Article
Mathematics, Applied
Takanori Ebata, Hiroki Ohwa
Summary: This paper proves the L-loc(1) stability of sequence of piecewise constant solutions constructed by the wave front tracking method for conservation laws. As a result, we can concretely observe the convergence rate of the sequence of piecewise constant solutions constructed by the wave front tracking method for conservation laws.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2023)
Article
Mathematics, Applied
Adrian M. Ruf
Summary: We prove the stability of adapted entropy solutions for scalar conservation laws with discontinuous flux with respect to changes in the flux. This is under the assumption that the flux is strictly monotone in u and the spatial dependency is piecewise constant with finitely many discontinuities. We use this result to establish a convergence rate for the front tracking method, a widely used numerical method in the field of conservation laws with discontinuous flux. These results are the first of their kind in the literature on conservation laws with discontinuous flux. Numerical experiments are also conducted to validate the convergence rate and compare the numerical solutions obtained from the front tracking method with finite volume approximations.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2022)
Article
Multidisciplinary Sciences
A. R. El Dhaba, S. Mahmoud Mousavi
Summary: A reduced micromorphic model subjected to external static load in a plane was studied using the finite element method. The complexity of the microstructure was avoided by analyzing the reduced micromorphic model to reveal the interaction of the microstructure and the external loading. Classical and nonclassical deformation measures were demonstrated and discussed for the first time for a material employing the reduced micromorphic model.
SCIENTIFIC REPORTS
(2021)
Article
Mathematics, Applied
Li Wang, Zhong-Rong Lu, Jike Liu
Summary: This paper presents a systematic and rigorous analysis on the convergence rates of the Harmonic balance Method (HB) for general smooth and non-smooth systems. The study found that when the restoring forces are discontinuous or of some special low smoothness, the convergence rates of HB become different from those of Fourier truncation. Numerical examples are studied and the results well verify the present theoretic convergence rates.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2021)
Article
Mechanics
B. E. Saunders, R. J. Kuether, R. Vasconcellos, A. Abdelkefi
Summary: In this study, the suitability of the harmonic balance method (HBM) to predict periodic solutions of a one-dimensional forced Duffing oscillator with freeplay nonlinearity is investigated. The convergence behavior of the HBM is understood by studying the route to impact, which involves a parametric study of increasing contact stiffness. The accuracy of nonlinear periodic responses is evaluated by comparing HBM results with time-integration results. Additionally, this study performs convergence and stability analysis specifically for isolas generated by the non-smooth nonlinearity. Hill's method and Floquet theory are used to compute the stability of periodic solutions and identify bifurcation types in the system.
INTERNATIONAL JOURNAL OF NON-LINEAR MECHANICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Javad Alamatian
Summary: This study introduces a simple restart strategy for the kinetic dynamic relaxation (DR) method, determining the position of the restart point through kinetic energy modeling and formulating the displacement vector based on the finite difference method. The proposed R-point formulation enhances the convergence rate of the kinetic DR method, reducing the average number of required iterations by about 9% and 5% in linear and nonlinear analyses, respectively.
ENGINEERING WITH COMPUTERS
(2022)
Article
Mathematics, Applied
Ulrik Skre Fjordholm, Kjetil Olsen Lye
Summary: We prove the convergence rates of monotone schemes for conservation laws with initial data that have unbounded total variation but Holder continuous, given that the Holder exponent of the initial data is greater than 1/2. Additionally, for strictly Lip(+) stable monotone schemes, we demonstrate convergence for any positive Holder exponent. Numerical experiments are conducted to validate the theory.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Min Zhong, Lingyun Qiu, Wei Wang
Summary: This paper systematically investigates the convergence rate of Landweber-type iteration in Banach spaces for nonlinear ill-posed problems, and provides two novel stopping rules including the discrepancy principle and the heuristic Hanke-Raus rule.
APPLIED MATHEMATICS LETTERS
(2023)
Article
Computer Science, Interdisciplinary Applications
Can Li, Shuming Liu
Summary: This article investigates high-order numerical approximations of scalar conservation laws with nonlocal viscous term, and designs a semidiscrete local discontinuous Galerkin method to solve the nonlocal model. The stability and convergence of the semidiscrete LDG method in the L(2) norm are proven, showing optimal convergence order for the linear case and sub-optimal convergence order for the nonlinear case. Numerical tests with Burgers equation demonstrate the validity and accuracy of the scheme, with results showing convergence order accuracy in space for both linear and nonlinear cases. Overall, the numerical simulations provided demonstrate the robustness and effectiveness of the numerical scheme.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS
(2021)
Article
Automation & Control Systems
Yan Xia, Yi Wan, Xichun Luo, Zhanqiang Liu, Qinghua Song
Summary: An improved numerical integration method is proposed for predicting milling stability by dividing the milling dynamic model into free and forced vibration stages and constructing an efficient and accurate state transition matrix. Experimental results show that this method outperforms existing methods in terms of accuracy and computational efficiency.
INTERNATIONAL JOURNAL OF ADVANCED MANUFACTURING TECHNOLOGY
(2021)
Article
Mathematics, Applied
Renjun Duan, Haiyan Yin, Changjiang Zhu
Summary: This paper focuses on the initial-boundary value problem for ions over a half line in the full Euler-Poisson system. It establishes the existence of stationary solutions under the Bohm criterion and proves the large time asymptotic stability of small-amplitude stationary solutions, with the convergence rate towards stationary solutions. The proof is based on the energy method, emphasizing on capturing the positivity of temporal energy dissipation functional and boundary terms with suitable space weight functions.
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
(2021)
Article
Automation & Control Systems
Mariana Ballesteros, Rita Q. Fuentes-Aguilar, Isaac Chairez
Summary: This paper introduces a design of exponential function-based learning law for artificial neural networks with continuous dynamics, aiming to model systems non-parametrically and adjust weights of ANN using two adaptive algorithms for faster convergence to smaller dimensions invariant set.
IEEE-CAA JOURNAL OF AUTOMATICA SINICA
(2022)
Article
Mathematics, Applied
Qingguang Guan, Max Gunzburger, Xiaoping Zhang
Summary: This paper analyzes the collocation method for solving one-dimensional steady-state and time-dependent nonlocal diffusion equations, addressing the difficulty of singularity in the kernel. Introducing the Hadamard finite part integral and a balance term to discretize the nonlocal operator proves effective in overcoming this challenge, as validated by numerical results.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics
Siqi Lv, Zhihua Nie, Cuicui Liao
Summary: This paper discusses the stability and convergence analyses of the multi-symplectic variational integrator for the nonlinear Schrödinger equation. The variational integrator is proven to have unconditional linear stability using the von Neumann method. The a priori error bound for the scheme is provided based on the Sobolev inequality and the discrete conservation laws. Subsequently, the variational integrator is shown to converge at O(Δx2+Δt2) in the discrete L2 norm using the energy method. The numerical experimental results confirm the theoretical derivation.
Article
Mathematics
Neeraj Bhauryal, Ujjwal Koley, Guy Vallet
Summary: The study focuses on the Cauchy problem for a stochastic scalar parabolic-hyperbolic equation in any space dimension with nonlocal, nonlinear, and possibly degenerate diffusion terms. Nonlocality is introduced due to involvement of fractional diffusion operators. The uniqueness proof involves a new technical framework and the vanishing viscosity method.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Renjun Duan, Shuangqian Liu, Shota Sakamoto, Robert M. Strain
Summary: This paper proves the existence of small-amplitude global-in-time unique mild solutions to both the Landau equation including the Coulomb potential and the Boltzmann equation without angular cutoff. It introduces a new function space with low regularity in the spatial variable to handle the problem in cases when the spatial domain is either a torus or a finite channel with boundary, considering both boundary conditions. The paper also studies the large-time behavior of solutions and further justifies the property of propagation of regularity of solutions in the spatial variables.
COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Hongyun Peng, Zhi-An Wang, Changjiang Zhu
Summary: This paper investigates the well-posedness and large-time behavior of a two-dimensional PDE-ODE hybrid chemotaxis system describing the initiation of tumor angiogenesis. The system is transformed into a parabolic-hyperbolic system using a Cole-Hopf type transformation, and it is shown that the solution of the transformed system converges to a constant equilibrium state as time tends to infinity. The Cole-Hopf transformation is then reversed to obtain relevant results for the pre-transformed PDE-ODE hybrid system. The use of the effective viscous flux is a key ingredient in the proof, enabling the desired energy estimates and regularity for chemotaxis systems with initial data of low regularity.
SCIENCE CHINA-MATHEMATICS
(2022)
Article
Mathematics, Applied
Klemens Fellner, Michael Kniely
Summary: We investigate a recombination-drift-diffusion model coupled to Poisson's equation to study the transport of charge in semiconductors. Our main result establishes an explicit functional inequality between relative entropy and entropy production, leading to exponential convergence to equilibrium. The approach is uniformly applied assuming the lifetime of electrons on the trap level is sufficiently small.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
Zefu Feng, Guangyi Hong, Changjiang Zhu
Summary: This paper concerns the large-time behavior of solutions to the compressible Navier-Stokes equations for ideal reacting gases. It establishes the asymptotic stability of the constant equilibrium state with strictly positive constant density, temperature, and vanishing velocity, mass fraction of the reactant under suitable small initial perturbation.
Article
Mathematics
Renjun Duan, Wei-Xi Li, Lvqiao Liu
Summary: This paper studies the Cauchy problem on the non-cutoff Boltzmann equation in a torus, and improves the regularity of solutions by treating the commutator between the regularization operators and the Boltzmann collision operator.
ADVANCES IN MATHEMATICS
(2022)
Article
Mathematics, Applied
Nangao Zhang, Changjiang Zhu
Summary: In this paper, the asymptotic behavior of solutions to a system of hyperbolic conservation laws with damping is investigated. The system includes compressible Euler equations with damping, Ml-model, etc. Under certain smallness conditions on initial perturbations, it is proved that the solutions to the Cauchy problem of the system globally exist and time-asymptotically converge to the corresponding equilibrium state, with the optimal convergence rate being provided. The approach employed is the technical time-weighted energy method combined with the Green's function method.
ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK
(2022)
Article
Physics, Mathematical
Renjun Duan, Dongcheng Yang, Hongjun Yu
Summary: This paper focuses on the large time asymptotics toward the viscous contact waves for solutions of the Landau equation with physically realistic Coulomb interactions. By introducing a new time-velocity weight function, it is proved that the solution tends toward a local Maxwellian in large time, which is the first result on the dynamical stability of contact waves for the Landau equation.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2022)
Article
Mathematics
Nangao Zhang, Changjiang Zhu
Summary: This paper focuses on the asymptotic behavior of solutions of the M-1 model in radiative transfer fields. By introducing a more general system, the paper rigorously proves the existence of solutions to this system and their convergence to specific nonlinear diffusion waves. Compared to previous research, the paper has weaker initial conditions and sharper conclusions.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2022)
Article
Physics, Mathematical
Dingqun Deng, Renjun Duan
Summary: This paper proves that the linearized Boltzmann or Landau equation with soft potentials has a spectral gap when the space domain is bounded with an inflow boundary condition, contrary to the existing results. The author introduces a new Hilbert space with an exponential weight function, where the action of the transport operator induces an extra non-degenerate relaxation dissipation in large velocity. This compensates for the degenerate spectral gap and leads to exponential decay of solutions.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2023)
Article
Mathematics
Guangyi Hong, Xiaofeng Hou, Hongyun Peng, Changjiang Zhu
Summary: In this paper, the Cauchy problem of the three-dimensional isentropic compressible Navier-Stokes equations is investigated. The global existence and uniqueness of classical solutions with large initial energy and vacuum are proven, under the assumptions of nearly isothermal fluid and vacuum or near vacuum far-field density. This is the first result on the global existence of large-energy solutions with vacuum to the three-dimensional compressible Navier-Stokes equations for the cases of vacuum and nonvacuum far-field constant states, which generalizes the result by Huang, Li, and Xin (Commun Pure Appl Math 65:549-585, 2012) on classical solutions with vacuum and small energy (large oscillations).
MATHEMATISCHE ANNALEN
(2023)
Article
Mathematics
Renjun Duan, Zongguang Li
Summary: In this paper, the Boltzmann equation with an additional internal energy variable I and a parameter d = 2 is considered to model the motion of a polyatomic gas. Using a perturbation framework, the global well-posedness for bounded mild solutions near global equilibria on torus is established. The proof relies on the L-2 to L-8 approach, where the linearized equation's decay property, the iteration technique for the linear integral operator, and the Duhamel's principle are employed to obtain the desired results.
INTERNATIONAL JOURNAL OF MATHEMATICS
(2023)
Article
Physics, Mathematical
Minyi Zhang, Changjiang Zhu
Summary: This paper investigates the asymptotic stability of a planar stationary solution to a initial-boundary value problem for a two-dimensional hyperbolic-elliptic coupled system of the radiating gas on half space. It is shown that the solution to the problem converges to the corresponding planar stationary solution as time tends to infinity under small initial perturbation. The proof is based on the standard L-2-energy method and the div-curl decomposition. Moreover, the convergence rate of the solution (u, q) to the corresponding planar stationary solution is t(-a/2-1/4) for the non-degenerate case and t(-1/4) for the degenerate case, which is proved using the time and space weighted energy method.
JOURNAL OF MATHEMATICAL PHYSICS
(2023)
Article
Mathematics, Applied
Huancheng Yao, Changjiang Zhu
Summary: This study examines the large-time-asymptotic behavior of a combination of a viscous contact wave and two rarefaction waves in the NavierStokes-Maxwell equations, taking into account electrodynamic effects. By analyzing the specific structure of the Maxwell equations in Lagrangian coordinates, it is proven that the composite wave pattern is time-asymptotically stable under certain conditions. This is the first result regarding the nonlinear stability of two different wave patterns in the compressible Navier-Stokes-Maxwell equations.
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
(2021)
Article
Mathematics, Applied
Renjun Duan, Haiyan Yin, Changjiang Zhu
Summary: This paper focuses on the initial-boundary value problem for ions over a half line in the full Euler-Poisson system. It establishes the existence of stationary solutions under the Bohm criterion and proves the large time asymptotic stability of small-amplitude stationary solutions, with the convergence rate towards stationary solutions. The proof is based on the energy method, emphasizing on capturing the positivity of temporal energy dissipation functional and boundary terms with suitable space weight functions.
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
(2021)
Article
Mathematics, Applied
Klemens Fellner, Jeff Morgan, Bao Quoc Tang
Summary: This study proves uniform-in-time bounds of nonnegative classical solutions to reaction-diffusion systems in all space dimensions. It advances recent research on global existence of reaction-diffusion systems dissipating mass. As an application, skew-symmetric Lotka-Volterra systems are shown to have unique classical solutions with uniformly bounded trajectories in all dimensions.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S
(2021)
Article
Mathematics, Applied
Miguel Brozos-Vazquez, Diego Mojon-Alvarez
Summary: We study the geometric structure of weighted Einstein smooth metric measure spaces with weighted harmonic Weyl tensor. A complete local classification is provided, showing that either the underlying manifold is Einstein, or decomposes as a warped product in a specific way. Moreover, if the manifold is complete, then it either is a weighted analogue of a space form, or it belongs to a particular family of Einstein warped products.
JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES
(2024)
Article
Mathematics, Applied
Domenec Ruiz-Balet, Enrique Zuazua
Summary: Inspired by normalising flows, we analyze the bilinear control of neural transport equations using time-dependent velocity fields constrained by a simple neural network assumption. We prove the L1 approximate controllability property, showing that any probability density can be driven arbitrarily close to any other one within any given time horizon. The control vector fields are explicitly and recursively constructed, providing quantitative estimates of their complexity and amplitude. This also leads to statistical error bounds when only random samples of the target probability density are available.
JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES
(2024)
