlinear Riesz space-fractional diffusion equation is developed and tested.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Shanshan Wang
Summary: This paper constructs split-step quintic B-spline collocation (SS5BC) methods for nonlinear Schrodinger equations in various dimensions. The proposed methods are verified to be convergent and efficient through numerical tests and comparisons. Furthermore, the SS5BC scheme is also successfully applied to compute Bose-Einstein condensates.
Article
Mathematics, Applied
Hossein Fazli, HongGuang Sun, Juan J. Nieto
Summary: The solvability of fractional differential equations involving the Riesz fractional derivative is considered by reducing the problem to a nonlinear mixed Volterra and Cauchy-type singular integral equation and using fractional calculus theory. By establishing compactness of the Riemann-Liouville fractional integral operator and using Krasnoselskii's fixed point theorem, it is shown that at least one solution exists. An example is included to demonstrate the theory's applicability.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
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
Engineering, Mechanical
Shreya Mitra, Sujoy Poddar, A. Ghose-Choudhury, Sudip Garai
Summary: Using conformable fractional space and time derivatives, a novel class of traveling wave solutions for the Hirota-Schrodinger (HS) equation and the nonlinear Schrodinger equation (NLSE) with quadratic-cubic nonlinearity (QCN) has been obtained. The obtained solutions show interesting dispersive corrections to the propagating waves, with different fractional powers displaying phase shifting, singularity and a flattening of the propagating pulse. Bright one-soliton and singular soliton solutions for the NLSE with QCN have also been discussed. These findings are likely to have significant relevance in the propagation of optical pulses in a highly nonlinear dispersive media.
NONLINEAR DYNAMICS
(2022)
Article
Mathematics, Applied
Junjie Wang
Summary: The paper presents high-order conservative schemes for the space fractional nonlinear Schrodinger equation, demonstrating their effectiveness through numerical experiments and proving the convergence of approximate solutions and the preservation of mass and energy conservation laws.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Mathematics
Fan Qin, Wei Feng, Songlin Zhao
Summary: This paper investigates a time-fractional derivative nonlinear Schrodinger equation with the Riemann-Liouville fractional derivative. The study includes Lie symmetry analysis, derivation of reduced equations, presentation of exact solutions using the invariant subspace method, and application of a new conservation theorem to construct conservation laws for the equation.
Article
Mathematics, Applied
Hongyu Qin, Fengyan Wu, Deng Ding
Summary: In this study, a linearized compact ADI numerical method is developed to solve the nonlinear delayed Schrodinger equation in two-dimensional space. The convergence of the fully-discrete numerical method is analyzed using discrete energy estimate method, showing a numerical scheme of order O(Delta t(2) + h(4)) with time stepsize Delta t and space stepsize h. Several numerical examples are presented to confirm the theoretical analyses.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Mathematics, Applied
Changhong Guo, Shaomei Fang
Summary: This paper studied the Crank-Nicolson difference scheme for the derivative nonlinear Schrodinger equation with the Riesz space fractional derivative. The existence of the difference solution is proved by the Brouwer fixed point theorem, and its convergence in the L-2 norm with second order accuracy in both temporal and space directions is investigated. When the fractional order equals to two, the results for the difference solution are in accordance with those for the non-fractional derivative Schrodinger equation.
JOURNAL OF APPLIED ANALYSIS AND COMPUTATION
(2021)
Article
Mathematics, Interdisciplinary Applications
Chuanjin Zu, Xiangyang Yu
Summary: This paper re-examines the time fractional Schrodinger equation and investigates the effects of different fractional derivatives and treatments of imaginary unit i. By considering the physical meaning of imaginary unit i, it is concluded that the fractional order of imaginary unit i is inappropriate. The time fractional Schrodinger equation with a limit-based fractional derivative is found to be more in line with the existing physical world.
CHAOS SOLITONS & FRACTALS
(2022)
Article
Mathematics, Applied
F. Abdolabadi, A. Zakeri, A. Amiraslani
Summary: In this paper, a split-step Fourier pseudo-spectral method is proposed for solving the space fractional coupled nonlinear Schrodinger equations. The method splits the equations into two subproblems, with one of them being linear. The solution for the nonlinear subproblem is computed exactly, and the Riesz space fractional derivative is approximated using a Fourier pseudo-spectral method. The stability, convergence, discrete charge, and multi-symplectic preserving properties of the proposed method are investigated, and it is extended for solving two-dimensional problems. Numerical experiments are conducted to validate the theoretical analysis and demonstrate the efficiency of the proposed scheme.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Mathematics, Applied
Hengfei Ding, Qian Yi
Summary: The main goal of this paper is to construct high-order numerical differential formulas for approximating the Riesz derivative and apply them to the numerical solution of the nonlinear space fractional Ginzburg-Landau equations. The paper introduces a novel second-order fractional central difference operator and a novel fourth-order fractional compact difference operator. It also develops new techniques and important lemmas to prove the unique solvability, stability, and convergence of the proposed difference scheme. Numerical examples demonstrate the efficiency and accuracy of the methods.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(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
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
Meng Li, Chengming Huang, Wanyuan Ming
NUMERICAL ALGORITHMS
(2020)
Article
Mathematics, Applied
Mingfa Fei, Chengming Huang
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(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
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)
Article
Mathematics, Applied
Junfeng Cao, Ke Chen, Huan Han
Summary: This paper proposes a two-stage image segmentation model based on structure tensor and fractional-order regularization. In the first stage, fractional-order regularization is used to approximate the Hausdorff measure of the MS model. The solution is found using the ADI scheme. In the second stage, thresholding is used for target segmentation. The proposed model demonstrates superior performance compared to state-of-the-art methods.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Dylan J. Oliver, Ian W. Turner, Elliot J. Carr
Summary: This paper discusses a projection-based framework for numerical computation of advection-diffusion-reaction (ADR) equations in heterogeneous media with multiple layers or complex geometric structures. By obtaining approximate solutions on a coarse grid and reconstructing solutions on a fine grid, the computational cost is significantly reduced while accurately approximating complex solutions.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Nathan V. Roberts, Sean T. Miller, Stephen D. Bond, Eric C. Cyr
Summary: In this study, the time-marching discontinuous Petrov-Galerkin (DPG) method is applied to the Vlasov equation for the first time, using backward Euler for a Vlasov-Poisson discretization. Adaptive mesh refinement is demonstrated on two problems: the two-stream instability problem and a cold diode problem.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Yizhi Sun, Zhilin Sun
Summary: This work investigates the convexity of a specific class of positive definite probability measures and demonstrates the preservation of convexity under multiplication and intertwining product. The study reveals that any integrable function on an interval with a polynomial expansion of fast absolute convergence can be decomposed into a pair of positive convex interval probabilities, simplifying the study of interval distributions and discontinuous probabilistic Galerkin schemes.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Bhagwan Singh, Komal Jangid, Santwana Mukhopadhyay
Summary: This paper examines the prediction of bending characteristics of nanoscale materials using the Moore-Gibson-Thompson thermoelasticity theory in conjunction with the nonlocal strain gradient theory. The study finds that the stiffness of the materials can be affected by nonlocal and length-scale parameters, and the aspect ratios of the beam structure play a significant role in bending simulations.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Guoliang Wang, Bo Zheng, Yueqiang Shang
Summary: This paper presents and analyzes a parallel finite element post-processing algorithm for the simulation of Stokes equations with a nonlinear damping term, which integrates the algorithmic advantages of the two-level approach, the partition of unity method, and the post-processing technique. The algorithm generates a global continuous approximate solution using the partition of unity method and improves the smoothness of the solution by adding an extra coarse grid correction step. It has good parallel performance and is validated through theoretical error estimates and numerical test examples.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Hao Xu, Zeng-Qi Wang
Summary: Fluid flow control problems are crucial in industrial applications, and solving the optimal control of Navier-Stokes equations is challenging. By using Oseen's approximation and matrix splitting preconditioners, we can efficiently solve the linear systems and improve convergence.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Zhengya Yang, Xuejuan Chen, Yanping Chen, Jing Wang
Summary: This paper focuses on the high-order stable numerical solutions of the time-space fractional diffusion equation. The Fourier spectral method is used for spatial discretization and the Spectral Deferred Correction (SDC) method is used for numerical solutions in time. As a result, a high-precision numerical discretization scheme for solving the fractional diffusion equation is obtained, and the convergence and stability of the scheme are proved. Several numerical examples are presented to demonstrate the effectiveness and feasibility of the proposed numerical scheme.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
