This paper addresses the problems of blow-up for a weakly dissipative generalized Dullin–Gottwald–Holm equation. A new sufficient condition on the initial data is provided to ensure the finite time local-in-space blow-up of strong solutions, which improves the local-in-space blow-up result of Novruzov and Yazar [1].

A collection of test integrals introduced by Genz (1984) has remained popular to this day for assessing the robustness of high-dimensional numerical integration algorithms. However, the explicit solutions to these integrals do not appear to be readily available in the existing literature: typically the true values of the test integrals are simply approximated using “overkill” numerical solutions. In

In this paper, we study a temporally distributed memory-based diffusion model with a weak kernel and nonlocal delay effect. Without diffusion, we present results on the stability and Hopf bifurcation of the positive constant steady state. With the inclusion of diffusion, further results on the stability and steady state bifurcation are derived. Finally, these findings are applied to a population model

We consider the semilinear heat equation ut−Δu=|u|p−1u+λu,onRnwhere p>1, and λ∈R is a parameter. When λ=0, the equation reduces to the classical heat equation. We reveal that the parameter λ in the linear term plays an important role in the blow-up conditions. Although the solution may blow up in finite time due to the cumulative effect of the nonlinearities, interestingly, we find that for λ>n2, all

The ‘bad’ Jaulent–Miodek (JM) equation describes the wave evolution of inviscid shallow water over a flat bottom in the presence of shear, which is ill-posed and unstable so that its general initial problem on the zero plane is difficult to solve through traditional mesh-based numerical methods. In this paper, using the Darboux transformation, we find a relation between the ‘bad’ JM equation and the

In this paper, we introduce an efficient numerical algorithm for solving the Lifshitz–Petrich equation on closed surfaces. The algorithm involves discretizing the surface with a triangular mesh, allowing for an explicit definition of the Laplace–Beltrami operator based on the neighborhood information of the mesh elements. To achieve second-order temporal accuracy, the backward differentiation formula

This paper addresses an inverse problem involving the simultaneous identification of the fractional order, potential coefficient, initial value and source term in a time-fractional Cattaneo equation. Utilizing the method of Laplace transformation, we demonstrate that the multiple unknowns can be uniquely determined from observational data collected at two boundary points.

With the modeling of the biquaternion algebra in multidimensional signal processing, it has become possible to address issues such as data separation, denoising, and anomaly detection. This paper investigates the singular value decomposition of biquaternion matrices (SVDBQ), establishing an SVDBQ theorem that ensures unitary matrices formed by the left and right singular vectors, while also introducing

This paper firstly studies exact solutions to the atmospheric equations of motion in the f-plane and β-plane approximations while considering centripetal forces. The obtained solutions are shown in Lagrangian coordinates. Additionally, we derive the dispersion relations and perform a qualitative analysis of density, pressure, and vorticity.

This paper considers the global asymptotic stability of a model with epidemic model with high risk and vaccinated class, and extends the related methods to two case of reaction–diffusion equations. The results presented here generalize those from Movahedi (2024).

In the non-local reaction–diffusion equation, the form of the kernel function has an important effect on the dynamics of the equation. In this paper, we study the spatiotemporal dynamics of a class of non-local reaction–diffusion equation where the non-locality is described by the top-hat function with the perceptual radius. The perceptual radius establishes a bridge between the local equation and

Dual quaternion matrix decompositions have played a crucial role in fields such as formation control and image processing in recent years. In this paper, we present an eigenvalue decomposition algorithm for dual quaternion Hermitian matrices. The proposed algorithm is founded on the structure-preserving tridiagonalization of the dual matrix representation of dual quaternion Hermitian matrices through

In this paper, we study the existence of infinitely many positive periodic solutions to a class of second order functional differential equations which cannot be applied directly to the fixed point theorem in cone. With suitable deformations, we construct the operator whose fixed point is closely related to the periodic solution of the original equation and show that the problem has infinitely many

This paper is concerned with the HLS lower critical Choquard equation with inverse-power potential and square-root-type nonlinearity. After giving a novel proof of subadditivity of the constraint minimizing problem and establishing the Brézis–Lieb lemma for square-root-type nonlinearity, we not only prove the existence of normalized solutions but also give its energy estimate.

Using the method of characteristics and defining one auxiliary function, we prove the existence of global attractor for a general age-structured HIV model, which can be used to solve the uniformly persistence problem in the Kumar and Abbas (2022).

Non-uniform distributions of various biological factors can be essential for tissue growth control, morphogenesis or tumor growth. The first model describing the emergence of such distributions was suggested by A. Turing for the explanation of cell differentiation in a growing embryo. In this model, diffusion-driven instability of the homogeneous in space solution appears due to the interaction of

We study the existence of infinitely many sign-changing solutions to the following nonlinear scalar Schrödinger equation −Δu+λu=f(u)inRNwith a prescribed mass ∫RN|u|2dx=a. Here f∈C1(R,R), a>0 is a given constant and λ∈R is an unknown parameter appearing as a Lagrange multiplier. Jeanjean and Lu have established the existence of infinitely many sign-changing normalized solutions in [Nonlinearity 32

This paper explores the spatiotemporal dynamics of a three-component predator–prey model with prey-taxis. We mainly show the existence of the steady state bifurcation and the bifurcating solution. Of most interesting discovery is that only the repulsive type prey-taxis could establish the existence of the steady state bifurcation and spatial pattern formation of the system. There are no steady state

We investigate a local modification of a variable-order time-fractional wave equation, which models the vibrations of a viscoelastic bar along its longitudinal axis. Under suitable assumptions regarding the variable order at t=0, we prove that the original model is equivalent to a multiscale wave equation. Furthermore, we analyze the well-posedness of its weak solution. Numerical experiments are implemented

This paper is concerned with the retarded reaction–diffusion equation ∂tu−Δu=f(u)+G(t,ut)+h(x) in a bounded domain. We allow both the nonlinear terms f and G to be supercritical, in which case the solutions may blow up in finite time, making it difficult to obtain global estimates. Here we employ some appropriate structure conditions to deal with this problem. In particular, we establish detailed global

Density functional theory calculations involve complex nonlinear models that require iterative algorithms to obtain approximate solutions. The number of iterations directly affects the computational efficiency of the iterative algorithms. However, for complex molecular systems, classical self-consistent field iterations either do not converge, or converge slowly. To improve the efficiency of self-consistent

In this paper, we present a quadrature formula on triangular domains based on a set of simplex points. This formula is defined via the constrained mock-Waldron least squares approximation. Numerical experiments validate the effectiveness of the proposed method.

The primary purpose of this work is to consider a (2+1)-dimensional generalized KP equation via ∂̄-dressing method. Using the Fourier transform and Fourier inverse transform, we give the expression of the Green function for spatial spectral problem. Then, we choose two linear independent eigenfunctions and calculate the ∂̄ derivative, a ∂̄ problem arises naturally. Based on the symmetry of the Green

We are interested in the following problem Δ2u+λu=g(u)inRN,∫RN|u|2dx=c,where N≥5, c>0 and λ∈R appears as a Lagrange multiplier. When g(u) satisfies a class of general mass supercritical conditions, we introduce one more constraint and consider the corresponding infimum. After showing that the new constraint is natural and verifying the compactness of the minimizing sequence, we obtain the existence

Existing studies have shown that asymptomatic cases might be related to short-term immunity on a timescale of weeks to months, which could have a significant impact on cholera epidemic transmission. In this paper, we are concerned with the global dynamical behavior of a cholera model with temporary immunity, which is characterized by discrete delay. The basic reproduction number of the model and the

In this paper, we establish the global stability of the spatially nonhomogeneous steady state solution of a reaction diffusion equation with nonlocal delay under the Dirichlet boundary condition. To achieve this, we obtain the global existence and nonnegativity of solutions and give an extensive study on the properties of omega limit sets.

The classical Haddock conjecture is extended to a kind of non-autonomous neutral functional differential equations (NFDEs) incorporating time-varying delays in this paper. By using the Dini derivative theory and inequality analyses, without requiring the strictly monotonically increasing property of the delay feedback function, it is demonstrated that every solution of the considered NFDEs is bounded

Alzheimer’s disease (AD) is characterized by the progressive deposition of β-amyloid (Aβ) plaques in the brain, where the Aβ oligomers have been confirmed to produce the critical cytotoxicity during the disease process. In this study, a model is established to describe the effect of Aβ oligomers on the interplay between Aβ and Ca2＋. Mathematical analysis demonstrates the existence and stability of

In this paper, we consider a class of Fisher–KPP equations with delays appearing in both diffusion and reaction terms. By employing some differential inequality analyses, we prove that the delayed Fisher–KPP equation possesses a pair of quasi-upper and quasi-lower solutions which have absolutely continuous derivatives. Based on this, we apply the monotone iteration method and the Perron’s theorem to

We prove a decay estimate for solutions to a transmission problem for wave equations with different propagation speeds in an infinite waveguide. The problem represents the wave propagation in unbounded and layered composite materials in which different properties are given. It is a new composition of a waveguide problem and a transmission problem, motivated by a unit cell model for CFRP. The proof

By complexifying the independent variables of the Kaup–Kuperschmidt (KK) equation, we derive the 4+2 integrable extension of the KK equation and its Lax pair. The construction of the associated nonlinear Fourier transform pair comprising both direct and inverse transforms is accomplished by conducting a spectral analysis of the t-independent part of the Lax pair. In the final section, the nonlinear

In this paper, we introduce novel class of Legendre spectral volume (LSV) methods for solving Allen–Cahn equations. Each spectral volume (SV) is refined with k Gauss–Legendre points to define an arbitrary order control volume (CV). Moreover, the second derivative is handled using the direct discontinuous Galerkin (DDG) approach. Furthermore, four numerical experiments are detailed including 1D and

This paper presents a novel product integration method that provides an appropriate numerical solution for nonlinear weakly singular Volterra integral equations (WSVIEs). Extensive research in the literature has focused on studying the existence and uniqueness of solutions to these equations. However, when solving the WSVIEs, the solution may exhibit a singular behavior near the initial point of the

We consider this equation σk(Au)=up−n+2n−2k,where n≥3 and p∈nn−2,n+2n−2. Here σk denotes the kth elementary symmetric function of the eigenvalues of Au, and Au=−2n−2u−n+2n−2D2u+2n(n−2)2u−2nn−2∇u⊗∇u−2(n−2)2u−2nn−2|∇u|2I,where ∇u denotes the gradient of u and D2u denotes the Hessian of u. This equation has fruitful backgrounds in geometry and physics. We then obtain Schoen’s Harnack type inequality in

In this paper, a linearized Euler Galerkin scheme is studied and the unconditionally optimal error estimate in L2-norm is obtained for Schrödinger equation with cubic nonlinearity without any time-step restriction. The key to the analysis is to bound the H1-norm between the numerical solution and the Ritz projection of the exact solution by mathematical induction for two cases rather than the error

In this paper, we show that collocation matrices of the Cauchy–Bernstein basis can be decomposed as products of a Cauchy–Vandermonde matrix and a block diagonal matrix. A useful application of this result is that the explicit expression of the determinant for the collocation matrices is presented. Consequently, an algorithm is provided to accurately compute the determinants. Numerical experiments confirm

This paper investigates a fully parallel decoupled approach of the discrete stabilized finite element method for the time-dependent Stokes–Darcy problem. By introducing an auxiliary function, we rigorously demonstrate that the parallel algorithm is unconditionally stable.

The present work concerns with the higher dimensional Rayleigh–Plesset equation for describing the nonlinear dynamics of gas-filled hyper-spherical bubbles in an ideal fluid. A strict qualitative analysis is made by means of the bifurcation theory of dynamic system, indicating that the bubble oscillation type is periodic. An analytical approach based on elliptic function is suggested to construct parametric

In this paper, we consider the following elliptic problem −Δu=f(x,u),inΩ,u=0,on∂Ω,(P) where the nonlinearity f satisfies the Ambrosetti–Rabinowitz condition. Using an additional growth condition of f at a bounded region, we can obtain five nontrivial solutions of (P) by applying homological linking arguments and Morse theory.

We consider the following fractional Schrödinger–Poisson system (−Δ)su+V(x)u+ϕu=f(u),inR3,(−Δ)sϕ=u2,inR3,where s∈(12,1) is a fixed constant, f is continuous, sublinear at the origin and subcritical at infinity. Applying the Clark’s theorem and truncation method, we can obtain a sequence of negative energy solutions.

In this article, we propose a novel second-order shifted convolution quadrature (SCQ) formula including both a shifted parameter θ and a new variable parameter δ. We prove the second-order truncation error of the novel formula for the time-fractional derivative, and derive the nonnegative property of the formula’s weights. Combining the novel formula with the finite element method, we develop a high

Explicit, conformal symplectic, exponential time differencing (ETD) methods have numerous advantages over other well-known and commonly used methods, including structure-preservation, high stability, ease of implementation, and computational efficiency. Such methods are constructed with second and fourth order accuracy through composition techniques using a simple first order scheme. For modeling Josephson

This article is concerned with the uniqueness of simultaneously determining the fractional order of the derivative in time, diffusion coefficient, and Robin coefficient, in one-dimensional time-fractional diffusion equations with derivative order α∈(0,1) and non-zero boundary conditions. The measurement data, which is the solution to the initial–boundary value problem, is observed at a single boundary

The building of the national communication infrastructure and growing demand for data traffic both depend heavily on the advancement of optical soliton communication technology. In particular, by studying the interaction of optical solitons, some methods of controlling optical solitons can be explored to design more stable and efficient optical communication systems. In this paper, the interactions

We prove that, under generic conditions, the covariance matrix of a 3D point cloud with rotational symmetry has a simple eigenvalue, whose associated eigenvector provides the direction of the axis of rotation, and a double eigenvalue. The direction of the axis of rotation can also be computed from higher order tensors related to the point cloud, which is useful in pathological cases. This leads to

We consider the existence of positive solutions for following Kirchhoff equation −a+b∫Ω|∇u|2dxΔu+u=|u|p−2uinΩ,u=0on∂Ω, where a,b>0, Ω={x∈RN:|x|>1} is the exterior of the unit ball in RN and N≥2. It is well-known that if 4

Based on the phase-field theory, a new wetting boundary condition (WBC) scheme is proposed to describe the fluid–solid interaction of binary fluids. Different from the common linear, cubic and sine form of surface energy wetting conditions, we adopt a mixed cubic and sine form of free energy in the present scheme. Two conditions are given to ensure that the spurious film at the solid surface disappears

In this study we consider a spline-based collocation method to approximate the solution of fractional convection–diffusion equations which include fractional derivatives in both space and time. This kind of fractional differential equations are valuable for modeling various real-world phenomena across different scientific disciplines such as finance, physics, biology and engineering.

In this paper, we study a class of Schrödinger–Bopp–Podolsky systems with a negative potential V(x) in R3. By using Mountain-Pass argument and detailed analysis of the energy level value, we obtain a normalized solution with positive energy under suitable assumptions on V(x). Moreover, we also prove that there is no normalized solutions with negative energy.

Under investigated in this paper is a Lakshmanan-Porsezian-Daniel equation that describes the nonlinear spin excitations in a (1+1)-dimensional isotropic biquadratic Heisenberg ferromagnetic spin chain with the octupole-dipole interaction or the propagation of the ultrashort pulses in a long-distance and high-speed optical fiber transmission system. Under certain parameter conditions, we simultaneously

The paper is concerned with the three-dimensional micropolar fluid equations. By using the successive approximation and Littlewood–Paley theory, we prove existence and uniqueness of time-periodic mild solutions of the three-dimensional micropolar fluid equations with external forces in Besov spaces.

A time-delayed dengue virus transmission model has been developed, which takes into account vaccination failure and the presence of exposed mosquitoes. This model also incorporates the survival probability of infected individuals during the incubation period to provide a clearer understanding of how latency affects the control reproduction number Rc. Furthermore, by employing the Lyapunov functional

In this paper, we propose high asymptotic order numerical methods for solving highly oscillatory second order ODEs with large initial data, where the total energy of the system becomes unbounded as the oscillation frequency grows. The existing asymptotic-numerical solvers are especially designed for the classical energy bounded oscillatory equations, offering no insight into their performance with

We establish uniform error bounds of the L1 discretization of the Caputo derivative of Hölder continuous functions. The result can be understood as: error = degree of smoothness - order of the derivative. We present an elementary proof and illustrate its optimality with numerical examples.

Several novel rational solutions with nonzero boundary condition for the Kundu equation, which is an important physical model, are derived using the technique of generalized Darboux transformation. It is the first time that a systemic analysis has been conducted on such rational solutions for the Kundu equation. For the 1-order rational solutions with nonzero boundary conditions, our findings reveal

A self-interacting dynamics that mimics the standard Consensus-Based Optimization (CBO) model is introduced. This single-particle dynamics is shown to converge to a unique invariant measure that approximates the global minimum of a given function. As an application, its connection to CBO with Personal Best introduced by C. Totzeck and M.-T. Wolfram (Math. Biosci. Eng., 2020) has been established.

In this paper, we focus on a class of non-autonomous Kirchhoff equations, that is, −(a+b∫R3|∇u|2dx)Δu−λu=K(x)|u|p−2u in R3, where a,b>0 are constants, λ∈R is unknown and appears as a Lagrange multiplier, 2

In this paper we consider a chemotaxis system with signal consumption and degenerate diffusion of the form ut=∇⋅(D(u)∇u−uS(u)∇v)+f(u,v),vt=Δv−uv,in a bounded domain Ω⊂RN with smooth boundary subjected to no-flux and homogeneous Neumann boundary conditions. Herein, the diffusion coefficient D∈C0([0,∞))∩C2((0,∞)) is assumed to satisfy D(0)=0, D(s)>0 on (0,∞), D′(s)≥0 on (0,∞) and that there are s0>0

Babenko’s equation describes traveling water waves in holomorphic coordinates. It has been used in the past to obtain properties of Stokes waves with smooth profiles analytically and numerically. We show in the deep-water limit that properties of Stokes waves with peaked profiles can also be recovered from the same Babenko’s equation. In order to develop the local analysis of singularities, we rewrite