In this paper, we propose a novel iterative optimization method to determine the best parameter c for the Radial basis function finite difference (RBF-FD) method based on the Double Operator Error (DOE). This method is a general iterative optimization approach that can rapidly determine the optimal c for a given problem and reduce the numerical error significantly, for any type of radial basis functions

The paper studies the stability of impulsive stochastic functional differential equations. At first, we obtained a new generalization of the Halanay inequality. Then, by applying the generalized Halanay inequality, the exponential stability results of impulsive stochastic functional differential equations are obtained.

In the paper, we consider the existence of normalized solutions for following fractional Schrödinger equation (FSE)(−Δ)su+V(x)u=K(x)f(u)+λu,inRNwhere N≥2, s∈(0,1), λ∈R is a parameter and (−Δ)s is the fractional Laplacian operator. We prove existence of normalized solutions for equation (FSE) under hypotheses on the potentials V and K. Moreover, we also obtain λ=0 is a bifurcation point.

In this paper, we study the following fourth-order elliptic equation with p-Laplacian, steep potential well and sublinear perturbation: Δ2u−Δpu+μV(x)u=f(x,u)+ξ(x)|u|q−2u,x∈RN,where N≥5, Δ2≔Δ(Δ) is the biharmonic operator, Δpu=div|∇u|p−2∇u with p>2, μ>0 is a parameter, f∈CRN×R,R, ξ∈L22−qRN with 1≤q<2, we have the potential V∈C(RN,R), and V−1(0) has nonempty interior. Under certain assumptions on V and

This paper deals with the Neumann problem for the two-species chemotaxis system (0.1)ut=Δu−χ∇⋅(u∇w),x∈Ω,t>0,vt=Δv+ξ∇⋅(v∇z),x∈Ω,t>0,0=Δw−w+v,x∈Ω,t>0,0=Δz−z+u,x∈Ω,t>0, in a smoothly bounded domain Ω⊂RN (N≤3) with nonnegative initial data. The parameters χ and ξ are positive. The present work asserts that this problem admits a globally defined bounded classical solution. We use a new method to obtain

Conventionally, the numerical stability of finite difference approximations of nonlinear Kawarada problems is shown only via frozen source terms, that is, ignoring potential jeopardization from quenching nonlinearities. The approach leaves an inadequacy behind even in the sense of localized stability analysis. This paper provides a much improved analysis of the numerical stability without freezing

The aim of this paper is to answer the question left in Gui and Liu (2015). We prove that given initial data u0∈Hs(R) with s>32 and for some T>0, the solution of the Camassa–Holm equation converges strongly in L∞(0,T;Hs(R)) to the inviscid Burgers equation as the filter parameter α tends to zero.

We study the 3D generalized magnetohydrodynamics (gMHD) equations with dissipation terms −(−Δ)αu and −(−Δ)βb. It is proved that a weak solution (u,b) to gMHD equations is smooth on R3×(0,T] if u, ∇u or (−Δ)m/2u belongs to Lq(0,T;Lp(R3)) with p,q and m=min{α,β} satisfying the generalized Ladyzhenskaya–Prodi–Serrin type conditions.

We prove a.s. (almost sure) unisolvency of interpolation by continuous random sampling with respect to any given density, in spaces of multivariate a.e. (almost everywhere) analytic functions. Examples are given concerning polynomial and RBF approximation.

In this paper, we develop a stochastic population-toxicant model with cross-diffusion and study the local boundedness of strong solutions. The existence of global martingale solution is obtained in a Hilbert space by Galerkin approximation method, the tightness criterion and the energy estimation.

In this paper, we prove some optimal regularity criteria for the 3D incompressible Navier–Stokes equations involving the gradient of one velocity component in the framework of anisotropic Lebesgue spaces satisfying Prodi–Serrin condition. These results cover the previous ones established by Wolf (2015) and O (2023).

In this paper, we study the existence and uniqueness of the weak solutions for a truncated system of a class of time-fractional reaction–diffusion–advection systems with discontinuous diffusion. Our approaches are based upon the classical Galerkin’s method and some new compactness criteria for proving the existence of weak solutions to time-fractional PDEs.

This paper is devoted to the study of stability of curved fronts in space–time periodic media established by Bu and Wang (2016). Under the condition of initial perturbations decay exponentially with an appropriate rate in the lower cone, we can prove that the curved front is exponentially stable by using the super-sub solutions technique.

The Landweber-type iteration method is one of the most prominent regularization methods for solving ill-posed problems when the data is corrupted by noise. By extending the previous results proposed by de Hoop et al. (2012), the current work systematically investigates the convergence rate of Landweber-type iteration in Banach spaces for nonlinear ill-posed problems. Two types of stopping rules including

In this work, general rogue wave solutions in the AB system are constructed by means of the Kadomtsev–Petviashvili hierarchy reduction method and explicit representations of these solutions are explored in terms of determinants whose matrix elements are fundamental Schur polynomials. Moreover, the dynamics of these rogue waves are investigated graphically by different choices of the free parameters

In this paper, we show the stability of the inverse source problem for the Maxwell equations with conductivity. The tangential components of the electric and magnetic fields on the boundary at multiple frequencies are required as the data for the analysis. The stability estimate consists of the Lipschitz type data discrepancy and the high frequency tail of the source function, where the latter decreases

In this paper, we mainly concern with the bifurcation problem of positive steady states for a degenerate reaction–diffusion host–pathogen model with spatial heterogeneous environment. A criterion on the existence of bifurcation of positive steady states from the disease-free steady state is established. Our results reveal that the mortality rate of the incubation host affects the existence of the positive

In this work, the continuity with respect to the Hurst parameter of solutions to stochastic evolution equations is studied. Compared with recent studies on such continuity property, the model here is considered in a different point of view in which the equations are of SPDEs type, the solution and the fractional Brownian motion take value on a Hilbert space. The main contribution is to investigate

In this paper, the exact solutions of the (3+1)-dimensional nKdV-nCBS equation are investigated by various approaches. Firstly, according to Lie group theory, the (3+1)-dimensional nKdV-nCBS equation is reduced to two (1+1)-dimensional partial differential equations. Secondly, three different types of exact solutions are obtained by the homoclinic test approach, including the singular kinked-type periodic

This paper investigates the stability of strong exponential attractors with respect to dissipative index θ∈[1/2,1) for the Kirchhoff wave model with structural nonlinear damping: utt−ϕ(‖∇u‖2)Δu+σ(‖∇u‖2)(−Δ)θut+f(u)=g(x). It proves that for each θ0∈[1/2,1), there exists a family of strong bi-space exponential attractors, which are also the standard exponential attractors of optimal regularity and are

The periods of the periodic wave solutions for nearly parallel vortex filaments model are discussed. By the transformation of variables, the vortex filaments model is reduced to the planar Hamiltonian system whose Hamiltonian function includes a logarithm term. We successfully handle the logarithm term in the study of the monotonicity of the period function of periodic solutions. Moreover, the asymptotic

Two localization sets for Pareto eigenvalues of matrices are established to provide some checkable sufficient conditions for the copositivity of matrices, which also answer the open question (c) in the paper Seeger (1999). Finally, numerical experiments are reported to show the efficiency of the proposed localization sets.

The diffusion in the comb model is an important kind of anomalous diffusion which is described by a governing equation containing the Dirac delta function and the solution’s domain is infinite. Two key problems are solved to analyze the mass transfer mechanism in the comb model. One is to construct the appropriate and reasonable boundaries for treating the infinite boundaries. The exact absorbing boundary

This letter devotes to the design of efficient prediction–correction numerical methods which produce high-order approximations of the solutions while preserving mass, or energy, or both of them, for the semiclassical Schrödinger equation with small Planck constant ɛ. The prediction step involves an explicit temporal fourth-order exponential Runge–Kutta method which allows the ɛ-oscillatory solution

Kaczmarz method is a traditional and widely used iterative method for solving consistent linear equations while its randomized version recently attracts much more attention due to its linear convergence rate in expectation. A general scheme of surrogate hyperplane Kaczmarz method is proposed to generate a new hyperplane by combining a number of selected hyperplanes. In particular, the residual-based

For a singularly perturbed convection diffusion problem, we study the supercloseness of the nonsymmetric interior penalty Galerkin (NIPG) method on a Bakhvalov-type mesh. In this process, a new composite interpolation is designed, which consists of Gauß Radau projection outside the layer and Gauß Lobatto projection inside the layer. Then by choosing the penalty parameters at different mesh points,

This paper proposes a linearly stabilized convolution quadrature method for solving the time-fractional Allen–Cahn equation. The stability condition is explicitly given such that the method is unconditionally stable for any time step size. The space is discretized by the central difference method. We prove that the fully discrete scheme preserves the discrete maximum principle, the discrete energy

This paper studies the stability and large time behavior to the 2D anisotropic Navier–Stokes equations with mixed partial dissipation. We establish the uniform upper bounds and the global stability of solutions, and obtain the optimal decay properties to these global solutions and their higher order derivatives without any small assumptions on the initial data.

In this paper, we investigate the nonlinear dynamics of an interesting class of vector solitons in the two-component modified Korteweg–de Vries (mKdV) equation. We construct the nondegenerate solitons and the breather solutions of the two-component mKdV equation by applying a non-standard form of the Hirota direct method. Our study shows that the nondegenerate solitons and the breather solutions of

This paper is concerned with the following planar Schrödinger–Newton equation −Δu+V(x)u+12π(ln(|⋅|)∗|u|p)|u|p−2u=f(x,u),x∈R2,where p≥2, V∈C(R2,[0,∞)) is axially symmetric and f∈C(R2×R,R) is of critical exponential growth in the sense of Trudinger–Moser. Under mild assumptions, we obtain the existence of axially symmetric solutions to the above equation for p≥2 by using some new techniques. In particular

In this paper, we investigate Kukles homogeneous systems ẋ=−y,ẏ=x+Qn(x,y), where Qn(x,y) is a homogeneous polynomial of degree n. There are two conjectures on the center-focus problem and isochronous center problem of the above systems. These two conjectures are claimed to be proven in Giné et al. (2015,2017). However, the proofs may have some gaps, hence they are still open. The gaps of Giné et al

We consider initial boundary value problem for the time–space fractional parabolic–elliptic Keller–Segel model 0CDtβu=−(−Δ)α2(ρ(v)u),(t,x)∈(0,T]×Ω(−Δ)α2v+v=u,(t,x)∈(0,T]×Ωin a bounded domain Ω⊂Rn(n≥3) with smooth boundary, where β∈(0,1),α∈(1,2) and ρ stands for a signal-dependent motility. It is shown that for some special initial datum, there exists the uniform-in-time upper bound for v such that

We extend Darboux-dressing transformation with expansion theorem to study the coupled Gerdjikov–Ivanov (cGI) equation. We successfully find its novel vector breather wave solutions and Nth-order rouge wave solutions. By considering an expansion theory, we first construct the Darboux-dressing transformation, which could iterate with the same spectral parameters for finding interesting exact solutions

It is proved that if the damped periodically forced Newtonian pendulum does not have periodic solutions, the same happens for the relativistic version of the problem for high values of the parameter c representing the speed of light in the vacuum.

In this paper, we consider the periodic traveling waves for a time–space periodic and degenerate reaction–diffusion dengue model. By establishing two comparison theorems and using comparison methods, we prove the globally exponential stability of time–space periodic traveling fronts. Our result is a continuation of Fang et al. (2020).

In this paper, we present the asymptotic “solutions” of the general Heun equation in terms of the orthogonal polynomials with respect to certain perturbed “classical” Jacobi weights. Our approach employs a double limit, which includes changes in variables or scaling. We construct the second-order differential equations satisfied by the orthogonal polynomials generated by the given Jacobi-type weights

The main objective of this paper is to study the simple zero soliton solutions and multiple-zero soliton solutions of the reverse-time Manakov system by using Riemann–Hilbert method. It is worth noting that the symmetry of discrete scattering data for the reverse-time Manakov system is very different from the local Manakov system. In addition, in order to better show the remarkable characteristics

In this paper, we analyze the existence of positive solution to a biharmonic equation with Navier boundary conditions, and formulate simple uniqueness conditions for the positive solution. In addition, a necessary condition for the existence of the positive solution is also derived.

This paper deals with the Neumann initial–boundary value problem for the chemotaxis-May–Nowak model with volume filling sensitivity ut=Δu−χ∇⋅f(u)∇v−uw+κ−u,x∈Ω,t>0,vt=Δv−v+uw,x∈Ω,t>0,wt=Δw−w+v,x∈Ω,t>0in a smooth bounded domain Ω⊂Rn (n≥1), where χ∈R, κ>0 and f∈C1([0,∞)) fulfilling that f(0)=0 and f≡0 on [m,∞) with some m>0. We prove that the problem admits a unique global bounded classical solution.

We study the averaging principle for a class of semilinear stochastic partial differential equations perturbed by space–time white noise. Using the factorization method and Burkholder’s inequality, the estimation of stochastic integral involving the heat kernel is obtained. Under suitable assumptions, we show that the original stochastic systems can be approximated by the averaged equations.

In this paper, we propose a stochastic SEI epidemic model in which the transmission rates are general functions and satisfy the log-normal Ornstein–Uhlenbeck (OU) process. We first theoretically prove that there is a unique positive global solution of this stochastic model. By constructing several suitable Lyapunov functions, the sufficient condition R0s>1 is established for the existence of stationary

A stochastic single species population model with Allee effect is considered in this paper. By using the theory of Random Dynamical System (RDS), the complete classification of the global dynamics of stochastic system is given. The theoretical results show that the dynamics of the system is completely determined by the threshold λ=2r−σ2: if λ≤0, the population will extinct, i.e., the solutions of the

A network is introduced to Lotka predator–prey model, where the network structure describes the movement directions between every two nodes. By using monotone iterative approach, we study the role of hunting strength on persistence and extinction. In the case of strong hunting, solutions converge uniformly to the semipositive equilibrium such that the predator survives while the prey becomes extinct

An impulsive stochastic pest management system with insecticide residue effects and group defense behavior is proposed to study how environmental noise affects the dynamics of pest populations. Firstly, we show that the proposed system exists a unique global positive solution with upper bounded in terms of expectation. Secondly, threshold criteria ensuring the extinction of pests and natural enemies

An integrable time-discretization of the Ito equation is presented. By use of Hirota’s bilinear method, the Bäcklund transformation, Lax pair and soliton solutions to the semi-discrete system are also derived. Since the integrable time-discrete system converges to the continuous Ito equation when the step size tends to zero and does not destroy the conserved quantities, we design a numerical scheme

A novel (2+1)-dimensional nonlinear Schördinger equation is constructed from (1+1)-dimensional Schördinger equation based on a deformation algorithm. The integrability of the obtained (2+1)-dimensional Schördinger equation is guaranteed by its Lax pair obtained directly from the Lax pair of the (1+1)-dimensional Schördinger equation. Because of the effects of the deformation, the (2+1)-dimensional

This paper focuses on a coupled model of heat conduction and heat radiation within bilayer textiles, which is decoupled to a parabolic model with nonlinear source term of functional type. Uniform boundedness and stability estimate of the solution to the nonlinear bilayer model are derived by energy estimates and Young’s inequality skills.

In this article, we rigorously prove the homogenization limit of the linear Boltzmann equation when the scattering term is highly oscillating with respect to the velocity variable. We prove that the limit equation keeps, in a suitably extended phase space, the same structure as the non-homogenized one. This situation does not coincide with what happens in standard phase space, where the appearance

We investigate an extensive generalized shift-splitting (EGSS) iterative method for solving three-by-three block saddle point problems (SPPs), which occur in the finite element discretization of stationary magnetohydrodynamics models and Maxwell equations. Mathematical analysis suggests that the EGSS iterative method has unconditional convergence. The performance of the proposed method is compared

This paper investigates the slightly compressible Brinkman–Forchheimer equations (BFEs): ∂tu−Δxu+∇xp+f(u)=g with D−1(t)∂tp+divu=0 in a bounded 3D domain with Dirichlet boundary conditions. The features of this problem is that, formally, this system is partially dissipative, and will recover to incompressible BFEs when the time-dependent coefficient D(t) goes to infinity as t→∞. The well-posedness and

This paper is concerned with the following planar Schrödinger-Poisson system −Δu+V(x)u+ϕu=f(u),x∈R2,Δϕ=u2,x∈R2,where V∈C(R2,[0,∞)) is axially symmetric and f∈C(R,R) has critical exponential growth in the sense of Trudinger-Moser. This system can be converted into the integro-differential equation with logarithmic convolution potential. We prove the existence of axially symmetric solutions by some new

We present a three-dimensional differential equation, which robustly displays a degenerate Andronov–Hopf bifurcation of infinite codimension, leading to a center, i.e., an invariant two-dimensional surface that is filled with periodic orbits surrounding an equilibrium. The system arises from a three-species bimolecular chemical reaction network consisting of four reactions. In fact, it is, up to a

We have recently introduced a very high order Discontinuous Galerkin (DG) method for the solution of stiff ordinary differential equations (ODEs) and delay differential equations (DDEs). In this communication we focus on the detection of breaking points frequently arising in the solution of stiff DDEs. Breaking points are discontinuities appearing in the solution of DDEs and/or in some of its derivatives

In this paper, we consider the fifth-order Camassa–Holm type equation which is integrable and admits the single pseudo-peakons and multi-pseudo-peakons. We discuss the orbital stability of single pseudo-peakons.

In this short paper, we give an improved estimate of the infinite time blowup solution for the semilinear wave equation with logarithmic nonlinearity. Comparing the previous work, this improved growth estimate not only provides the information about the relationship between the power of the nonlinearity and the growth, but also gives a faster growth estimate. The proof of the main conclusions is carried

In this paper, we present a meshless radial basis function method to solve conservative Allen–Cahn equation on smooth compact surfaces embedded in R3, which can inherits the mass conservation property. The proposed method is established on the operator splitting scheme. We approximate the surface Laplace–Beltrami operator by an iterative radial basis function approximation method and discretize the

In this paper, we study the Liouville-type theorem of the stationary Navier–Stokes equations on an oscillation growth condition of the potential tensors. Based on the oscillation growth condition, we prove Poincaré-type inequalities on different balls, which play a decisive role to get the result.

This paper deals with the monotonicity and uniqueness of traveling waves in a class of delayed diffusion equations with degenerate bistable nonlinearity. We use a new method to show that the wave speed of traveling waves that may equal to zero is unique. In addition, with the help of asymptotic behavior of traveling waves, the monotonicity and uniqueness up to a translation of traveling waves are established

A systematic method is developed to obtain localized wave solutions of the Fokas–Lenells equation on theta-function backgrounds. First, using the properties of the theta-functions, we find a theta-function seed solution of the Fokas–Lenells equation. Next, based on the Riccati equation of Lax pair, we construct a Darboux transformation of the Fokas–Lenells equation. Then, the Kaup–Newell type spectral

This paper considers periodic perturbations of a class of nonlinear degenerate systems. By the technique of introducing external parameters, the implicit function theorem and the theory of Brouwer topological degree, it is proved that the perturbed systems have response solutions if the perturbations are sufficiently small.