Article
Mathematics, Applied
L. Cesbron, A. Mellet, M. Puel
Summary: In this study, we investigate the fractional diffusion approximation of a kinetic equation set with diffusive reflection conditions at the boundary. In a specific singular limit, corresponding to small Knudsen number and long time asymptotic, we prove that the asymptotic density function is the unique solution to a fractional diffusion equation with Neumann boundary condition. This analysis completes previous work by the same authors, in which a limiting fractional diffusion equation was identified on the half-space but the uniqueness of the solution could not be established.
ASYMPTOTIC ANALYSIS
(2022)
Article
Mathematics, Applied
Shaoqiang Tang, Gang Pang
Summary: This paper presents a new method for numerical boundary treatment of the Riesz space fractional diffusion equation, achieving accurate solutions to the Cauchy problems in one and two space dimensions through numerical evaluation of kernel functions.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Computer Science, Interdisciplinary Applications
Qingyun Yao, Haibing Wang
Summary: A numerical scheme for solving an initial-boundary value problem for the time-fractional diffusion equation is developed based on the boundary integral equation method. This scheme involves stable discretization for layer potentials, rewriting layer potential operators as generalized Abel integral operators in time and establishing a stable time discretization scheme using the composite trapezoidal rule. The efficiency and accuracy of the proposed numerical scheme are demonstrated through several numerical examples.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Multidisciplinary Sciences
Andang Sunarto, Praveen Agarwal, Jackel Vui Lung Chew, Jumat Sulaiman
Summary: This study focused on the numerical solution of a space-fractional parabolic partial differential equation using the half-sweep preconditioned successive over relaxation method, demonstrating its efficiency in solving SFDE problems.
Article
Mathematics, Applied
Wael W. Mohammed
Summary: This paper considers the approximate solutions of time-fractional reaction-diffusion equations forced by multiplicative noise on a bounded domain. When the diffusion is large, the solutions of the stochastic time-fractional reaction-diffusion equations with polynomial term can be approximated by the solutions of a stochastic time-fractional ordinary equations. Our results are illustrated by applying to time-fractional logistic and time-fractional Ginzburg-Landau equations.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics
Keng Deng, Yixiang Wu
Summary: This paper investigates a reaction-diffusion equation with continuous delay and spatial variable coefficients, establishing a sharp threshold dynamic result for the global attractivity of positive steady state solutions. By analyzing the omega-limit set of the equation and proving it to be a singleton, the method is applied to demonstrate the global attractivity of the positive steady state of a spatially nonlocal diffusive logistic model.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Marcelo F. Furtado, Joao Pablo P. da Silva
Summary: The critical problem is studied using a variational approach, obtaining nonnegative nonzero solutions based on the parameter lambda value.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2021)
Article
Mathematics, Applied
Feimin Huang, Yong Wang
Summary: We study the steady Boltzmann equation in half-space with diffusive reflection boundary conditions. We establish the existence of a boundary layer solution for both linear and nonlinear Boltzmann equations in half-space with diffusive reflection boundary condition when the far-field Mach number of the Maxwellian is zero. The continuity and spacial decay of the solution are obtained, and uniqueness is established under some constraint conditions.
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
(2022)
Article
Automation & Control Systems
Jianping Huang, Hua-Cheng Zhou
Summary: This article focuses on the boundary stabilization problem of time-space fractional diffusion equation. By decomposing the system into two subsystems and verifying the Kalman rank condition, a controller is proposed to stabilize the entire system.
EUROPEAN JOURNAL OF CONTROL
(2022)
Article
Physics, Multidisciplinary
Anderson L. de Jesus, Alan C. Maioli, Alexandre G. M. Schmidt
Summary: The study investigates the scattering of a plane wave in the hyperbolic plane, formulating the problem using the Lippmann-Schwinger equation and solving it exactly for barriers modeled as various geometric shapes.
Article
Engineering, Multidisciplinary
Yuanyuan Zhao, Mei Huang, Xiaoping Ouyang, Jun Luo, Yongqing Shen, Fang Bao
Summary: The paper introduces a method called half boundary method (HBM) for solving two-dimensional unsteady convection-diffusion equations. The method reduces the order of the equation by introducing new variables, making it faster and more efficient than the finite volume method when dealing with a large number of divisions.
ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS
(2022)
Article
Engineering, Multidisciplinary
Oscar P. Bruno, Tao Yin
Summary: This paper presents a windowed Green function method for the numerical solution of elastic scattering problems by locally-rough surfaces, demonstrating accurate and super-algebraically fast convergence as the window-size grows in numerical experiments.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2021)
Article
Mathematics, Applied
Biao Zhang, Weiping Bu, Aiguo Xiao
Summary: In this paper, a numerical method for solving the time-space fractional diffusion equation with Robin fractional derivative boundary condition is proposed, with detailed analysis of stability and convergence provided. The proposed method is validated to be effective through numerical experiments.
NUMERICAL ALGORITHMS
(2021)
Article
Mathematics, Applied
Hanna Okrasinska-Plociniczak, Lukasz Plociniczak
Summary: This study investigates a time-fractional porous medium equation that is important in applications such as hydrology and material sciences. The study reveals that solutions of the free boundary Dirichlet, Neumann, and Robin problems on the half-line satisfy a Volterra integral equation with a non-Lipschitz nonlinearity. Based on this result, the study proves the existence, uniqueness, and constructs a family of numerical methods that outperform the usual finite difference approach. Furthermore, the study demonstrates the convergence of these methods and supports the theory with several numerical examples.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Mathematics
Ryan Thiessen, Thomas Hillen
Summary: This paper investigates the network formation of endothelial cells and discovers new spatial criss-cross patterns due to competing cues. The study suggests that competing cues, such as tissue anisotropy and chemotaxis, may lead to the formation of patterns in different directions.
