Article
Mathematics
Fethi Bouzeffour
Summary: The aim of this work is to introduce a novel concept, Riesz-Dunkl fractional derivatives, within the context of Dunkl-type operators. It is particularly noteworthy that the Riesz-Dunkl fractional derivative smoothly reduces to both the well-known Riesz fractional derivative and the fractional second-order derivative when a specific parameter kappa equals zero. Furthermore, the concept of the fractional Sobolev space is introduced and characterized using the versatile framework of the Dunkl transform.
Article
Mathematics
Julian Bailey
Summary: The article introduces classes of weights for which certain operators are bounded on weighted Lebesgue spaces. It also proves the boundedness of Lv-Riesz potentials and examines different generalized forms of Schrodinger operators. Finally, necessary conditions for weights to satisfy in order for certain operators to be bounded are investigated.
JOURNAL OF FUNCTIONAL ANALYSIS
(2021)
Article
Mathematics
Peng Chen, Xuan Thinh Duong, Liangchuan Wu, Lixin Yan
Summary: This article aims to establish the exponential-square integrability of a function associated with an operator L whose square function is bounded, even extending to the Laplace operator on Euclidean spaces R-n. The results obtained include estimates for the norm on LP as p increases, weighted norm inequalities for square functions, and eigenvalue estimates for Schrodinger operators on R-n or Lipschitz domains of R-n.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2021)
Article
Mathematics
Ting Chen, Wenchang Sun
Summary: The paper investigates the properties of multilinear fractional integral operators in Lebesgue spaces, specifically discussing the linear extension and boundedness of such operators for functions in mixed-norm Lebesgue spaces. It provides a complete characterization of exponents under less restrictive conditions on linear maps, and gives necessary and sufficient conditions for the boundedness of the operator in cases where m = 1 or n = 1.
MATHEMATISCHE ANNALEN
(2021)
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
Farzaneh Safari, Qingshan Tong, Zhen Tang, Jun Lu
Summary: This paper discretizes the fractional Galilei invariant advection-diffusion equation and its more general version with nonlinear source term by combining the weighted and shifted Grunwald difference approximation formulae and Crank-Nicolson technique. A new version of the backward substitution method is proposed for numerical approximation, and various basis functions are used. Finally, the effectiveness of the method is verified through numerical experiments.
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
J. A. Tenreiro Machado
Summary: This paper introduces a conceptual experiment combining the bouncing ball model and the GL fractional derivative formulation, interpreting the results in the light of classical physics. The mechanical experiment offers a physical perspective and straightforward visualization. This approach not only motivates students to learn fractional calculus, but also sparks discussions on the use of fractional models in mechanics.
FRACTIONAL CALCULUS AND APPLIED ANALYSIS
(2021)
Article
Mathematics, Interdisciplinary Applications
Jacek Gulgowski, Dariusz Kwiatkowski, Tomasz P. Stefanski
Summary: This paper discusses wave propagation in a medium described by a fractional-order model, investigating the causality of the system and transfer function under different conditions. Results are demonstrated and illustrated through numerical simulations and analyses, with additional comments on the Kramers-Kronig relations for the transfer function logarithm.
FRACTAL AND FRACTIONAL
(2021)
Article
Mathematics, Applied
Chu-Hee Cho, Youngwoo Koh, Jungjin Lee
Summary: This paper demonstrates that a local space-time estimate implies a global space-time estimate for dispersive operators. By considering a Littlewood-Paley type square function estimate and a generalization of Tao's epsilon removal lemma in mixed norms for dispersive operators in a time variable, the authors obtain sharp global space-time estimates with optimal regularity from previously known local ones.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2022)
Article
Mathematics, Applied
Changping Xie, Shaomei Fang
Summary: This paper investigates a second order scheme for approximating solutions of space-fractional diffusion equations with fractional Neumann boundary conditions. The proposed scheme is based on the Crank-Nicholson method in time and the shifted Grunwald-Letnikov operator and Taylor expansion method in space, utilizing Riesz fractional derivatives. The convergence, solvability, and stability of the scheme are proven, with a new numerical approach introduced to handle boundary conditions. The accuracy and efficiency of the method are demonstrated through two numerical examples.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Physics, Mathematical
Chris Kottke, Frederic Rochon
Summary: We provide a pseudodifferential characterization of the limiting behavior of certain Dirac operators associated to a fibered boundary metric as k tends to 0, and use this characterization to derive a pseudodifferential characterization of the low energy limit of the resolvent of the operator. We also prove that the Dirac operator is Fredholm when acting on suitable weighted Sobolev spaces.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2022)
Article
Mathematics, Applied
Jocelyn Sabatier, Christophe Farges
Summary: The paper demonstrates that the Caputo definition of fractional differentiation may lead to issues when used in defining a time fractional model that considers initial conditions. Simple examples are used to illustrate this, and the analysis is also extended to other definitions such as Riemann-Liouville and Grunwald-Letnikov. These findings raise questions about the validity of results in the analysis of time fractional models involving initial conditions.
Article
Mathematics
Yanhui Wang
Summary: This article establishes estimates of L-p(R-n) and the weak type (1,1) for the Riesz transform del L-2(-1/4) related to Schrodinger-type operators, where the nonnegative potential V belongs to the reverse Holder class RHs. The research focuses on the characteristics and properties of Schrodinger-type operators, including the estimates and weak type of the Riesz transform related to L-2.
BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY
(2021)
Article
Mathematics, Applied
Golsa Sayyar, Seyed Mohammad Hosseini, Farinaz Mostajeran
Summary: This paper introduces a high-order approach for solving time-space fractional diffusion equations, demonstrating its stability and convergence through numerical examples.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(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
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)
Article
Mathematics, Applied
M. S. Bruzon, T. M. Garrido, R. de la Rosa
Summary: We study a family of generalized Zakharov-Kuznetsov modified equal width equations in (2+1)-dimensions involving an arbitrary function and three parameters. By using the Lie group theory, we classify the Lie point symmetries of these equations and obtain exact solutions. We also show that this family of equations admits local low-order multipliers and derive all local low-order conservation laws through the multiplier approach.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Dohee Jung, Changbum Chun
Summary: The paper presents a general approach to enhance the Pade iterations for computing the matrix sign function by selecting an arbitrary three-point family of methods based on weight functions. The approach leads to a multi-parameter family of iterations and allows for the discovery of new methods. Convergence and stability analysis as well as numerical experiments confirm the improved performance of the new methods.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Abhishek Yadav, Amit Setia, M. Thamban Nair
Summary: In this paper, we propose a Galerkin's residual-based numerical scheme for solving a system of Cauchy-type singular integral equations using Chebyshev polynomials. We prove the well-posedness of the system and derive a theoretical error bound and convergence order. The numerical examples validate the theoretical results.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Fernando Chacon-Gomez, M. Eugenia Cornejo, Jesus Medina, Eloisa Ramirez-Poussa
Summary: The use of decision rules allows for reliable extraction of information and inference of conclusions from relational databases, but the concepts of decision algorithms need to be extended in fuzzy environments.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Ilhame Amirali, Gabil M. Amiraliyev
Summary: This paper considers the one-dimensional initial-boundary problem for a pseudoparabolic equation with a time delay. To solve this problem numerically, a higher-order difference method is constructed and the error estimate for its solution is obtained. Based on the method of energy estimates, the fully discrete scheme is shown to be convergent of order four in space and of order two in time. The given numerical results illustrate the convergence and effectiveness of the numerical method.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Tong-tong Shang, Guo-ji Tang, Wen-sheng Jia
Summary: The goal of this paper is to investigate a class of linear complementarity problems over tensor-spaces, denoted by TLCP, which is an extension of the classical linear complementarity problem. First, two classes of structured tensors over tensor-spaces (i.e., T-R tensor and T-RO tensor) are introduced and some equivalent characterizations are discussed. Then, the lower bound and upper bound of the solutions in the sense of the infinity norm of the TLCP are obtained when the problem has a solution.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Fabio Difonzo, Pawel Przybylowicz, Yue Wu
Summary: This paper focuses on the existence, uniqueness, and approximation of solutions of delay differential equations (DDEs) with Caratheodory type right-hand side functions. It presents the construction of the randomized Euler scheme for DDEs and investigates its error. Furthermore, the paper reports the results of numerical experiments.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Priyanka Roy, Geetanjali Panda, Dong Qiu
Summary: In this article, a gradient based descent line search scheme is proposed for solving interval optimization problems under generalized Hukuhara differentiability. The innovation and importance of these concepts are presented from practical and computational perspectives. The necessary condition for existence of critical point is presented in inclusion form of interval-valued gradient. Suitable efficient descent direction is chosen based on the monotonic property of the interval-valued function and specific interval ordering. Mathematical convergence of the scheme is proved under the assumption of Inexact line search. The theoretical developments are implemented with a set of interval test problems in different dimensions. A possible application in finance is provided and solved by the proposed scheme.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Zhongqian Wang, Changqing Ye, Eric T. Chung
Summary: In this paper, the constrained energy minimizing generalized multiscale finite element method (CEM-GMsFEM) with mixed boundary conditions for elasticity equations in high contrast media is developed. The method offers advantages such as independence of target region's contrast from precision and significant impact of oversampling domain sizes on numerical accuracy. Furthermore, this is the first proof of convergence of CEM-GMsFEM with mixed boundary conditions for elasticity equations. Numerical experiments demonstrate the method's performance.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Samaneh Soradi-Zeid, Maryam Alipour
Summary: The Laguerre polynomials are a new set of basic functions used to solve a specific class of optimal control problems specified by integro-differential equations, namely IOCP. The corresponding operational matrices of derivatives are calculated to extend the solution of the problem in terms of Laguerre polynomials. By considering the basis functions and using the collocation method, the IOCP is simplified into solving a system of nonlinear algebraic equations. The proposed method has been proven to have an error bound and convergence analysis for the approximate optimal value of the performance index. Finally, examples are provided to demonstrate the validity and applicability of this technique.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Almudena P. Marquez, Maria L. Gandarias, Stephen C. Anco
Summary: A generalization of the KP equation involving higher-order dispersion is studied. The Lie point symmetries and conservation laws of the equation are obtained using Noether's theorem and the introduction of a potential. Sech-type line wave solutions are found and their features, including dark solitary waves on varying backgrounds, are discussed.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Susanne Saminger-Platz, Anna Kolesarova, Adam Seliga, Radko Mesiar, Erich Peter Klement
Summary: In this article, we study real functions defined on the unit square satisfying basic properties and explore the conditions for generating bivariate copulas using parameterized transformations and other constructions.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Lulu Tian, Nattaporn Chuenjarern, Hui Guo, Yang Yang
Summary: In this paper, a new local discontinuous Galerkin (LDG) algorithm is proposed to solve the incompressible Euler equation in two dimensions on overlapping meshes. The algorithm solves the vorticity, velocity field, and potential function on different meshes. The method employs overlapping meshes to ensure continuity of velocity along the interfaces of the primitive meshes, allowing for the application of upwind fluxes. The article introduces two sufficient conditions to maintain the maximum principle of vorticity.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Cheng Wang, Jilu Wang, Steven M. Wise, Zeyu Xia, Liwei Xu
Summary: In this paper, a temporally second-order accurate numerical scheme for the Cahn-Hilliard-Magnetohydrodynamics system of equations is proposed and analyzed. The scheme utilizes a modified Crank-Nicolson-type approximation for time discretization and a mixed finite element method for spatial discretization. The modified Crank-Nicolson approximation allows for mass conservation and energy stability analysis. Error estimates are derived for the phase field, velocity, and magnetic fields, and numerical examples are presented to validate the proposed scheme's theoretical results.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Mingyu He, Wenyuan Liao
Summary: This paper presents a numerical method for solving reaction-diffusion equations in spatially heterogeneous domains, which are commonly used to model biological applications. The method utilizes a fourth-order compact alternative directional implicit scheme based on Pade approximation-based operator splitting techniques. Stability analysis shows that the method is unconditionally stable, and numerical examples demonstrate its high efficiency and high order accuracy in both space and time.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
