This paper proves the existence of nontrivial solution for two classes of quasilinear systems of the type { − ∆ Φ 1 u = F u ( x , u , v )+ λ R u ( x , u , v ) in Ω − ∆ Φ 2 v = − F v ( x , u , v ) − λ R v ( x , u , v ) in Ω u = v = 0 on ∂ Ω where λ>0 is a parameter, Ω is a bounded domain in R N ( N≥2) with smooth boundary ∂Ω. The first class we drop the ∆ 2 -condition of the functions Φ ̵̃ i ( i=1 ,2) and assume that F has a double criticality. For this class, we use a linking theorem without the Palais-Smale condition for locally Lipschitz functionals combined with a concentration–compactness lemma for nonreflexive Orlicz-Sobolev space. The second class, we relax the ∆ 2 -condition of the functions Φ i ( i=1 ,2). For this class, we consider F=0 and λ=1 and obtain the proof based on a saddle-point theorem of Rabinowitz without the Palais-Smale condition for functionals Fréchet differentiable combined with some properties of the weak ∗ topology.

In this paper, we use a stochastic partial differential equation (SPDE) as a model for the density of a population under the influence of random external forces/stimuli given by the environment. We study statistical properties for two crucial parameters of the SPDE that describe the dynamic of the system. To do that we use the Galerkin projection to transform the problem, passing from the SPDE to a system of independent SDEs; in this manner, we are able to find the Maximum likelihood estimator of the parameters. We validate the method by using simulations of the SDEs. We prove consistency and asymptotic normality of the estimators; the latter is showed using the Malliavin-Stein method. We illustrate our results with numerical experiments.

In comparison to quaternions, split quaternions exhibit a more intricate algebraic structure, characterized by the presence of nontrivial zero factors. Furthermore, in various fields such as geometry and electromagnetism, split quaternions serve as more convenient research tools than quaternions. This paper aims to establish the necessary and sufficient conditions for the existence of (anti-) η-Hermitian solutions to a novel system of matrix equations formulated over split quaternions. We also provide the general expression of such solutions when the system is solvable, along with the least squares (anti-) η-Hermitian solution in the case of inconsistency. To illustrate the key findings of this paper, numerical examples are presented.

In the previous work [6], we classified the solutions to a family of systems of Klein-Gordon equations with non-negative energy below the ground state into two parts: one blows up in finite time while the other extends to a global solution. In the present work, we strengthen this result, showing that these global solutions are indeed scattering in the energy space. Here we adapted Kenig-Merle’s concentration-compactness approach to the system.

We investigate the motion of closed, smooth non-self-intersecting curves that evolve in space R 3 . The geometric evolutionary equation for the evolution of the curve is accompanied by a parabolic equation for the scalar quantity evaluated over the evolving curve. We apply the direct Lagrangian approach to describe the geometric flow of 3D curves resulting in a system of degenerate parabolic equations. We prove the local existence and uniqueness of classical Hölder smooth solutions to the governing system of nonlinear parabolic equations. A numerical discretization scheme has been constructed using the method of flowing finite volumes. We present several numerical examples of the evolution of curves in 3D with a scalar quantity. In this paper, we analyze the flow of curves with no torsion evolving in rotating and parallel planes. Next, we present examples of the evolution of curves with initially knotted and unknotted curves.

Marriage and divorce are pivotal aspects of societal dynamics, influencing family structures and individual well-being. This research introduces an advanced mathematical model to comprehend the intricate dynamics of marital relationships. Unlike conventional models, which often overlook the unpredictability inherent in human interactions, our model integrates stochastic elements and real-world factors such as mental distress and family issues. Mathematical modeling unveils the dynamics of marriage and divorce, guiding interventions to foster relationship stability and societal cohesion amidst multifaceted influences. However, the deterministic approach is limited due to the randomness of virus behavior. To address this issue, a more realistic stochastic model for computer viruses is considered, which can better capture the unpredictability of virus spread and establish effective preventive measures. The positivity and uniqueness of the model are checked, and the extinction and persistence theorems are proved using Ito's method. Numerical simulation shows that when the value of increases the divorces increases in the population while by increasing the value of the broken individuals recovered from their conflict and came back to its married lifestyle.

In this paper, we study the initial-boundary value problem of three-dimensional viscous, compressible, and heat conductive magnetohydrodynamic equations. Local existence and uniqueness of strong solutions is established with any such initial data that the initial compatibility conditions do not be required. The analysis is based on some suitable prior estimates for the strong coupling term u·∇ H and strong nonlinear term curl H × H . Our proof of the existence and uniqueness of solutions is in the Lagrangian coordinates first and then transformed back to the Euler coordinates.

The goal of this note is to study the spectrum of a self-adjoint convolution operator in L 2 ( R d ) with an integrable kernel that is perturbed by an essentially bounded real-valued potential tending to zero at infinity. We show that the essential spectrum of such operator is the union of the spectrum of the convolution operator and of the essential range of the potential. Then we provide several sufficient conditions for the existence of a countable sequence of discrete eigenvalues. For operators having non-connected essential spectrum we give sufficient conditions for the existence of discrete eigenvalues in the corresponding spectral gaps.

In this article, a two prey-one predator model in which prey and predator disperse simultaneously in a heterogeneous environment with n patches is proposed and analyzed. We prove that the solution of the system is positive and uniformly ultimately bounded. Meanwhile, we use the monotonic theory of spectral bounds to investigate the effect of the dispersal rate on population dynamics. To be precise, we discuss the stability behaviour for the trivial equilibrium and semitrivial equilibrium as well as the uniform persistence of the system. Furthermore, we prove the global asymptotic stability of the positive equilibrium by constructing a global Lyapunov function which applies the results from graph theory. Some numerical simulations are provided to show the effectiveness of the theoretical results.

The paper aims to apply the trigintaduonion spaces to explore the physical properties of four interactions simultaneously, including the electromagnetic fields, gravitational fields, weak nuclear fields, and strong nuclear fields. J. C. Maxwell first applied the algebra of quaternions to study the physical properties of electromagnetic fields. It inspired some subsequent scholars to introduce the quaternions, octonions, sedenions, and trigintaduonions to research the electromagnetic fields, gravitational fields, weak nuclear fields, strong nuclear fields, quantum mechanics, gauge fields, and curved spaces and so forth. The algebra of trigintaduonions is able to discuss the physical quantities of four interactions, including the field potential, field strong, field source, linear momentum, angular momentum, torque, and force. In the field theories described with the algebra of trigintaduonions, the weak nuclear field is composed of three types of fundamental fields. These three fundamental fields, related to weak nuclear fields, can describe the physical properties of weak nuclear fields collectively. This is consistent with the conclusion of the electroweak theory. Meanwhile the strong nuclear field consists of three types of fundamental fields. These three fundamental fields relevant to strong nuclear fields may investigate the physical properties of strong nuclear fields mutually. It is coincident with the deduction of quark theory. According to the properties of trigintaduonions, one can deduce the Yang-Mills equation related to the gauge fields. It means that the electromagnetic field occupies a quaternion space. The gravitational field owns one different quaternion space. The weak nuclear fields occupy three mutually independent quaternion spaces. The properties of weak nuclear fields are different from those of electromagnetic fields or gravitational fields. According to the multiplicative closure, the strong nuclear fields also own three quaternion spaces independent of each other. These explorations further deepen the understanding of the physical properties of weak and strong nuclear fields.

In this paper, we study the stability and convergence of a conservative Crank-Nicolson finite difference scheme applied to the Korteweg-De Vries (KdV) equation endowed with initial data. We design a three-point average scheme associated to the convective term and the dispersion term is discretized by certain discrete operators along with the Crank-Nicolson scheme for the temporal discretization to establish that the proposed scheme is L 2 -conservative. The convergence analysis reveals that utilizing inherent Kato’s local smoothing effect, the proposed scheme converges to a classical solution for sufficiently regular initial data u 0 ∈ H 3 ( R ) and to a weak solution in L 2 ( 0 , T ; L loc 2 ( R ) ) for non-smooth initial data u 0 ∈ L 2 ( R ) . Optimal convergence rates in both space and time for the devised scheme are derived. The theoretical results are justified through several numerical illustrations.

This paper presents the dynamic response of cantilever beam on fractional-order nonlinear viscoelastic foundation subjected to Gaussian white noise. The control equations of the cantilever beam on viscoelastic foundation are established using Hamilton's principle. The problem is solved using the shifted Chebyshev polynomial algorithm, and the control equations are transformed into a system of nonlinear algebraic equations. Numerical examples analyse the correction error and the second norm error, confirming the effectiveness and accuracy of the algorithm in solving such problems. Furthermore, the algorithm's robustness was verified by comparing the responses of Gaussian white noise and non-Gaussian white noise cantilever beam on the viscoelastic foundation. The study examined the impact of various loads and parameters on the cantilever beam, as well as the effect of different harmonic loads on its stress. The research results are in line with the existing literature. These studies offer valuable guidance for practical engineering and enhance comprehension of the dynamic response of cantilevers on foundation in complex environments.

In this paper, we consider a predator-prey model with Cosner type functional response and combined harvesting. First, we explore the existence and stability of the equilibria. Then using the center manifold theorem and normal form theory, we investigate codimension one and codimension two bifurcations of the model. The analysis shows that the system has a variety of bifurcation phenomena including transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation, Bogdanov-Takens bifurcation and homoclinic bifurcation. Our findings indicate that the dynamics with harvesting are significantly richer than the system without harvesting. Finally, numerical simulations are provided to support the analytical results.

We address the problem of existence, uniqueness and approximation of the solution of a large class of fractional integral differential equations with boundary conditions. Based on fixed point techniques with Boyd-Wong type conditions, the existence and uniqueness of the solution is investigated. This, together with the use of certain types of biorthogonal systems, allows us to propose a method to approximate the solution which is tested with several examples.

In this paper, we develop and analyze a mathematical model for spreading malaria, including treatment with Transmission Blocking Drugs (TBDs). The paper’s main aim is to demonstrate the impact the chosen model for demographic growth has on the disease’s transmission and the effect of its treatment with TBDs. We calculate the model’s control reproduction number and equilibria, and perform a global stability analysis of the disease-free equilibrium point. The mathematical analysis reveals that, depending on the model’s demography, the model can exhibit forward, backward and even some unconventional types of bifurcation, where disease elimination can occur for both small and large values of the reproduction number. We also conduct a numerical analysis to explore the short-time behavior of the model. A key finding is that for one type of demographic growth, the population experienced a significantly higher disease burden than the others, and when exposed to high levels of treatment with TBDs, only this population succeeded in effectively eliminating the disease within a reasonable timeframe.

This paper proposes a dynamic programming approach targeted to solve a natural resource problem of water storage management for irrigation in an environmentally and socially sustainable way. The problem we examine in our formulation, which deals with the control of water storage in tanks, incorporates assumptions that are milder compared to those previously discussed in the literature concerning similar approaches. Specifically, we consider a time periodic optimal control problem whose performance functional to be maximized merely satisfies a contraction assumption (in the sense of Boyd and Wong) weaker than the one usually considered in the pertinent literature. By using an appropriate fixed point theorem, a time periodic value function can be constructed to enable the definition of optimal feedback control strategies. After showing the insufficiency of Banach contraction frameworks, and proving the underlying auxiliary mathematical results, we show our main result under conditions that are weaker than the previous related ones. Simulation results illustrate the performance of our approach.

This paper discusses the Maxwell system of electrodynamics in the context of distributions. It is used to establish boundary conditions for fields at the interface when the charge and current densities are concentrated measures on the interface. The paper derives the generalized Snell's law from this analysis.

We consider an alternative approach to investigate three-dimensional exterior mixed boundary value problems (BVP) for the steady state oscillation equations of the elasticity theory for isotropic bodies. The unbounded domain occupied by an elastic body, Ω − ⊂ R 3 , has a compact boundary surface S = ∂ Ω − , which is divided into two disjoint parts, the Dirichlet part S D and the Neumann part S N , where the displacement vector (the Dirichlet type condition) and the stress vector (the Neumann type condition) are prescribed respectively. Our new approach is based on the classical potential method and has several essential advantages compared with the existing approaches. We look for a solution to the mixed boundary value problem in the form of a linear combination of the single layer and double layer potentials with densities supported on the Dirichlet and Neumann parts of the boundary respectively. This approach reduces the mixed BVP under consideration to a system of boundary integral equations, which contain neither extensions of the Dirichlet or Neumann data nor the Steklov-Poincaré type operator involving the inverse of a special boundary integral operator, which is not available explicitly for arbitrary boundary surface. Moreover, the right-hand sides of the resulting boundary integral equations system are vector-functions coinciding with the given Dirichlet and Neumann data of the problem in question. We show that the corresponding matrix integral operator is bounded and coercive in the appropriate L 2 -based Bessel potential spaces. Consequently, the operator is invertible, which implies unconditional unique solvability of the mixed BVP in the class of vector-functions belonging to the Sobolev space [ W 2 , loc 1 ( Ω − ) ] 3 and satisfying the Sommerfeld-Kupradze radiation conditions at infinity. We also show that the obtained matrix boundary integral operator is invertible in the L p -based Besov spaces and prove that under appropriate boundary data a solution to the mixed BVP possesses C α -Hölder continuity property in the closed domain Ω − ‾ with α = 1 2 − ε , where ε>0 is an arbitrarily small number.