Search

This paper is concerned with the well‐posedness of a diffusion–reaction system for a susceptible‐exposed‐infected‐recovered (SEIR) mathematical model. This model is written in terms of four nonlinear partial differential equations with nonlinear diffusions, depending on the total amount of the SEIR populations. The model aims at describing the spatio‐temporal spread of the COVID‐19 pandemic and is a variation of the one recently introduced, discussed, and tested in a paper by Viguerie et al (2020). Here, we deal with the mathematical analysis of the resulting Cauchy–Neumann problem: The existence of solutions is proved in a rather general setting, and a suitable time discretization procedure is employed. It is worth mentioning that the uniform boundedness of the discrete solution is shown by carefully exploiting the structure of the system. Uniform estimates and passage to the limit with respect to the time step allow to complete the existence proof. Then, two uniqueness theorems are offered, one in the case of a constant diffusion coefficient and the other for more regular data, in combination with a regularity result for the solutions. [ABSTRACT FROM AUTHOR]

The present paper deals with mathematical modeling of a fishery resource system in an aquatic atmosphere consisting of two zones: a free fishing zone and a reserved zone, where fishing is strictly prohibited. The dynamics of the system is studied in the presence of bird predator. In this paper, the quadratic harvesting of the fish species in unreserved zone has been considered whereas bird predator species is subjected to linear harvesting. All the possible biological and bionomic equilibria of the system are studied extensively for their existence, local as well as global stability. We have found various ranges of harvesting parameter for maintaining the Sustainability in the proposed ecosystem. Optimal harvesting is discussed using Pontryagin's maximum principle. Numerical simulations are done to support the theoretical results obtained. [ABSTRACT FROM AUTHOR]

Recently, an infinitesimal approach for finding reciprocal transformations has been proposed. The method uses the group analysis approach and consists of similar steps as for finding an equivalence group of transformations. The new method provides a systematic tool for finding classes of reciprocal transformations (group of reciprocal transformations). Similar to the classical group analysis, this approach can be also applied for finding all reciprocal transformations (not only composing a group) of the equations under study. The present paper provides this algorithm. As an illustration, the method is applied to the two‐dimensional stationary gas dynamics equations. Equivalence group, group of reciprocal transformations, and completeness of all discrete reciprocal transformations are presented in the paper. The results are stated in form of a theorem. [ABSTRACT FROM AUTHOR]

In this paper, we consider the asymptotic behavior of the solutions of the quadratic wave equations with perturbed oscillating coefficients which are very important mathematical models in describing the oscillation of particles imposed with time‐dependent electromagnetic fields. By the application of microlocal analysis and stochastic analysis, this paper makes a delicate classification of various types of time‐dependent oscillating coefficients on the principal quantum harmonic oscillator part and explores the corresponding regularity behavior and L2$$ {L}^2 $$‐estimates of the solution of the equation. It is interesting to find out that weak or mild oscillations will not cause any energy growth, while regular and strong oscillations will lead to polynomial or exponential growth of the energy. This is meaningful to help us explore the energy conversion mechanism in the electromagnetic field. Furthermore, in order to demonstrate the optimality of the L2$$ {L}^2 $$‐estimates, typical counterexamples with periodic coefficients will be constructed to show the lower bound of growth rate by the application of harmonic analysis and instability arguments. Compared with formal literatures focusing on the constant coefficients, current results in this paper reveal the essential relation between oscillation types and energy growth. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove the existence of weak solutions to thermo‐visco‐elastic system of equations. In considered problem, the temperature term is included in the elastic constitutive equations (generalized Hooke's law) as well as in describing the evolution of visco‐elastic strain. The main idea of presented proof is to analyze reformulated model, in which visco‐elastic strain (internal variable) is eliminated. For such system of equations, we propose a two‐level Galerkin approximation, which allows us to show a non‐negativity of temperature during the entire deformation process. In order to characterize the weak limits in the nonlinearities occurring in the system at different levels of approximation, the following tools are used: Truncation methods for heat equation proposed in the paper of D. Blanchard [Nonlinear Analysis, 21(10):725‐743, 1993], modified Boccardo‐Gallouët's approach, and Minty‐Browder trick. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem of the d‐dimensional magneto‐micropolar equations (d=2$$ d=2 $$ or d=3$$ d=3 $$) with general fractional dissipation. The aim of this paper is to obtain the existence and uniqueness of solutions in the weakest possible inhomogeneous Besov spaces. Using the technical tools of Litttlewood–Paley decomposition and Besov spaces theory, we obtain the local existence in the functional setting of inhomogeneous Besov spaces. Furthermore, such solutions are unique only in 2D case. [ABSTRACT FROM AUTHOR]

In this paper, using the Faà di Bruno formula and some properties of the Bell polynomials of the second kind, we obtain a new explicit formula for the generalized Humbert–Hermite polynomials. We provide determinantal representations for the ratio of two differentiable functions. We obtain a recursive relation for the generalized Humbert–Hermite polynomials. As a practice, we derive an alternative recursive relation for generalized Humbert–Hermite polynomials via the Hessenberg determinant. Finally, we derive several families of multilinear and multilateral generating functions for the generalized Humbert–Hermite‐type polynomials and other polynomials which are mentioned in this paper. Our results also include many well‐known polynomials in the literature. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the influence of time delay effect and steady flow control strategy on traffic flow. Consider that the time delay in sensing flux information during the operation of traffic system is inevitable. Meanwhile, a new method is applied to model and analyse traffic system. A novel discrete‐time lattice hydrodynamic model considering time delay is introduced in this paper. Based on the proposed model with constant and varying time‐delay, the asymptotic stabilities are investigated and corresponding stable conditions are derived. The traffic flow evolution trend with and without time delay is conducted and compared by numerical simulations. Also, under the influence of different control gain conditions, density wave characteristics for traffic flow system with time delay are intuitively shown. The results indicate that time delay effect in sensing density and flux information can aggravate traffic congestion, and traffic congestion will be effectively relieved by considering steady flow control signal. [ABSTRACT FROM AUTHOR]

This paper concerns the mixed H∞$$ {H}_{\infty } $$ and passivity‐based filtering problem for networked nonlinear systems with randomly occurring parameter uncertainties, multipath data packet dropouts, time‐varying delay, and quantization effects. The discrete‐time nonlinear plant considered in this paper is modeled as a Takagi–Sugeno fuzzy system with plant rules. The data missing phenomenon is considered in both measurement and performance output signals due to the unreliable nature of communication links. Further, to reduce the network bandwidth utilization, the measurement quantization is also employed for both the outputs before transmitting through the communication channel. Also, additive filter gain fluctuation is taken into account to reflect the imprecision in filter implementation. Stochastic variables obeying the Bernoulli distribution are incorporated in the model description to characterize the random occurrence of parameter uncertainties and data packet dropouts. A set of sufficient conditions assuring the stochastic stability of the filtering error system with mixed H∞$$ {H}_{\infty } $$ and passivity performance is derived according to Lyapunov stability theory. Finally, three numerical examples including mass–spring–damper and tunnel diode circuit model are established to establish the validity of the developed filter design algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we aim to establish the uniform stability criteria for fractional‐order time‐delayed gene regulatory networks with leakage delays (FOTDGRNL). First, we establish the existence and uniqueness of the considered systems by using the Banach fixed point theorem. Second, the delay‐dependent uniform stability and robust uniform stability of FOFGRNLT are investigated with the help of certain analysis techniques depending on equivalent norm techniques. Finally, the paper comes up with two numerical examples to justify the applicability of our theoretical results. [ABSTRACT FROM AUTHOR]

During past decades, the study of the interaction between predator and prey species has become one of the most exciting topics in computational biology and mathematical ecology. In this paper, we aim to investigate the stability of a diseased model of susceptible, infected prey and predators around an internal steady state. To this end, the fractional derivatives based on the Mittag‐Leffler kernels in the Liouville‐Caputo concept has been taken into consideration. The existence and uniqueness of the acquired solutions to the model are also studied in this paper. In order to investigate the effects of the fractional‐order along with other existing parameters in the model, several possible scenarios have been examined. As it is seen in the proposed graphical simulations, the employed fractional operator is capable of capturing all anticipated theoretical features of the model. The numerical technique employed in this contribution is precise and efficient and can be easily adopted to investigate many fractional‐order models in biology. It is found that new proposed operators of fractional‐order can describe the real‐world phenomena even better than integer‐order differential equations because of their memory‐related properties. [ABSTRACT FROM AUTHOR]

We consider the incremental subgradient algorithm employing dynamic step sizes for minimizing the sum of a large number of component convex functions. The dynamic step size rule was firstly introduced by Goffin and Kiwiel [Math. Program., 1999, 85(1): 207‐211] for the subgradient algorithm, soon later, for the incremental subgradient algorithm by Nedic and Bertsekas in [SIAM J. Optim., 2001, 12(1): 109‐138]. It was observed experimentally that the incremental approach has been very successful in solving large separable optimizations and that the dynamic step sizes generally have better computational performance than others in the literature. In the present paper, we propose two modified dynamic step size rules for the incremental subgradient algorithm and analyse the convergence and complexity properties of them. At last, the assignment problem is considered and the incremental subgradient algorithms employing different kinds of dynamic step sizes are applied to solve the problem. The computational experiments show that the two modified ones converges dramatically faster and more stable than the corresponding one in [SIAM J. Optim., 2001, 12(1): 109‐138]. Particularly, for solving large separable convex optimizations, we strongly recommend the second one (see Algorithm 3.3 in the paper) since it has interesting computational performance and is the simplest one. [ABSTRACT FROM AUTHOR]

This paper is concerned with a mixed type differential operator Lu=kyuxx−uyy+byux+qyu,$$ Lu=k(y){u}_{xx}-{u}_{yy}+b(y){u}_x+q(y)u, $$which is initially defined with C0,π∞Ω‾$$ {C}_{0,\pi}^{\infty}\left(\overline{\Omega}\right) $$, where Ω‾={x,y:−π≤x≤π,−∞

In this paper, a class of fractional stochastic differential equations (SFDEs) with impulses is considered. By virtue of Mönch's fixed point theorem and Banach contraction principle, we explore the existence and uniqueness of solutions to the addressed system. Furthermore, with the aid of the Jensen inequality, Hölder inequality, Burkholder–Davis–Gundy inequality, Grönwall–Bellman inequality, and some novel assumptions, the averaging principle of our considered system is obtained. At the end of this paper, an example is provided to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper numerically investigates the effect of pulsating flow on the settling dynamics of rigid circular particles. This is an interdisciplinary subject and spans several areas ranging from mathematical and numerical modeling to fluid mechanics. For this purpose, pulsatile flow characteristics are embedded in the combination of the direct‐forcing immersed boundary method and the split‐forcing lattice Boltzmann method. Inter‐collision forces between the solid boundaries (particles and boundaries) and the added mass force due to acceleration are considered. Adequate verification tests are done to ensure the credibility of the findings. The critical parameters of pulsating flow, such as amplitude and frequency of pulsation, are investigated in detail. The paper especially puts emphasis on the interaction between particles and studies the well‐known drafting, kissing, and tumbling (DKT) phenomena. Two different scenarios are taken into account and also compared with the stationary flow. The first case is when the pulsating flow is in the direction of gravity (co‐flow), while in the latter, there is an opposing flow (counterflow). The sedimentation manners of 12 particles in a vertical channel are also presented. The findings shed light on the importance of pulsating flow and the extension of the proposed computational method for such problems. It is also revealed that pulsation and its variables can alter DKT by either postponing or speeding up the process. Also, in some cases, the cycle of DKT can be maintained incompletely, and particles would just stick together. The results can be useful for various engineering problems like filtration and particle sorting. [ABSTRACT FROM AUTHOR]

In this paper, we study a predator–prey system with the simplified Holling IV functional response and antipredator behavior such that the adult prey can attack vulnerable predators. The model has been investigated by Tang and Xiao, and the existence and stability of all possible equilibria are determined. In addition, they performed a bifurcation analysis and showed that the system undergoes a Codimension 2 Bogdanov–Takens bifurcation. In this paper, for the same model, we further show that the cusp‐type Bogdanov–Takens bifurcation can be of Codimension 3, which acts as an organizing center for the whole bifurcation set. In addition, we propose the existence of Hopf bifurcation of Codimension 2 and the coexistence of stable limit cycle and unstable limit cycle. In particular, we show that the antipredator behavior has great effect on the dynamics of the model, it may cause the predator population to extinct while the prey population will increase up to the carrying capacity. Numerical simulations including bifurcation diagrams and phase portraits are performed to illustrate and confirm the theoretical results. These results may enrich the dynamics of predator–prey systems. [ABSTRACT FROM AUTHOR]

This paper proposed a new class of stochastic cellular neural networks (SCNNs) with distributed time delays. It is driven by time‐varying Brownian motion. By constructing the Lyapunov functions with a more concise form and using the time‐varying Itô formula, combining some inequality techniques, this paper proved the almost surely exponential stability and the p$$ p $$‐th moment exponential stability of the solution of the SCNNs. The numerical example verifies the theoretical results of this paper. [ABSTRACT FROM AUTHOR]

The paper focuses on applying the octonions to explore the influence of the external torque on the angular momentum of fluid elements, revealing the interconnection of the external torque and the vortices of vortex streets. J. C. Maxwell was the first to introduce the quaternions to study the physical properties of electromagnetic fields. The contemporary scholars utilize the quaternions and octonions to investigate the electromagnetic theory, gravitational theory, quantum mechanics, special relativity, general relativity, curved spaces, and so forth. The paper adopts the octonions to describe the electromagnetic and gravitational theories, including the octonionic field potential, field strength, linear momentum, angular momentum, torque, and force. In case the octonion force is equal to zero, it is able to deduce eight independent equations, including the fluid continuity equation, current continuity equation, and force equilibrium equation. Especially, one of the eight independent equations will uncover the interrelation of the external torque and angular momentums of fluid elements. One of its inferences is that the direction, magnitude, and frequency of the external torque must impact the direction and curl of the angular momentum of fluid elements, altering the frequencies of Karman vortex streets within the fluids. It means that the external torque is interrelated with the velocity circulation, by means of the liquid viscosity. The external torque is able to exert an influence on the direction of downwash flows, improving the lift and drag characteristics generated by the fluids. [ABSTRACT FROM AUTHOR]

This paper is dedicated to obtaining closed‐form solutions of linear difference equations which are asymptotically close to the self‐adjoint Euler‐type difference equation. In this sense, the equation is related to the Euler–Cauchy differential equation y′′+λ/t2y=0$$ {y}^{\prime \prime }+\lambda /{t}^2y=0 $$. Throughout the paper, we consider a system of sequences which behave asymptotically as an iterated logarithm. [ABSTRACT FROM AUTHOR]

The paper deals with a new type of inverse spectral problems for second‐order quadratic differential pencils when one of the boundary conditions involves arbitrary entire functions of the spectral parameter. Although various aspects of the inverse spectral theory for the pencils have been of a special interest during the last decades, such settings were considered before only in the particular case of a Sturm–Liouville equation. We develop an approach covering also the quadratic dependence on the spectral parameter in the differential equation, which is based on the completeness and basisness of certain functional systems. By this approach, we obtain a uniqueness theorem and an algorithm for solving the inverse problem along with sufficient properties of the mentioned systems. The presented results give a universal tool for studying a number of important specific situations, including various Hochstadt–Lieberman‐type inverse problems both on an interval and on geometrical graphs, which is illustrated as well. [ABSTRACT FROM AUTHOR]

Acupuncture is an important part of traditional Chinese medicine (TCM). Although the efficacy of acupuncture has been widely accepted, the scientific basis behind it is still lack of exploration. Some experimental studies have shown that acupuncture can lead to changes in cell membrane potential, thus affecting the transmission of nerve signals to a certain extent. The change of cell membrane potential mainly depends on a large number of voltage dependent ion channels on the cell membrane. Relevant studies have proved that such voltage dependent ion channels have certain mechanical sensitivity, but the principle is not clear. Based on the theoretical model of cell action potential excitability proposed by Hodgkin–Huxley, taking rat skeletal muscle cell fibers as an example, this paper makes a rough stability analysis of the nonlinear system, explores the conditions of repeated discharge of cell membrane, and simulates the law of cell membrane potential frequency variation under different stimulating currents by numerical calculation. Based on the results of numerical simulation, we propose a hypothesis: Mechanical stimulation will produce a certain amount of stimulating current on the cell membrane, making the cell membrane potential in the state of repeated discharge, so as to block or interfere with the transmission of nerve signals and achieve the analgesic effect and so forth. [ABSTRACT FROM AUTHOR]

This paper is dedicated to solving the model reconstruction problem of Boolean networks (BNs) based on incomplete information. By resorting to semi‐tensor product, the issue of finding logical functions is converted into designing corresponding structure matrices. In line with desired attractors, two circumstances, standard BNs and delayed BNs, are taken into consideration, and several new results are presented. For the model reconstruction of standard BNs, an algebraic condition is put forward and an algorithm is developed for reconstructing unknown logical functions. For the model reconstruction of delayed BNs, an augmented system is deduced, based on which, several criteria and the corresponding algorithm are proposed. Finally, as an application to gene regulatory networks, the results obtained in this paper are used to analyze the WNT5A network and the apoptosis network. [ABSTRACT FROM AUTHOR]

This paper deals with a fourth‐order parabolic equation involving variable exponents, subject to Navier boundary conditions. We firstly give an optimal classification on the initial energy for blow‐up or global solutions, which is characterized by the sign of the Nehari energy and the difference between the initial energy and the depth of the potential well. Then, large time estimates of solutions are obtained. The results of the paper extend and complete the ones in "Applied Mathematics Letters 78(2018)141–146." [ABSTRACT FROM AUTHOR]

This document is an editorial summarizing the Applied Geometric Algebras in Computer Science and Engineering (AGACSE) conference held in Brno, Czech Republic in September 2021. The conference aimed to promote the use of geometric algebra in fields such as image processing, robotics, and quantum computing. The conference proceedings were published in the journal Mathematical Methods in the Applied Sciences. The editorial provides a list of accepted papers, covering topics such as applied geometry, technological applications, algebra, and quantum phenomena. One specific paper explores the use of geometric algebra in teaching rotations through neural networks. The document is a compilation of research papers showcasing the applications of geometric algebra in various fields, including robotics, control systems, image processing, cryptography, and physics. Each paper presents a specific problem or application and proposes a unique approach or solution using geometric algebra. The authors compare their methods with existing techniques and provide mathematical analysis to support their claims. Overall, the papers demonstrate the versatility and effectiveness of geometric algebra in different domains. [Extracted from the article]

As the evolution of species relies on not only the current state but also the past information, it is more reasonable and realistic to take delay into an ecological model. This paper deals with a stochastic predator–prey model that considers the distribution delay and assume that the intrinsic growth rate and the death rate in the model are governed by Ornstein–Uhlenbeck process to simulate the random factors in the environment. Based on the existence and uniqueness of the global solution to the model and the boundedness of the p$$ p $$ order moments of the solution, several conditions are established to analyze the survival of the species. Specifically, a criteria for the existence of the stationary distribution to the stochastic system is established by constructing some suitable Lyapunov functions. And the analytical expression of the probability density function of the model around the quasi‐equilibrium is obtained. Furthermore, the extinction of species in the model is also explored. Finally, numerical simulations are carried out to illustrate the theoretical results obtained in this paper. [ABSTRACT FROM AUTHOR]

This paper provides a new robust control approach to uncertain mechanical manipulators in the computed torque framework. With respect to the fact that model uncertainties occurring from the framework could make the overall system unstable although the resulting nominal plant is stabilized, this paper develops a readily applicable method for estimating such uncertain elements. More precisely, we propose a time‐delay compensation method, by which prior values relevant to the model uncertainties in a sufficiently small time instant are used for such an estimation. The theoretical effectiveness of the proposed method is validated by establishing the two different arguments on operator‐theoretic exponential stability and Lypaunov‐based disturbance input‐to‐state stability. Finally, some simulation and experiment results are provided to demonstrate the overall arguments developed in this paper for both the theoretical and practical aspects. [ABSTRACT FROM AUTHOR]

In the paper mentioned in the title, we studied the semi‐Markovian random walk processes with jumps and delaying screen in zero. More precisely, we obtained a mathematical modeling of the semi‐Markov random walk processes with a delaying screen in zero, given in general form by means of fractional differential equation. In this note, we provide a correction to some minor technicality in the proof of Theorem 2 in the aforementioned paper. [ABSTRACT FROM AUTHOR]

This paper is concerned with the multiplicity results to a class of p$$ p $$‐Kirchhoff type elliptic equation with the homogeneous Neumann boundary conditions by the abstract linking lemma due to Brézis and Nirenberg. We obtain the twofold results in subcritical and critical cases, which is a meaningful addition and completeness to the known results about Kirchhoff equation. At the same time, this paper also gives a method to deal with p$$ p $$‐Laplacian, Kirchhoff equation, and some Kirchhoff type equation in a unified variational framework. [ABSTRACT FROM AUTHOR]

We consider the following fractional NLS with focusing inhomogeneous power‐type nonlinearity: i∂tu−(−Δ)su+|x|−b|u|p−1u=0,(t,x)∈ℝ×ℝN,$$ i{\partial}_tu-{\left(-\Delta \right)}^su+{\left|x\right|}^{-b}{\left|u\right|}^{p-1}u=0,\kern0.30em \left(t,x\right)\in \mathrm{\mathbb{R}}\times {\mathrm{\mathbb{R}}}^N, $$where N≥2$$ N\ge 2 $$, 1/2

In this paper, the authors investigate the flow of an incompressible Boussinesq fluid in a semi‐infinite plane channel. An explicit spatial decay estimate in terms of the distance from finite end of the channel is obtained from an integro‐differential inequality for an energy integral. To make the result explicit, an upper bound for total energy is also obtained. The results of this paper may be thought of as a version of Saint‐Venant principle. [ABSTRACT FROM AUTHOR]

We study the periodicity of multipatch dispersal predator‐prey system with Holling type‐II functional response in this paper. By providing a new method, we overcome the difficulty to get the priori bounds estimation of unknown solutions of operator equation Lu=λNu. Graph theory with coincidence degree theory is used, and a sufficient criterion for the periodicity of the system is obtained. The criterion presented in this paper is closely related with topological structure of dispersal network and can be verified easily. Finally, a numerical example is also provided to verify the effectiveness of theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, the problem of attractivity of solutions for functional hereditary integral equations is initiated. A hereditary integral equation is converted into a second kind Volterra integral equation of convolution type by use of the property of convolution. Sufficient conditions for the existence of attractive solutions are established according to the fixed point theorem. The conclusions presented in this paper extend some published results. The effectiveness of the proposed results is illustrated by three numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we study the full parabolic attraction–repulsion Keller–Segel model with a fractional diffusion in ℝn$$ {\mathbb{R}}^n $$ for n=2$$ n=2 $$ or 3. We are more interested in the question that whether the solutions exist globally or blow up in finite time, which was studied in the classical attraction‐repulsion Keller‐Segel model by Jin and Wang (J. Differential Equations, 2016) through constructing a suitable energy functional. However, for the fractional attraction–repulsion Keller–Segel model, it is challenging to find a similar energy functional to study the existence of the solutions. In the present paper, under the condition of ξγ=χα$$ \xi \gamma =\chi \alpha $$, we introduce the Lp$$ {L}_p $$‐ Lq$$ {L}_q $$ estimates of the fractional derivative modulus of the solution to carry out the first step of the problem. Armed with a priori estimate, it is sufficient for us to obtain the global existence of the solutions by the Moser–Alikakos iterative method and finally arrive at the decay estimates of the solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the stabilization of a laminated beam system under the action of a new thermal coupling. Motivated by the work of Almeida Júnior et al. in the context of the Timoshenko system, we consider a thermoelastic coupling acting on the shear force. The present work constitutes a generalization of Almeida Júnior's paper because our study model considers an interfacial slip. More precisely, if we make the parameter γ$$ \gamma $$ of our system go to infinity, we obtain the Timoshenko beam system studied by the authors. We show that the weakly dissipative system of our manuscript is exponentially stable if, and only if, the wave propagation speeds are equal. Otherwise, we show that the system is polynomially stable and that the associated semigroup decays with rate t−1/2$$ {t}^{-1/2} $$. Furthermore, we show that this decay rate is optimal. [ABSTRACT FROM AUTHOR]

In this paper, we continue to study an initial boundary value problems to a model describing the evolution in time of diffusive phase interfaces in sea ice growth. In a previous paper, the global existence and the large‐time behavior of weak solutions in one space was studied under Dirichlet boundary conditions. Here, we show that the global existence of weak solutions and the large‐time behavior are also studied under Neumann boundary condition. In this paper, we study in space dimension lower than or equal to 3. [ABSTRACT FROM AUTHOR]

In this paper, we consider the effects of singular convection term and lower order term on the existence and regularity of solutions to the following elliptic Dirichlet problem: −divM(x)∇u=−divuE(x)+f(x)uγ,x∈Ω,u(x)=0,x∈∂Ω,$$ \left\{\begin{array}{rl}-\operatorname{div}\left(M(x)\nabla u\right)=-\operatorname{div}\left(uE(x)\right)+\frac{f(x)}{u^{\gamma }},& x\in \Omega, \\ {}\kern45pt u(x)=0,& x\in \mathrm{\partial \Omega },\end{array}\right. $$where γ>0,Ω⊂ℝN(N>2)$$ \gamma >0,\Omega \subset {\mathbb{R}}^N\left(N>2\right) $$ is a bounded smooth domain with 0∈Ω,f∈Lm(Ω)$$ 0\in \Omega, f\in {L}^m\left(\Omega \right) $$ with m≥1$$ m\ge 1 $$ is a non‐negative function. The main results of this paper show the combined effects of singular convection term and lower order term on the existence and regularity of solution to this problem. [ABSTRACT FROM AUTHOR]

In this paper, we take a fresh look at the determination of the transition matrix for a homogeneous, biisotropic particle employing the Null‐field approach. The condition for passivity of the material is discussed. In particular, we focus on some previously unproved properties of the Beltrami spherical vector waves, such as completeness, orthogonality, and linear independence, which are all instrumental in the analysis of the scattering problem with the Null‐field approach. Some numerical examples illustrate the approach. [ABSTRACT FROM AUTHOR]

In this review paper, we discuss some of our recent results concerning the rigorous analysis of initial boundary value problems (IBVPs) and newly discovered effects for certain evolution partial differential equations (PDEs). These equations arise in the applied sciences as models of phenomena and processes pertaining, for example, to continuum mechanics, heat‐mass transfer, solid–fluid dynamics, electron physics and radiation, chemical and petroleum engineering, and nanotechnology. The mathematical problems we address include certain well‐known classical variations of the traditional heat (diffusion) equation, including (i) the Sobolev–Barenblatt pseudoparabolic PDE (or modified heat or second‐order fluid equation), (ii) a fourth‐order heat equation and the associated Cahn–Hilliard (or Kuramoto–Sivashinsky) model, and (iii) the Rubinshtein–Aifantis double‐diffusion system. Our work is based on the synergy of (i) the celebrated Fokas unified transform method (UTM) and (ii) a new approach to the rigorous analysis of this method recently introduced by one of the authors. In recent works, we considered forced versions of the aforementioned PDEs posed in a spatiotemporal quarter‐plane with arbitrary, fully non‐homogeneous initial and boundary data, and we derived formally effective solution representations, for the first time in the history of the models, justifying a posteriori their validity. This included the reconstruction of the prescribed initial and boundary conditions, which required careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. In each IBVP, the novel formula was utilized to rigorously deduce the solution's regularity properties near the boundaries of the spatiotemporal domain. Importantly, this analysis is indispensable for proving (non)uniqueness of solution. These works extend previous investigations. The usefulness of our closed‐form solutions will be demonstrated by studying their long‐time asymptotics. Specifically, we will briefly review some asymptotic results about Barenblatt's equation. [ABSTRACT FROM AUTHOR]

In this paper, we provide the normal form for the Hopf bifurcation of a class of the reaction‐diffusion equation with memory‐based diffusion and nonlocal effect, where the delay is present in the differential term, similar to the chemotaxis model with time delay. The eigenvalue problems and the decomposition of the phase space are discussed in detail. Through a series of variable transformations, we obtain the third‐order truncated normal form of the model constrained on the central manifold and its equivalent equation in polar coordinates. Then, with the help of the dynamic analysis for the finite dimensional equations, the key parameters for determining the direction and stability of the Hopf bifurcation are given. These theoretical results are applied to the Bazykin's model, the stability, Turing bifurcation and Hopf bifurcation of the equilibrium are demonstrated through both theoretical and numerical methods. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the conservation of total energy and cross‐helicity for the weak solutions in the ideal magnetohydrodynamic (MHD) equations. In the spirit of recent works of Berselli (J. Differ. Equ. 368 (2023), 350–375.) and Berselli‐Georgiadis (NoDEA Nonlinear Differ. Equ. Appl. 31 (2024), 33), by establishing a new generalized Constantin‐E‐Titi type commutator estimate to allow us to make full use of the total energy, we extend the previous classical results to a wider range of exponents. These results indicate the role of the time integrability, spatial integrability, and differential regularity of the velocity (magnetic field) in the conserved quantities of weak solutions in the ideal MHD equations. [ABSTRACT FROM AUTHOR]

The Weyl expansion technique, also known as the angular spectrum expansion, expresses an outgoing spherical wave as a linear combination of plane waves. The scalar spherical waves are the solutions of the homogeneous Helmholtz equation and therefore have direct relation to the scalar multipole fields. This paper gives the Weyl expansion of multipole fields, scalar and vector, of any degree and order for spherical wave functions. The expressions are given in closed form for the scalar, ψℓm(τ)$$ {\psi}_{\mathit{\ell m}}^{\left(\tau \right)} $$, and vector, Mℓm(τ),Nℓm(τ)$$ {\mathbf{M}}_{\mathit{\ell m}}^{\left(\tau \right)},{\mathbf{N}}_{\mathit{\ell m}}^{\left(\tau \right)} $$, Lℓm(τ)$$ {\mathbf{L}}_{\mathit{\ell m}}^{\left(\tau \right)} $$, multipole fields, evaluated across a plane orthogonal to any given direction. In the case of scalar spherical multipoles, the spherical gradient operator has been used, while for the vector spherical multipoles, the vector spherical wave operator has been constructed. [ABSTRACT FROM AUTHOR]

In this paper, we study block Jacobi operators on ℤ$$ \mathrm{\mathbb{Z}} $$ with quasi‐periodic meromorphic potential. We prove the nonperturbative Anderson localization for such operators in the large coupling regime. [ABSTRACT FROM AUTHOR]

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 are 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$$ u\cdotp \nabla H $$ and strong nonlinear term curlH×H$$ \operatorname{curl}\kern0.1em H\times H $$. Our proof of the existence and uniqueness of solutions is in the Lagrangian coordinates first and then transformed back to the Euler coordinates. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic expansion for the homogeneous model of wind‐driven oceanic circulation with the nonslip boundary condition when both of the beta‐plane parameter and the Reynolds number go to infinity. By asymptotic analysis under certain constraints on wind tensor, we derive a formal expansion including the geophysical boundary layer in the large beta‐plane parameter and high Reynolds number limit, from which one sees that the external force, wind tensor, has an important effect on the behavior of the boundary layers. Moreover, when the external force and initial data satisfy certain constraints, to exclude the appearance of strong boundary layers, we construct approximate solutions with steady boundary layers to the homogeneous model of wind‐driven oceanic circulation in the large beta‐plane parameter and high Reynolds number limit. By using an energy method, we obtain the L∞$$ {L}^{\infty } $$‐stability of small perturbation around this approximate solutions, which verifies the validity of the asymptotic expansion of weak boundary layers in the large beta‐plane parameter and high Reynolds number limit. [ABSTRACT FROM AUTHOR]

The aim of this paper is twofold. The first goal is to give well‐posedness results for a Robin–transmission problem and a Robin–Dirichlet problem for the Brinkman system in bounded Lipschitz domains in Rn,n≥2$$ {\mathbb{R}}^n,n\ge 2 $$. The Robin–Dirichlet problem appears as a special limit case of the Robin–transmission problem. This aim is achieved by employing the layer potential theoretical methods. The second goal is to provide a well‐posedness result for a Robin–transmission problem and an existence result for the Robin–Dirichlet problem for a Navier–Stokes type system in bounded Lipschitz domains in Rn,n=2,3$$ {\mathbb{R}}^n,n=2,3 $$. Again, the Robin–Dirichlet problem appears as a special limit case of the Robin–transmission problem. This aim is achieved by using the well‐posedness results in the linear case combined with a fixed point argument. In order to illustrate our theoretical results, we also consider a special case of boundary value problems of Robin–Dirichlet type, which models a fluid flow in a porous square cavity. We solve numerically this problem. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study an age‐structured epidemiological compartment model and its respective problem, applied but not limited to the COVID‐19 pandemic, in order to investigate the role of the age of the individuals in the evolution of epidemiological phenomena. We investigate the well‐posedness of the model, as well as the global dynamics of it in the sense of basic reproduction number, via constructing Lyapunov functions. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the Boussinesq equations with magnetohydrodynamics convection, in which the temperature obeys degenerate damped wave equations. Under the influence of an equilibrium, we prove that the 2D magnetic Bénard system remains stable in the whole space ℝ2$$ {\mathrm{\mathbb{R}}}^2 $$. In particular, we obtain the decay rate and large‐time behavior. Without the coupling, the 2D MHD system with only vertical dissipation is not known to be stable. It is revealed that both the thermal damping and the horizontal background magnetic field are indispensable. [ABSTRACT FROM AUTHOR]

This paper develops an efficient combined interpolation/finite element approach for solving a three‐dimensional Black‐Scholes problem with stochastic volatility. The technique consists to approximate the time derivative by interpolation whereas the space derivatives are approximated using the finite element method. Both stability and error estimates of the new algorithm are deeply analyzed in the L∞$$ {L}^{\infty } $$‐norm. The proposed method is explicit, unconditionally stable, temporal second‐order accurate and fourth‐order convergence in space. This result suggests that the constructed scheme is faster and more efficient than a broad range of numerical methods widely studied in the literature for the Black‐Scholes models. Some numerical experiments are carried out to confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]