An inverse spectral problem for Sturm–Liouville operators with frozen argument irrationally proportioned to the interval length is studied in this paper. We present a constructive procedure for reconstructing the potential from the spectrum.

In this paper, we give a different method from Ao and Zou (2019) and Chen and Zou (2012) to consider the following nonlinear Schrödinger system with one critical exponent and one subcritical exponent: −Δpu+μ|u|p−2u=|u|q−2u+αλ|u|α−2u|v|βinRN,−Δpv+ν|v|p−2v=|v|p∗−2v+βλ|u|α|v|β−2vinRN,where N≥max{3,p},μ,ν,λ>0,α≥1,β≥1,2≤pμ0, there exists λμ,ν∈μ−μ0ααpνββp,(μα)αp(νβ)βp such that if λ>λμ,ν, the above system

In this letter we present a Keller–Segel model with logistic growth dynamics arising in the study of chemotactic pattern formation. We prove the existence of a minimum wave speed for which the model exhibits nonnegative traveling wave solutions at all speeds above this value and none below. The exact value of the minimum wave speed is given for all biologically relevant parameter values. These results

The regularized least-squares radial basis approximation is a kernel-based method to approximate a set of scattered data by a least-squares fit based on an optimization procedure that balances a tradeoff between smoothness of approximation and closeness to the data via a smoothing parameter. This paper suggests the generalized regularized least-squares radial basis approximation for noisy data and

This paper aims at the global well-posedness problem to the n-dimensional magneto-micropolar equations with hyperdissipation. By fully exploiting the special structure of the system and the energy methods, we establish the global existence and uniqueness of solution for the system with hyperdissipation. The result obtained her e improves the known global regularity results in this direction.

Computation of rotation tensor is essential in the analysis of deformable bodies. This paper propose an explicit expression for rotation tensor R of deformation gradient F, and successfully establishes an intrinsic relation between the exponential mapping Q=expA and the deformation F. As an application, Truesdell’s simple shear deformation is revisited. For easy use of our formulation, we provide a

In this paper, we consider the oscillation of a class of third-order nonlinear neutral differential equations with distributed deviating arguments by using comparison principles. The obtained criteria essentially extend and improve related results in the literature by removing some restrictive conditions, which can easily be extended to more general third-order differential equations. Examples are

In this paper, an unconditionally stable fully discrete numerical scheme for the two-dimensional (2D) time fractional variable coefficient diffusion equations with non-smooth solutions is constructed and analyzed. The L2-1σ scheme is applied for the discretization of time fractional derivative on graded meshes and anisotropic finite element method (FEM) is employed for the spatial discretization. The

In this paper, we considered a class of fourth order equation modeling epitaxial thin film growth. We establish a blow up result for the initial and boundary value problem. Furthermore, the lower bound of the blow up time and the blow up rate are derived.

In this paper, we study the following fractional Schrödinger-Poisson system (−Δ)su+V(x)u+ϕu=f(x,u)+λ|u|p−2u,inR3,(−Δ)tϕ=u2,inR3,where s∈(34,1), t∈(0,1), λ>0 is a real parameter, (−Δ)s is the fractional Laplace operator, p≥2s∗=63−2s. Under some suitable assumptions on V and f, by using Moser iterative method and truncation technique, we prove that the above equation have a ground state solution and

In this paper, we are concerned with a diffusive cholera epidemic model with nonlinear incidence rate. By constructing suitable Lyapunov functionals, we investigate the global stability of the disease free equilibrium and the endemic equilibrium.

We present a supplementary variable method (SVM) for developing structure-preserving numerical approximations to a partial differential equation system with deduced equations. The PDE system with deduced equations constitutes an over-determined, yet consistent and structurally unstable system of equations. We augment a proper set of supplementary variables to the over-determined system to make it well-determined

In this paper, a spatial–temporal generalized finite difference method (GFDM) with an additional condition is proposed for two-dimensional (2D) transient heat conduction analysis of functionally graded materials (FGMs). The 2D time dependent problem is explored as the three-dimensional (3D) stationary problem by treating the temporal direction as the third dimension. Based on the time-marching procedure

The bilinear form of a fourth-order nonlinear generalized Boussinesq water wave equation is presented by using the Hirota’s bilinear method. Based on the presented bilinear form, some rational solutions and interaction solutions are obtained by choosing four class of test functions. The features such as the extreme values and asymptotics of the solutions are investigated and the three-dimensional plots

The wick-type stochastic fractional nonlinear Schrödinger equation is considered, and some analytical white noise functional solutions are found by the fractional F-expansion method with the Hermite transformation. The chirped and chirp-free fractional bright and dark soliton solutions with the Brownian motion function are constructed.

The Hirota bilinear method is studied in a lot of local equations, but there are few work to solve nonlocal equations with external potential by Hirota bilinear method. In this letter, we succeed to bilinearize the generalized nonlocal Gross–Pitaevskii (NGP) equation with an arbitrary time-dependent linear potential through a nonstandard procedure and present more general bright soliton solutions,

In this paper, we develop a Nitsche’s type interface stabilized method for fully mixed Stokes–Darcy problem with original Beavers–Joseph conditions. The method introduces two strongly consistent interface stabilization terms to guarantee the stability of the fully mixed finite element method. The stability and optimal error estimates of this new stabilized method are also derived.

We apply the ∂̄-dressing method to study a coupled Gerdjikov–Ivanov (GI) equation. Compared with results on the classical coupled GI equation, more general spatial and a time singular spectral problems associated with GI equation are derived from a local 2 × 2 matrix ∂̄-equation via two linear constraint equations. A more general GI hierarchy with source is proposed by using recursive operator. The

In this paper, by applying the direct method of moving planes, the authors study the radial symmetry of standing waves for nonlinear fractional Laplacian Schrödinger systems with Hardy potential. Firstly, under the condition of infinite decay, the radial symmetry of the solution is established. Secondly, under the condition of no decay, the radial symmetry and non-existence of solution are established

The current paper is devoted to the stochastic damped higher-order KdV equation driven by white noise. We prove the existence of invariant measure for stochastic damped (2n+1)-th order KdV equation with random initial value, and the ergodicity with deterministic initial data.

By applying variational arguments and the lower and upper solution method, we obtain several criteria for the existence of positive solutions to a class of singular discrete Dirichlet problems with a parameter. Our results cover the cases when the nonlinear function in the equation is superlinear, asymptotically linear, and sublinear. We provide several examples to illustrate our results.

We investigate a diffusive predator–prey system with discontinuous harvesting function under Neumann boundary conditions. Then the existence and some estimates of the positive solution of the system are acquired. Based on the differential inclusions theory, we discuss the existence of constant equilibrium. Finally, by constructing a suitable Lyapunov functional, we study the global asymptotic stability

In this paper, a class of nonlinear reaction–diffusion equations is studied, and Abelian integral method is applied to show the existence of a unique periodic traveling wave solution. Simulation is presented to verify the theoretical prediction.

In this paper, we study a discontinuous Galerkin (DG) method for solving an elliptic hemivariational inequality for semipermeable media. A priori error analysis is established, and we show that the DG scheme with the linear element reaches optimal convergence order under appropriate solution regularity assumptions. One numerical example is presented to support the theoretical analysis.

In this paper, we consider the following supercritical nonlinear Schrödinger equation: (0.1)−Δu+λV(x)u=P(x)|u|p−2u+μ|u|q−2u,inRN,where μ>0, N≥3, 2

The orthogonality sampling method (OSM) is a recently developed non-iterative technique for imaging and identifying targets in inverse scattering problems. In this paper, a set of short sound-soft open arcs is obtained by OSM in the limited-aperture inverse scattering problem. We introduce an indicator function of OSM and explore its mathematical structure. In this formulation, the identification largely

A novel numerical technique is developed in this paper to accurately and efficiently resolve the inverse source problem of the nonlinear time-fractional wave equation. Based on all given conditions, the homogenization function of nonlinear time-fractional wave equation can be derived, and then a family of homogenization functions is obtained. Furthermore, a numerical model is established by the superposition

This article investigates the uniqueness of identifying the fractional-order, potential, and Robin coefficient simultaneously in one-dimensional time-fractional diffusion equation with non-homogeneous boundary condition. By using one boundary measurement, we prove that the fractional-order, potential on the entire interval, and Robin coefficient are determined simultaneously from asymptotic properties

In this paper, we investigate the initial value problem for the three dimensional nematic liquid crystal flows with two directional viscosity. The main purpose of this paper is to establish global well-posedness of classical small solutions. The proof is mainly based on the energy estimate, anisotropic type Sobolev inequality and bootstrapping argument.

In this paper, we are concerned with the Cauchy problem of the 3D incompressible micropolar equations with partial viscosity and damping. We establish the global existence and uniqueness of the solution with a general initial data.

In this contribution, we investigate an initial–boundary value problem for a fractional diffusion equation of distributed order where the coefficients of the elliptic operator are dependent on spatial and time variables. We consider a homogeneous Dirichlet boundary condition. Using a classical variational approach, we establish the existence of a unique weak solution to the problem in C[0,T],H01(Ω)∗∩L∞(0

The mean-square stability of the θ-method for neutral stochastic delay integro-differential equations (NSDIDEs) is considered in this paper. We construct two classes of θ-methods, i.e. the stochastic linear theta (SLT) method and the split-step theta (SST) method for NSDIDEs. Under the one-sided growth condition and contractive condition, we show that both methods are mean-square exponentially stable

In this paper, we remove the bounded total variation condition on the initial data and the restriction of the concentration of a fixed background charge being a constant in the paper “Relaxation of the Isothermal Euler–Poisson System to the Drift-Diffusion Equations,” (Quart. Appl. Math., 58 (2000), 511–521), and obtain the zero relaxation time limits to isothermal hydrodynamic model for semiconductor

In this paper, we study the existence of positive solutions for a class of reaction–diffusion equations in spatially heterogeneous environment. We also determine the asymptotic profiles of positive solutions when some parameter goes to zero. Our analytical results suggest that the spatial heterogeneity has obvious influence on the positive solutions.

Degree elevation is a typical corner-cutting algorithm. It refers to the process transforming control polygons when embedding a polynomial space of some degree into any polynomial space of higher degree. Dimension elevation similarly refers to the transformation of control polygons when embedding an Extended Chebyshev space possessing a Bernstein basis into another one, of higher dimension. Unlike

Explicit exponential stability tests are obtained for the scalar neutral differential equation ẋ(t)−a(t)ẋ(g(t))=−∑k=1mbk(t)x(hk(t)), together with exponential estimates for its solutions. Estimates for solutions of a non-homogeneous neutral equation are also obtained, they are valid on every finite segment, thus describing both asymptotic and transient behavior. For neutral differential equations

Neither any monotonicity condition nor any Ambrosetti–Rabinowitz growth condition required, some novel existence results of ground state solutions for a class of nonlinear fractional Schrödinger systems have been discovered. Lack of these conditions would entail some difficulties that we would overcome.

We find a dichotomy between the system of difference equations un+1=(a+cvn)∕un and vn+1=(b+dun+1)∕vn, n=0,1,2,…, and its perturbed system un+1=(a+cvn)∕un and vn+1=(b+dun+1+ηvn2)∕(vn+ηvn), n=0,1,2,…, where a,b,c and d are arbitrary positive real numbers, η≥0 and the initial values u0,v0>0, which originate from the Lyness difference equation with period-two coefficients. Namely, there are infinitely

We propose a new computational approach for constructing the refinement matrix mask of interpolating Hermite splines of any order and with general dilation factor. Our strategy exploits the refinability properties of cardinal B-splines with simple knots and simplifies the constructive procedures proposed so far.

This paper is concerned with a fourth-order exponential wave integrator Fourier pseudo-spectral method for solving the Klein–Gordon equation. We suggest a new numerical integration formula to approximate the time integral term in the phase space. The error estimate shows the suggested numerical method is fourth-order accurate in time and spectral accurate in space. The theoretical findings are confirmed

This paper is dedicated to investigating large time decay of weak solutions for the 2D Oldroyd-B model of non-Newtonian flows. Due to the inconsistencies of the dissipative effect between Laplacian velocity viscosity and linear stress damping, either the classic Fourier splitting methods or Kato’s methods for both velocity u and stress tensor τ are not available directly. Based on a new observation

In this article we propose a second order, linear, unconditionally stable, implicit–explicit scheme based on the Crank–Nicolson–Leapfrog discretization and the artificial compression method for solving phase field models of two-phase incompressible flows. We show that the scheme is unconditionally long-time stable. Numerical examples are provided to demonstrate the accuracy and long-time stability

We consider nonlinear wave type PDEs with delay of the form utt+H1(u)ut=[G(u)ux]x+H2(u)ux+F(u,w),where u=u(x,t) is the unknown function, w=u(x,t−τ), and τ is the delay time. The source function F(u,w) depends on one or several arbitrary functions of one argument. Using the modified method of functional constraints, we obtain a number of new exact solutions with generalized and functional separation

This article studies the existence of global smooth solution for the Chaplygin gas equations with source terms in Rd, d≥1. We establish the existence and uniqueness of global smooth solution provided the initial data is sufficiently small, which tells us that the damping terms can prevent the development of singularity in small amplitude.

This paper is concerned with conical traveling fronts for reaction–diffusion equations with combustion nonlinearity in R3. By constructing a sequence of pyramidal traveling fronts and taking its limit, we establish the existence and qualitative properties of conical traveling fronts.

In this paper, we prove the global in time regularity for the 2D micropolar Rayleigh–Bénard convection problem with the zero diffusivity.

In this paper, we consider a fractional hepatitis B epidemic model. We illustrate the solvability through solutions for the model. The basic reproduction number which is represented by R0 is obtained. The disease-free and endemic equilibria of the model will be obtained. Finally, by fractional Barbalat’s Lemma, global dynamics of the model is studied.

In this paper, we study a class of damped vibration problems with nonlinearity of linear growth. We obtain the existence and multiplicity of nontrivial periodic solutions by using the Morse theory and a critical point theorem. Compared with the existing work, our results do not require the system to be asymptotically linear at infinity.

We study the occurrence of traveling wave solutions for a modified Burgers’ equation with diffusivity D and a time delay τ. Monotonic traveling waves with speed c are shown to exist when τc2∕D≤e−1. Graphs are provided to help characterize the behavior of these solutions compared to those for the familiar Burgers’ equation without delay.

Inspired by Chang (1981) and Long (1990), we consider the existence of periodic solutions for second-order difference equations with asymptotically linear nonlinearity via Conley index theory. Moreover, when we study the following one-dimensional autonomous difference equations with form Δ2xn−1+f(xn)=0,n∈Z,xn∈R,we can derive at least three periodic solutions under suitable assumptions.

In this paper we study the advanced differential equations of the form x′(t)−q(t)xσ(t)=0,t≥t0,where q is a function of nonnegative real numbers and σ is a function of positive real numbers such that σ(t)>t, for all t≥t0. Based on a recursive sequence we have constructed, we improve the well known oscillation condition lim supt→∞∫tσ(t)q(s)ds>1 by lim supt→∞∫tσ(t)q(s)ds≥1.An example illustrating our

In this paper we provide a sufficient condition for regularity of solutions to the 3D nematic liquid crystal flows. More precisely, we prove that if the horizontal velocity component uh=(u1,u2) satisfies ∇uh∈Lq(0,T;Lp(R3)),3p+2q≤32,2≤p≤∞, then the solution (u,d) is regular.

This paper is devoted to study the controllability of a one-dimensional fluid-particle interaction model where the fluid follows the viscous Burgers equation and the point mass obeys Newton’s second law. We prove the null controllability for the velocity of the fluid and the particle and an approximate controllability for the position of the particle with a control variable acting only on the particle

The reproducing kernel functions (RKFs) and related theory have been employed to deal with the numerical solutions of boundary value problems (BVPs). The existing RKFs based approaches for BVPs have global convergence order two and three in the reproducing kernel Hilbert spaces (RKHSs) W3 and W4, respectively. The global convergence order of the present new method is four and six in the RKHSs W3 and

In this paper, we propose a phase-field model for the shape transformation and its simple numerical method. The proposed phase-field model is based on the Allen–Cahn equation, which is a second-order nonlinear parabolic partial differential equation and originally arises from modeling phase separation in alloys. The numerical scheme is a hybrid method using the operator splitting method. To validate

Consider the boundary value problem u′′+u=αu−+p(t), u(0)=0=u(π), α≤0. We show that a necessary and sufficient condition for the problem to be solvable is that ∫0πp(t)sintdt≥0. We thus answer positively a counter part of a long-standing open problem posed by S. Fučík. A more general problem is also considered and the sufficient condition for the existence of solution is obtained.

We confirm in this paper that the number q∗(α)=p(N+α)N−p with α>0 is exactly the critical exponent for the embedding from the weighted Sobolev space Wr1,p(RN) of radial functions into Lq(RN,|x|α) where 1

This paper concerns the long-time dynamics of a reaction–diffusion system with negative feedback and inhibition. We first derive a priori L∞-norm estimates for the solution. Then, by constructing two different types of Lyapunov functions, we prove that the positive equilibrium is globally attractive when the basic reproduction number R0>1, while the microorganisms-free equilibrium is globally attractive

This paper mainly focuses on the stationary probability distribution of stochastic differential equations (SDEs) under regime switching. We give the explicit expression of stationary distribution according to the limit probability distribution of SDEs under some conditions. As a special case, we obtain the stationary distribution of SDEs with Markovian switching as the weighted mean of the stationary

A classical two species Lotka–Volterra competition–diffusion–advection model is considered in this paper, where the diffusion coefficients, advection coefficients, resource functions and competition rates are all spatially heterogeneous. The global stability of nonhomogeneous steady states in the advective heterogeneous environment is studied by introducing a weighted Lyapunov functional associated