This article is dedicated to developing a circulant and skew-circulant splitting (CSCS) iterative method for addressing a specific class of diagonal-plus-asymmetric Toeplitz systems. Theoretically, we have analyzed that the spectral radius of the convergence factor of the proposed CSCS iterative method is strictly less than 1, which implies the unconditional convergence of the proposed iterative method

This short article studies a deterministic quasi-Monte Carlo lattice rule in weighted unanchored Sobolev spaces of smoothness 1. Building on the error analysis by Kazashi and Sloan, we prove the existence of unshifted rank-1 lattice rules that achieve a worst-case error of O(n−1/4(logn)1/2), with the implied constant independent of the dimension, under certain summability conditions on the weights

In this paper, a reaction–diffusion-advection model with nonlinear boundary conditions arising from algae species is investigated. By employing the super-sub solution method, we not only establish the existence of global solutions, but also obtain the existence and global asymptotical stability of positive steady-state solutions. It is observed that the positive steady-state solution is decreasing

The wide class of nonlinear Schrödinger equation of the general form is studied. These nonlinear partial differential equations, depending on arbitrary functions, are not integrable by the inverse scattering transform but have exact solutions. The approach is proposed that makes it possible to find nonlinear Schrodinger equations of the general form that have exact solutions. This approach is that

The basic hyperbolic–elliptic black-oil model describes oil–water displacement in a porous medium. Given its mathematical complexity, there is a need for particular simple solutions for validation of numerical methods. We present a class of stationary solutions, which are easy to compute, and in many cases are given by explicit formulae. These solutions are constructed by a nonlinear coupling of two

In order to improve the convergence speed of the fixed point iteration method, inspired by Li and Li (2023) , we add a scaling matrix to the fixed point method and construct a new two-step fixed point iteration (NTFPI) method to solve the tensor absolute value equation and analyze the convergence of the NTFPI method. Finally, two numerical examples are given to illustrate the feasibility and effectiveness

The Maxwell–Landau–Lifshitz equation with spin accumulation is studied in the paper. We prove the existence and uniqueness of global solutions using energy estimates method in two-dimensional space.

This work presents a novel class of balanced Euler methods designed for approximating stochastic Volterra integral equations. These methods aim to address certain numerical instabilities commonly encountered with the explicit Euler approach. The study derives the convergence order and stability characteristics of the proposed schemes in the mean-square sense. Additionally, a comprehensive analytical

This paper establishes an AIDS model that includes both asymptomatic and symptomatic infected individuals, and we assume the transmission rate follows the log-normal Ornstein–Uhlenbeck process, which allows us to develop a stochastic model. For the stochastic model, by constructing appropriate Lyapunov functions, we derive the disease will extinct when R0e<1. The critical value R0s>1 for existence

We investigate the two-dimensional space fractional Schrödinger-Poisson-Xα model, which incorporates fractional Laplacian operators to generalize classical quantum mechanics. By leveraging the Strang splitting Fourier spectral method, the model is solved effectively under periodic boundary conditions, ensuring high accuracy and computational efficiency. Numerical experiments confirm the second-order

In this paper, a study on the fast numerical analysis based on spatial nonuniform grids and inverse problem for the two-dimensional space–time fractional coupled equations with fractional Neumann boundary conditions are conducted. The second order L1+ method combined with the Crank–Nicolson (CN) method in time and the fractional block-centered finite difference (BCFD) method based on spatial nonuniform

In this paper, we consider the strong instability of standing waves for the following cubic nonlinear Schrödinger system with partial confinement −i∂tΦ1−ΔΦ1+V(x)Φ1=(μ1|Φ1|2+β|Φ2|2)Φ1,(t,x)∈R×R3,−i∂tΦ2−ΔΦ2+V(x)Φ2=(μ2|Φ2|2+β|Φ1|2)Φ2,(t,x)∈R×R3.When V(x)=x12+x22, Jia, Li and Luo (Discrete Contin. Dyn. Syst. 40, 2020, 2739-2766) investigated the existence of stable standing waves. When V(x)=x12/x22/x32

The stationary distribution, as a fundamental concept in stochastic processes, is of great significance for exploring the long-term behavior and stability of populations. In this paper, a stochastic reaction–diffusion predator–prey model with additional food, fear effect and anti-predator behavior is proposed, in which the stochastic fluctuations are characterized by a Ornstein–Uhlenbeck process. We

This article concerns the following Klein–Gordon–Maxwell system −Δu+V(x)u−(2ω+ϕ)ϕu=|u|s−2u+λf(u),x∈R3,Δϕ=(ω+ϕ)u2,x∈R3,where ω>0 is a constant, 4≤s<6, λ>0 is a parameter. When f only satisfies suplinear conditions but not satisfies subcritical growth and Ambrosetti–Rabinowitz conditions, the existence of positive solution can be proved via variational methods, Moser iteration and perturbation arguments

A stochastic single-species model subject to additive Allee effects and nonlinear stochastic perturbation is proposed and analyzed. First we demonstrate that this model has a unique positive solution for any positive initial value. Then, by analyzing the stability of invariant measures, we testify that there is a unique dynamical bifurcation point Λ to the equation, the sign of Λ determines the dynamical

Although many physical experiments and numerical simulations show that the magnetic field can stabilize and inhibit electrically conducting fluids, whether 2D incompressible MHD equations with only magnetic diffusion develop finite time singularities or not is one of the most challenging problems and remains open. Therefore, this issue has always attracted a lot of attention of mathematicians. Due

We consider a class of wave equations with strong damping, weak damping and nonlinear source term. By constructing the relationship between the blowup time and the coefficients of strong damping and weak damping, we exhibit and verify an interesting phenomenon that the local solution becomes the global solution as the coefficient of strong damping or weak damping goes to infinity.

In this paper, based on a novel integral inequality and the Lyapunov direct method, we propose a systematic approach to determining the global stability of the endemic steady states of age-structured epidemic models. The inequality makes it convenient to verify the negative (semi-)definiteness of the derivative of a Lyapunov functional candidate. The applicability of this approach is illustrated with

This paper focuses on a category of stochastic single population systems. Under mild assumptions, we provide a sufficient condition for the existence of stationary distribution in this system by employing variable substitution and the Krylov–Bogoliubov theorem. Furthermore, we demonstrate its proximity to being the sufficient and necessary condition by examining the system’s extinction.

We show that the following nonlinear system of difference equations of interest xn(l)=al∏j=1,j≠lkxn−1(j)f(xn−1(1),…,xn−1(k)),n∈N,l=1,k¯,where k≥2, aj,x0(j)∈ℂ∖{0},j=1,k¯, and the function f:ℂk→ℂ is homogeneous of degree k−2, is solvable in a closed form considerably extending some results in the literature.

For numerical methods applied to singularly perturbed reaction-diffusion problems, the balanced norm has emerged as an effective tool. In this manuscript, we analyze supercloseness properties in the balanced norm for the nonsymmetric interior penalty Galerkin (NIPG) method on a Bakhvalov-type mesh. To achieve this, we construct a novel interpolant that combines the Lagrange interpolant and a local

A complex integral formula provides an explicit solution of the initial value problem for the nonlinear scalar 1D equation ut+[f(u)]x=0, for any flux f(u) and initial condition u0(x) that are analytic. This formula is valid for some times t>0, u remaining analytic.

Periodicity has been widely recognised in a variety of areas including biology, finance and control theory. As an important class of non-autonomous SDEs, stochastic differential equations (SDEs) with periodic coefficients have thus been receiving great attention recently. In this paper, we study the strong convergence of the truncated Euler–Maruyama (EM) method to the superlinear SDEs with periodic

In this paper, the fundamental (1＋1)-dimensional nonlinear Schrödinger equation is extended to a novel (2＋1)-dimensional nonlocal nonlinear Schrödinger (NNLS) equation with a PT-symmetric term. We obtain the 1-soliton solution, 2-soliton solution, breather wave and strange wave solution of the (2＋1)-dimensional NNLS equation via the Hirota bilinear method. Some obtained solutions describe the interactions

β-amyloid (Aβ) oligomers have been increasingly shown to produce the crucial cytotoxicity during the progression of Alzheimer’s disease (AD). In this paper, we develop a stochastic AD model with Aβ oligomer effect, where Black–Karasinski process is introduced to describe the random fluctuations in neurobiological environment. First, the well-posedness and Markov–Feller property of the solution of the

In this paper, the local discontinuous Galerkin (LDG) method is used to solve the fourth-order time-fractional sub-diffusion model with the Caputo–Fabrizio fractional derivative. Based on the generalized alternating numerical flux, we derive the fully discrete LDG scheme, the convergence order of our discrete scheme is O(τ2+hk+1), where τ, h and k represent the time step size, space step size and polynomial

In this paper, by using the generalized Nehari manifold and the principle of symmetric criticality, we prove the existence of sign-changing solutions to a class of fractional Choquard system with strongly indefinite structure.

Considering the important effects of population diffusion and vaccine ineffectiveness on disease transmission, a stochastic SVI (Susceptible–Vaccinated–Infected) epidemic model with reaction–diffusion is mainly investigated in this paper. We prove the existence of the unique global positive strong solution by an innovative variable transformation. The sufficient conditions for disease persistence and

A linear matrix delayed discrete equation ΔX(k)=BX(k−m)+X(k−m)C+F(k),is considered, where m is a fixed positive integer, k=0,1,… is a discrete independent variable, X(k) is an n×n unknown variable matrix, Δ is the first order forward difference, B, C are given n×n constant matrices, and F(k) is a given n×n variable matrix. Using a special matrix function, under some commutativity conditions, formulas

In this paper, we present a within-host model incorporating time delays and drug control mechanisms to study the dynamics of infectious diseases. We begin by defining the basic reproduction number, R0, and subsequently prove that when R0<1, the disease-free equilibrium is globally attractive; while for R0>1, the disease persists uniformly. Numerical simulations are employed to validate our analytical

We construct the global large solutions for the compressible flow of liquid crystals in R3. This class of data relax the smallness restriction imposed on the the initial incompressible velocity. Particularly, our work improves upon previous studies by Hu and Wu (2013) and Zhai (2025).

We propose a practical tool for evaluating and comparing the accuracy of FDMs for the Helmholtz equation. The tool based on Fourier analysis makes it easy to find wavenumber explicit order of convergence, and can be used for rigorous proof. It fills in the gap between the dispersion analysis and the actual error with source term. We illustrate it for classical and some dispersion free schemes in 1D

A specific class of Ablowitz–Kaup–Newell–Segur (AKNS) matrix spectral problems is reduced using pairs of similarity transformations. The corresponding integrable hierarchies are derived from the reduced Lax pairs, extending the standard matrix AKNS integrable hierarchies. A few illustrative examples are provided to showcase the diversity of matrix integrable nonlinear Schrödinger equations.

The aim of this paper is to establish a version of the Feynman–Kac formula for a class of regime-switching general diffusion processes, in which the general diffusion part is a time-homogeneous Markov process (whose infinitesimal generator is the general diffusion including both Laplacian and Lévy operators). The classical method based on the Itô formula can no longer be used here due to the presence

This paper proposes a stochastic HIV/AIDS model that includes screening for virus carriers and infected individuals actively seeking treatment, with the average number of sexual partners k̄ controlled by a log-normal Ornstein–Uhlenbeck process. By constructing appropriate Lyapunov functions, the existence of a stationary distribution is obtained. Additionally, we establish sufficient condition for

The backward substitution method is a newly developed semi-analytical meshless method. This paper makes the first attempt to apply the backward substitution method for the simulation of piezoelectric structures. The numerical solution is divided into boundary approximation and domain approximation with correction functions. After obtaining the boundary approximation using a series of basis functions

We make a further step in the unisolvence open problem for unsymmetric Kansa collocation, proving almost sure nonsingularity of Kansa matrices with polyharmonic splines and random fictitious centers, for second-order elliptic equations with mixed boundary conditions. We also show some numerical tests, where the fictitious centers are local random perturbations of predetermined collocation points.

This article deals with the following singular parabolic–elliptic chemotaxis system (0.1)ut=Δu−χ∇⋅(uv∇v),x∈Ω,0=Δv−αv+μu,x∈Ω,under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂RN with N≥3, where parameters χ,α and μ are positive constants. Fujie, Winkler, and Yokota Fujie(2015) in 2014 and Fujie and Senba Fujie(2016) in 2016 proved that system (0.1) has a unique globally bounded

We study two-crested traveling Stokes waves on the surface of an ideal fluid with infinite depth. Following Chen & Saffman (1980), we refer to these waves as class II Stokes waves. The class II waves are found from bifurcations from the primary branch of Stokes waves away from the flat surface. These waves are strongly nonlinear, and are disconnected from small-amplitude solutions. Distinct class II

In this paper, a new framework for studying the existence of generalized or strongly generalized solutions to a wide class of inclusion systems involving double-phase, possibly competing differential operators, convection, and mixed boundary conditions is introduced. The technical approach exploits Galerkin’s method and a surjective theorem for multifunctions in finite dimensional spaces.

In the paper, we complement existing oscillation criteria for linear third-order delay differential equations by establishing novel sufficient conditions for the nonexistence of so-called Kneser solutions (nonoscillatory solutions with alternating signs of their derivatives). The significant extent of our improvement over known results is illustrated by the example provided. Furthermore, the technique

The cusp singularity, with only Hölder continuity, is a typical singularity formed in the quasilinear hyperbolic partial differential equations, such as the Hunter–Saxton and Camassa–Holm equations. We establish the global existence of Hölder continuous energy conservative weak solution for a family of Hunter–Saxton type equations, where the regularity of solution varies with respect to a parameter

In this paper, we investigate the Cauchy problem for a new integrable nonlocal fifth-order nonlinear Schrödinger (FONLS) equation with three free parameters. By solving a 2 × 2 matrix Riemann–Hilbert problem in the complex k-plane, we obtain the limit form solutions of the nonlocal FONLS equation. As an example, we provide an exact expression of the one-soliton solution for the nonlocal FONLS equation

In this paper, our main purpose is to prove the existence of the first eigenvalue for the polyharmonic operator with Navier boundary conditions. In addition, the corresponding eigenfunction is demonstrated to be positive. As an application, we will discuss a necessary condition for the existence of positive solutions to some polyharmonic problems on the first eigenvalue.

This paper explores the asymptotic profile of steady states in a partially degenerate Aedes aegypti population model within advective environments. By reducing the model to a scalar equation, we establish the existence and uniqueness of positive steady-state solutions using the method of upper and lower solutions. We analyze the interaction between diffusion and advection, focusing on their effects

In this paper, we propose conservative Crank–Nicolson-type and compact finite difference schemes for the nonlinear Schrödinger equation with point nonlinearity. To construct these schemes, we first transform the point nonlinearity into an interface condition, then decompose the computational domain along the interface into two subregions with a jump condition. Different discretization approximations

This note considers the computation of the logarithm of symmetric positive definite matrices using the Gauss–Legendre (GL) quadrature. The GL quadrature becomes slow when the condition number of the given matrix is large. In this note, we propose a technique dividing the matrix logarithm into two matrix logarithms, where the condition numbers of the divided logarithm arguments are smaller than that

In this work, we propose and analyze a stochastic SIQRS epidemic model with the disease transmission rate driven by a log-normal Ornstein–Uhlenbeck process. By establishing a series of Lyapunov functions, we derive sufficient criteria for the asymptotical stability of the positive equilibrium of the system which suggests the prevalence of the disease in the long term. This work provides a basis for

We consider the absolute value equation (AVE) Ax−|x|=b, where the diagonal entries of A∈Rn×n are all greater than 1 and 〈A〉−I is an irreducible singular M-matrix (〈A〉 is the comparison matrix of A). We investigate the existence and uniqueness of solutions for the AVE. The AVE does not necessarily have a unique solution for every b∈Rn, so most of the existing convergence results for various iterative

The paper investigates a pivotal condition for the Bobkov-Tanaka type spectrum for double-phase operators. This condition is satisfied if either the weight w driving the double-phase operator is strictly positive in the whole domain or the domain is convex and fulfils a suitable symmetry condition.

This study refines the two-stage social insect model of Kang et al. (2015) by incorporating a nonlinear egg cannibalism rate. The introduction of nonlinearity presents analytical challenges, addressed through the application of the compound matrix method to rigorously establish global stability. The analysis reveals complex dynamical behaviors, including two distinct types of bistability: one between

We consider a nonlinear eigenvalue problem driven by the nonautonomous (p,q)-Laplacian with unbalanced growth. Using suitable Rayleigh quotients and variational tools, we show that the problem has a continuous spectrum which is an upper half line and we also show a nonexistence result for a lower half line.

How does the delay function affect its decay rate for a stable stochastic delay differential equation with an unbounded delay? Under suitable Khasminskii-type conditions, an existence-and-uniqueness theorem for an SDDE with a general unbounded time-varying delay will be firstly given. Its decay rate will be discussed when the equation is stable. Given the unbounded delay function, it will be shown

In this paper, we consider the oscillation of second-order neutral advanced dynamic equations on time scales of the form (r(t)(zΔ(t))α)Δ+q(t)f(y(m(t)))=0, where z(t)=y(t)+p(t)y(τ(t)) and m(t)≥t. We consider two cases of τ(t)≥t and τ(t)≤t, respectively. Some new oscillatory results are based on the new comparison theorems that enable us to reduce problem of the oscillation of the second-order equations

This paper presents a multi-scale method for inhomogeneous fourth-order singular perturbation problems. This method guarantees a uniform high-order convergence rate, regardless of the presence of multi-scale coefficients or boundary layer effects. The numerical experiments in two and three dimensions confirm the theory.

A fast wavelet collocation method with compression techniques is proposed for solving the Steklov eigenvalue problem. Based on the potential theory, the Steklov eigenvalue problem is reformulated as a boundary integral equation with logarithmic singularity. By using the compression technique, the wavelet coefficient matrix is truncated into sparse. This technique leads to the algorithm faster. We show

This study presents a normalized time-fractional Fisher equation to resolve scaling inconsistencies associated with conventional time-fractional derivatives. A finite difference scheme is applied to numerically solve the equation. Computational experiments are conducted to investigate the impact of the fractional order on the system’s dynamics. The numerical results demonstrate the influence of memory

The fundamental solution is a particular solution of the inhomogeneous equation with Dirac delta function as the right hand side term. It holds significant importance in both applied and theoretical mathematics and physics. This study focuses on deriving 3D fundamental solutions in the frequency domain for wave propagation in a fluid-saturated porous medium in the context of Biot's theory. The approach

The method of complex variable functions is integrated to investigate the steady state problem wherein SH guided waves impinge upon two dimensional layered inhomogeneous piezoelectric media with a circular inclusion, and the corresponding analytical expressions are derived. Specifically, the guided wave expansion technique is employed to formulate the incident wave field of planar SH guided waves.

In numerical linear algebra, finding the low-rank approximation of large-scale nonsymmetric matrices is a core problem. In this work, we combine the generalized Nyström method and randomized subspace iteration to propose a new low-rank approximation algorithm, which we refer to as the generalized Nyström method with subspace iteration. Moreover, utilizing the projection theory, we perform an in-depth