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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]

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

Let μ$$ \mu $$ be a finite Borel measure on [0,1)$$ \left[0,1\right) $$. In this paper, we consider the generalized integral type Hilbert operator Iμα+1(f)(z)=∫01f(t)(1−tz)α+1dμ(t)(α>−1).$$ {\mathcal{I}}_{\mu_{\alpha +1}}(f)(z)={\int}_0^1\frac{f(t)}{{\left(1- tz\right)}^{\alpha +1}} d\mu (t)\kern0.30em \left(\alpha >-1\right). $$The operator Iμ1$$ {\mathcal{I}}_{\mu_1} $$ has been extensively studied recently. The aim of this paper is to study the boundedness (resp. compactness) of Iμα+1$$ {\mathcal{I}}_{\mu_{\alpha +1}} $$ acting from the normal weight Bloch space into another of the same kind. As consequences of our study, we get completely results for the boundedness of Iμα+1$$ {\mathcal{I}}_{\mu_{\alpha +1}} $$ acting between Bloch type spaces, and between logarithmic Bloch spaces. [ABSTRACT FROM AUTHOR]

Many challenges are still faced in bridging the gap between mathematical modeling and biological sciences. Measuring population immunity to assess the epidemiology of health and disease is a challenging task and is currently an active area of research. However, to meet these challenges, mathematical modeling is an effective technique in shaping the population dynamics that can help disease control. In this paper, we introduce a susceptible–infected–recovered (SIR) model and a susceptible–exposed–infected–recovered–deceased (SEIRD) model based on conformable space‐time partial differential equations (PDEs) for the coronavirus disease 2019 (COVID‐19) pandemic. As efficient analytical tools, we present new modifications based on the fractional exponential rational function method (ERFM) and an analytical technique based on the Adomian decomposition method for obtaining the solutions for the proposed models. These analytical approaches are more efficacious for obtaining analytical solutions for nonlinear systems of PDEs with conformable derivatives. The interesting result of this paper is that it yields new exact and approximate solutions to the proposed COVID‐19 pandemic models with conformable space‐time partial derivatives. [ABSTRACT FROM AUTHOR]

In this paper, we devise a novel method to solve Kawahara‐type equations numerically. In this novel method, for spatial discretization, we use delta‐shaped basis functions and generate differentiation matrices for spatial derivatives of the Kawahara‐type equations. For discretization of temporal variable, we utilize a high‐order geometric numerical integrator based on Lie group methods. For illustration of efficiency of the suggested method, we consider some test problems. We calculate errors and make some comparisons with other results that exist in literature. We also report changes in conservation laws during numerical simulations, and we indicate that the suggested method can preserve the conservation laws pretty good. Outcomes of numerical simulations indicate that the suggested method in this paper is reliable and effective for nonlinear partial differential equations (PDEs). [ABSTRACT FROM AUTHOR]

Nonstationary Gabor (NSG) frames for L2(ℝ)$$ {L}^2\left(\mathrm{\mathbb{R}}\right) $$ allow for flexible sampling and varying window functions and have found applications in adaptive signal analysis. Recently, the first author of this paper generalized the notion of NSG frames for L2(ℝ)$$ {L}^2\left(\mathrm{\mathbb{R}}\right) $$ to the space L2(ℝ,ℂK)$$ {L}^2\left(\mathrm{\mathbb{R}},{\mathrm{\mathbb{C}}}^K\right) $$ of vector‐valued signals and investigated the existence of vector‐valued NSG (VVNSG) frames. In this paper, we consider the reconstruction of any functions in L2(ℝ,ℂK)$$ {L}^2\left(\mathrm{\mathbb{R}},{\mathrm{\mathbb{C}}}^K\right) $$ from coefficients obtained from VVNSG frames. We provide different approximately dual frames of VVNSG frames and give the corresponding reconstruction error bounds. In particular, for a special class of VVNSG frames that are related to painless VVNSG frames, we show that the approximately dual frames carry the VVNSG structure, which is very important since there exist fast analysis and reconstruction algorithms for VVNSG frames. We also provide a method to construct the special class of VVNSG Bessel sequences. Finally, a constructive example is given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

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]

The previous published work by Othmani et al. invites a further useful discussion on how the viscoelastic rheological model affects the second harmonic and single modes of guided waves. However, this paper has two main objectives. The first objective is to establish a simple relationship between the Rayleigh waves velocity (VR), fundamental Lamb modes (A0 and S0), and the shear wave velocity (VS) in unidirectional viscoelastic carbon–epoxy at high frequency. The second objective is to study the second harmonic Lamb modes. Numerical codes based on Legendre polynomial and Newton–Raphson methods are developed and implemented to predict dispersion and attenuation curves. The second harmonic generation was investigated with the hysteretic and Kelvin–Voigt viscoelastic models. Results showed that rheological models have a much larger effect on the attenuation curves than the ones on the dispersion trajectories. Additionally, the optimal angles of Lamb and SH modes in unidirectional carbon–epoxy under Snell‐Descartes law was investigated. The present numerical results may render as a useful message for the rheological community. [ABSTRACT FROM AUTHOR]

In this paper, we study the long‐time dynamical behavior of the non‐autonomous velocity–vorticity–Voigt model of the 3D Navier–Stokes equations with damping and memory. We first investigate the existence and uniqueness of weak solutions to the initial boundary value problem for above‐mentioned model. Next, we prove the existence of uniform attractor of this problem, where the time‐dependent forcing term f∈Lb2(ℝ;H−1(Ω))$$ f\in {L}_b^2\left(\mathbb{R};{H}^{-1}\left(\Omega \right)\right) $$ is only translation bounded instead of translation compact. The results in this paper will extend and improve some results in Yue and Wang (Comput. Math. Appl., 2020) in the case of non‐autonomous and contain memory kernels which have not been studied before. [ABSTRACT FROM AUTHOR]

This paper presents a two‐warehouse inventory model for non‐instantaneous deteriorating items with a price‐dependent stochastic demand under advance sales discount and available capacity. In this model, the retailer offers a price discount to customers if they can commit their orders prior to the sales period. It encourages the retailer to order a large quantity that can be stored in an owned warehouse and then in a rented warehouse if there are excess items. Shortages are allowed and partially backlogged. The objective of this paper is to simultaneously determine the optimal price discount, the optimal order quantity, and the optimal replenishment schedule in such a way that the expected total profit is maximized. Two theorems are proved in order to verify that the expected total profit function is concave and to identify that an optimal replenishment decision exists. We further propose an algorithm to find the optimal replenishment policy by the retailer. Finally, we use numerical examples to illustrate the results presented herein. Moreover, based upon the sensitivity analysis, some important managerial implications are also discussed. [ABSTRACT FROM AUTHOR]

It is well‐known that discrete‐time Kolmogorov systems play important role in studying predator‐prey models with interactions. A topic of this paper deals with stochastic discrete‐time Kolmogorov systems which are called Lotka–Volterra operators. In the present paper, we provide a new class of LV operators, and establish that each of operator from this class has historical behavior, that is, their time averages diverge. [ABSTRACT FROM AUTHOR]

In this paper a thorough analysis is carried out of the type of order reduction that Lawson methods exhibit when used to integrate nonlinear initial boundary value problems. In particular, we focus on nonlinear reaction‐diffusion problems, and therefore, this study is important in a large number of practical applications modeled by this type of nonlinear equations. A theoretical study of the local and global error of the total discretization of the problem is carried out, taking into account both, the error coming from the space discretization and that due to the integration in time. These results are also corroborated by the numerical experiments performed in this paper. [ABSTRACT FROM AUTHOR]

In this paper, using the Gromov–Hausdorff distances between two global attractors (which may be in disjoint phase spaces) and two semi‐dynamical systems introduced by Lee et al. (2020), we consider the continuous dependence of the global attractors and the stability of the semi‐dynamical systems on global attractors induced by the Brinkman–Forchheimer equation under variation of the domain. The results of this paper improve on previous results, which can compare any two systems in different phase spaces without the process of "pull‐backing" the perturbed systems to the original domain. [ABSTRACT FROM AUTHOR]

The aim of this study is to explore the global stability of Cohen–Grossberg Clifford‐valued neutral‐type neural network models with time delays. In order to achieve the aim of this paper, and to solve the non‐commutativity problem caused by Clifford numbers multiplication, the original Clifford‐valued system is first decomposed into 2mn$$ {2}^mn $$‐dimensional real‐valued systems. Some sufficient criteria for the global stability of the addressed network models are established by constructing an appropriate Lyapunov functional. The established stability conditions have not been affected by the neutral delay and time delay values. The proposed method and results of this paper are new. The feasibility of the stability criteria obtained are verified using two numerical examples. [ABSTRACT FROM AUTHOR]

In this work, we consider the one‐dimensional Dirac operator DQ+Qsu(x)=1iσ2ddx+Q(x)+Qs(x)u(x),$$ {\mathfrak{D}}_{\mathrm{Q}+{\mathrm{Q}}_s}u(x)=\left(\frac{1}{\mathrm{i}}{\sigma}_2\frac{d}{dx}+\mathrm{Q}(x)+{\mathrm{Q}}_s(x)\right)u(x), $$where σ2$$ {\sigma}_2 $$ is Pauli's matrix, Q$$ \mathrm{Q} $$ is a 2×2$$ 2\times 2 $$‐matrix representing a regular potential that includes the electrostatic and scalar interactions as well as the anomalous magnetic momentum, Qs$$ {\mathrm{Q}}_s $$ is a singular potential consisting of N$$ N $$ delta distributions Qs(x)=∑j=1NAjδx−xj,$$ {\mathrm{Q}}_s(x)=\sum \limits_{j=1}^N{\mathrm{A}}_j\delta \left(x-{x}_j\right), $$Aj$$ {\mathrm{A}}_j $$ (j=1,⋯,N$$ j=1,\cdots, N $$) are 2×2$$ 2\times 2 $$‐matrices representing the strengths of Dirac deltas, and u=u1u2⊤$$ u={\left({u}^1\kern0.5em {u}^2\right)}^{\top } $$ is a two‐spinor. We associate to the operator DQ+Qs$$ {\mathfrak{D}}_{\mathrm{Q}+{\mathrm{Q}}_s} $$ an unbounded in L2ℝ,ℂ2$$ {L}^2\left(\mathbb{R},{\mathbb{C}}^2\right) $$ symmetric operator denoted by DQ,Ω$$ {\mathcal{D}}_{\mathrm{Q},\Omega} $$, where Ω=xjj=1N$$ \Omega ={\left\{{x}_j\right\}}_{j=1}^N $$ is the support of singular potential Qs$$ {\mathrm{Q}}_s $$. The operator DQ,Ω$$ {\mathcal{D}}_{\mathrm{Q},\Omega} $$ includes only the regular potential Q$$ \mathrm{Q} $$ together with certain interaction conditions at each point xj∈Ω$$ {x}_j\in \Omega $$. The paper presents a method for determining the discrete spectrum of the operator DQ,Ω$$ {\mathcal{D}}_{\mathrm{Q},\Omega} $$ for arbitrary potential Q$$ \mathrm{Q} $$ whose entries are given by L∞(ℝ)$$ {L}^{\infty}\left(\mathbb{R}\right) $$‐functions. The eigenvalues λn$$ {\lambda}_n $$ of the operator DQ,Ω$$ {\mathcal{D}}_{\mathrm{Q},\Omega} $$ are the zeros of a dispersion equation ηΩ(λ)=0$$ {\eta}_{\Omega}\left(\lambda \right)=0 $$, where the characteristic function ηΩ$$ {\eta}_{\Omega} $$ is determined explicitly in terms of power series involving the spectral parameter λ$$ \lambda $$. The construction of the characteristic function ηΩ$$ {\eta}_{\Omega} $$ from a set of monodromy matrices and the interaction conditions is presented in the paper. Moreover, its power series representation leads to an efficient numerical method for calculating the eigenvalues of the Dirac operator DQ,Ω$$ {\mathcal{D}}_{\mathrm{Q},\Omega} $$ from the zeros of certain approximate function η˜Ω$$ {\tilde{\eta}}_{\Omega} $$ which is obtained by truncating the series up to a finite number of terms. Several examples show the applicability and accuracy of the numerical method. [ABSTRACT FROM AUTHOR]

The dynamic transitions of the Brusselator model has been recently analyzed in Choi et al. (2021) and Ma and Wang (2011). Our aim in this paper is to address the relation between the pattern formation and dynamic transition results left open in those papers. We consider the problem in the setting of a 2D rectangular box where an instability of the homogeneous steady state occurs due to the perturbations in the direction of several modes becoming critical simultaneously. Our main results are twofold: (1) a rigorous characterization of the types and structure of the dynamic transitions of the model from basic homogeneous states and (2) the relation between the dynamic transitions and the pattern formations. We observe that the Brusselator model exhibits different transition types and patterns depending on the nonlinear interactions of the pattern of the critical modes. [ABSTRACT FROM AUTHOR]

The accuracy and repeatability of the color reproduction in print is determined by the fine‐tuning of the tone reproduction curves of the basic printing colorants (most often this is CMYK). However, the diversity of manufacturers of printing equipment and dyes introduces an element of significant uncertainty about color uniformity. In addition, the traditional approach does not take into account the effect of hue change when applying the original dyes, as well as the nonlinearity of the hue rise in high‐ and low‐density areas. Determining the color of base colorants that produces the most uniform tone change is an important engineering challenge. Previously, there was no scientific basis for such calculations. We recently proposed an alternative color correction model based on gradation trajectories as an analogue of gradation curves in the CIE Lab space. We have also described the extension of the approach to double color overlay (gradation surfaces) and its analytical and discrete implications. The trajectories are the geodetic lines on gradation surfaces. In this paper, we propose using the gradation trajectories to determine "ideal" or "true" initial printing dyes for electrophotography. To simplify calculations, natural color discretization in digital printing is used. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a generalization of Vieta theorem (Vieta's formulas) to the case of Clifford geometric algebras. We compare the generalized Vieta formulas with the ordinary Vieta formulas for characteristic polynomial containing eigenvalues. We discuss Gelfand–Retakh noncommutative Vieta theorem and use it for the case of geometric algebras of small dimensions. We introduce the notion of a simple basis‐free formula for a determinant in geometric algebra and prove that a formula of this type exists in the case of arbitrary dimension. Using this notion, we present and prove generalized Vieta theorem in geometric algebra of arbitrary dimension. The results can be used in symbolic computation and various applications of geometric algebras in computer science, computer graphics, computer vision, physics, and engineering. [ABSTRACT FROM AUTHOR]

This survey introduces 101 new publications on applications of Clifford's geometric algebras (GAs) newly published during 2022 (until mid‐January 2023). The selection of papers is based on a comprehensive search with Dimensions.ai, followed by detailed screening and clustering. Readers will learn about the use of GA for mathematics, computation, surface representations, geometry, image, and signal processing, computing and software, quantum computing, data processing, neural networks, medical science, physics, electric engineering, control and robotics. [ABSTRACT FROM AUTHOR]

Protein structure prediction (PSP) is the prediction of the three‐dimensional (3D) folding of a protein (its tertiary structure) starting from its amino acid sequence (its primary structure). The state of the art in PSP is achieved by deep learning pipelines that require several input features extracted from amino acid sequences. It has been demonstrated that features that grasp the relative orientation of amino acids positively impact the prediction accuracy of the 3D coordinates of atoms in the protein backbone. In this paper, we demonstrate the relevance of geometric algebra (GA) in instantiating orientational features for PSP problems. We do so by proposing two novel GA‐based metrics which contain information on relative orientations of amino acid residues. We then employ these metrics as additional input features to a graph transformer (GT) architecture to aid the prediction of the 3D coordinates of a protein, and compare them to classical angle‐based metrics. We show how our GA features yield comparable results to angle maps in terms of accuracy of the predicted coordinates. This is despite being constructed from less initial information about the protein backbone. The features are also fewer and more informative and can be (i) closely associated to protein secondary structures and (ii) more easily predicted compared to angle maps. We hence deduce that GA can be employed as a tool to simplify the modeling of protein structures and pack orientational information in a more natural and meaningful way. [ABSTRACT FROM AUTHOR]

In the present paper, we consider a convex combination of non‐Volterra quadratic stochastic operators defined on a finite‐dimensional simplex depending on a parameter θ$$ \theta $$ and study their trajectory behaviors. We showed that in the case m=3$$ m=3 $$ depending on the parameter θ∈[0,1]$$ \theta \in \left[0,1\right] $$ and an initial point, the trajectories of such operator converge to a fixed point. For m≥4$$ m\ge 4 $$, any trajectory of the operator converges to the center of the simplex. [ABSTRACT FROM AUTHOR]

In this paper, a one‐dimensional degenerate parabolic equation involving a multiplicative control with a nonlocal space term represented by an integral kernel is considered. After some well‐posedness results, we derive several properties of the control‐to‐state mapping. We establish the existence of at least one optimal control and derive the first and the second conditions for optimality, giving the local uniqueness of an optimal control. Finally, we prove that we may expect the global uniqueness of the optimal control under additional assumptions. [ABSTRACT FROM AUTHOR]

The current study is important from two perspectives. Firstly, in this article, we suggest a novel analytical technique for creating the exact solutions to nonlinear partial differential equations (NLPDEs). In order to study the dynamical behaviors of various wave phenomena, we can construct the several exact solutions in the form of Jacobi elliptic solutions, hyperbolic solutions, trigonometric solutions, and exponential solutions by using this method. Secondly, we consider a more generalized form of the (2+1)‐dimensional Korteweg–de Vries (KdV) and modified Korteweg‐de Vries (mKdV) equations that plays an important role in describing the shallow water. We successfully extract several soliton solutions for the examined equation. By choosing the appropriate parameters, some graphs of the discovered solutions have been represented in the figures. Every obtained solution has been demonstrated to satisfy the corresponding equation. The findings demonstrate that the method can be applied to solve a number of nonlinear evolution equations. The new solutions and the paper's findings could improve our understanding of how the waves move through shallow water and open up new research avenues. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel dual inertial Tseng's extragradient method for solving variational inequality problems in real Hilbert spaces, particularly those involving pseudomonotone and Lipschitz continuous operators. Our secondary method incorporates variable step‐size, updated at each iteration based on some previous iterates. A notable advantage of these algorithms is their ability to operate without prior knowledge of Lipschitz‐type constants and without the need for any line‐search procedure. We establish the convergence theorem of the proposed algorithms under mild assumptions. To illustrate the numerical behavior of the algorithms and to make comparisons with other methods, we conduct several numerical experiments. The results of these evaluations are showcased and thoroughly examined to exemplify the practical significance and effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, we propose and study a discretized one‐predator two‐prey system along with prey refuge and Michaelis–Menten‐type prey harvesting. The interaction among the species is considered as Holling type III functional response. Firstly, existence and local stability of all the fixed points are derived under certain parametric conditions. Furthermore, a special consideration is made to global asymptotic stability of the interior fixed point. Then, we have shown that the system undergoes different types of bifurcations including transcritical bifurcation, flip bifurcation, and Neimark–Sacker bifurcation by using center manifold theorem, bifurcation theory, and normal form method. Also, Feigenbaum's constant of the system is calculated. It is observed that both harvesting and refuge have a stabilizing effect on the system, and the stabilizing effect of harvesting dominates the stabilizing effect of refuge. Of most interest is the finding of coexisting attractors and multistability. In particular, optimal harvesting policy has been obtained by extension of Pontryagin's maximum principle to discrete system. Finally, some intriguing numerical simulations are provided to verify our analytic findings and rich dynamics of the three species system. [ABSTRACT FROM AUTHOR]

In this paper, we develop a hypercomplex operator calculus to treat fully analytically boundary value problems for the homogeneous and inhomogeneous fractional Helmholtz equation where fractional derivatives in the sense of Caputo and Riemann–Liouville are applied. Our method extends the recently proposed fractional reduced differential transform method (FRDTM) by using fractional derivatives in all directions. For the special separable case in three dimensions, we obtain completely explicit representations for the fundamental solution. This allows us to interpret and to understand the appearance of spatial steady‐state solutions or spatial blow‐ups of the fractional Helmholtz equation in a better way. More precisely, we were able to present explicit conditions for the parameters in the representation formulas of the fundamental solutions under which we obtain bounded or spatial decreasing steady‐solutions and when spatial blow‐ups occur. We also illustrate this with some representative numerical examples. Furthermore, we show that it is possible to recover the recently studied cases as well as the classical cases as particular limit cases within our more general setting. Using the hypercomplex operator approach also allows us to factorize the fractional Helmholtz operator and obtain some interesting duality relations between left and right derivatives, Caputo and Riemann–Liouville derivatives, and eigensolutions of antipodal eigenvalues in terms of a generalized Borel–Pompeiu formula. This factorization, in turn, allows us to tackle inhomogeneous fractional Helmholtz problems. [ABSTRACT FROM AUTHOR]

In this paper, we first develop the modified gradient‐based iterative (MGI) method for the complex conjugate and transpose matrix equations A1XB1+A2X‾B2+A3XTB3+A4XHB4=E$$ {A}_1X{B}_1+{A}_2\overline{X}{B}_2+{A}_3{X}^T{B}_3+{A}_4{X}^H{B}_4=E $$. By adopting the updated technique, we can make full use of the latest information to compute the next result, which leads to a faster convergence rate. In theory, we apply the real representation of a complex matrix and the vec‐operator to prove the convergence properties. Furthermore, we extend the MGI algorithm to solve the generalized complex conjugate and transpose matrix equations. Then, the necessary and sufficient conditions for convergence of the MGI algorithm are presented. Lastly, three numerical examples are introduced to testify the efficiency of our methods. [ABSTRACT FROM AUTHOR]

In this paper, the direct similarity reduction method is used to reduce the modified variable coefficient Zakharove–Kusnetsov equation (vcmZK), which represents the propagation of quantum dust acoustic waves (QDAWs) in an inhomogeneous magnetized dusty plasma field, into a nonlinear ordinary differential equation (NODE). Moreover, the Jacobi elliptic method is used to solve the NODE, where many novel double periodic, kink, and solitary wave solutions are obtained. Additionally, we have chosen to plot periodic, kink, and soliton QDAWs solutions according to some choices for the variable coefficients. According to the a chieved results, we can conclude that the choices of the functional forms of the variable coefficients can control the QDAWs propagation. Finally, the new discovered results may have many important applications in astrophysics, atmospheric science, nanotechnology, and industrial plasma physics. [ABSTRACT FROM AUTHOR]

This paper is concerned with the deduction of a mathematical model for the small longitudinal vibrations of a bar where one end is clamped and in the other end is glued a concentrated mass. The existence of global solutions of this model perturbed with a general nonlinear internal dissipation and with a general nonlinear boundary dissipation is derived. For a particular case, the exponential decay rate of the energy is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we develop the Hermitian refinement of symplectic Clifford analysis, by introducing a complex structure 핁 on the canonical symplectic manifold (ℝ2n,ω0)$$ \left({\mathrm{\mathbb{R}}}^{2n},{\omega}_0\right) $$. This gives rise to two symplectic Dirac operators Ds$$ {D}_s $$ and Dt$$ {D}_t $$ (in the sense of Habermann), leading to a u(n)$$ \mathfrak{u}(n) $$‐invariant system of equations on ℝ2n$$ {\mathrm{\mathbb{R}}}^{2n} $$. We discuss the solution space for this system, culminating in a Fischer decomposition for the space of (harmonic) polynomials on ℝ2n$$ {\mathrm{\mathbb{R}}}^{2n} $$ with values in the symplectic spinors. To make this decomposition explicit, we will construct the associated embedding factors using a transvector algebra. [ABSTRACT FROM AUTHOR]

This paper is devoted to analyze the zero‐Hopf bifurcation of a generalized three‐dimensional (3D) jerk system, the jerk function of this system has all quadratic and cubic terms. Due to the averaging method of second order, we show that at most three periodic orbits bifurcate form the zero‐Hopf equilibrium point of this jerk system, and this upper bound is sharp. Furthermore, by using the averaging method of third order, we show that three is also the maximal number of periodic orbits bifurcate from this zero‐Hopf equilibrium point. Finally, the numerical method is used to justify the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, the minimal observability of switching Boolean networks (SBNs) is investigated. Firstly, applying the semi‐tensor product (STP) method of matrices, a parallel extension system is constructed, based on which a necessary and sufficient condition to detect the observability of the SBNs is given. Secondly, when an SBN is unobservable, the specific steps to obtain the required measurements to make the system observable are given using the set reachable method; however, the measurements given in this part are not necessarily the fewest. Then, a criterion for determining the minimum number of measurements is further proposed through a constructed indicator matrix. Lastly, the effectiveness of the new results is verified by an example. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the existence and uniqueness of solutions for a sixth‐order differential equation involving nonlocal and integral boundary conditions. Firstly, we obtain the properties of the relevant Green's functions. The existence result of at least one nontrivial solution is obtained by applying the Krasnoselskii–Zabreiko fixed point theorem. Moreover, we also establish the existence of unique solution to the considered problem via Hölder and Minkowski inequalities and Rus's theorem. Finally, two numerical examples are included to show the applicability of our main results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the viscoelastic Kirchhoff plate equation with memory and time delay. Under appropriate conditions on the real numbers μ1$$ {\mu}_1 $$ and μ2$$ {\mu}_2 $$, we prove the existence of a compact global attractor with finite fractal dimension through a stabilizability estimate. Additionally, we demonstrate the existence of a fractal exponential attractor. To the best of our knowledge, these findings are novel for for this system. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Lipschitz stability of a perturbed impulsive differential system concerning the unperturbed system. We employ the variation of parameters or the constant of variation for impulsive differential systems with an initial time difference. [ABSTRACT FROM AUTHOR]

A system of time‐fractional diffusion equations posed in an exterior domain of ℝN$$ {\mathrm{\mathbb{R}}}^N $$( N≥3$$ N\ge 3 $$) under homogeneous Dirichlet boundary conditions is investigated in this paper. The time‐fractional derivatives are considered in the Caputo sense. Using nonlinear capacity estimates specifically adapted to the nonlocal properties of the Caputo fractional derivative, the geometry of the domain, and the boundary conditions, we obtain sufficient conditions for the nonexistence of a weak solution to the considered system. [ABSTRACT FROM AUTHOR]

The subject of KCC theory is a second‐order ordinary differential equation, it is sometimes difficult to convert the high dimensional system into an equivalent second‐order system because of the analytical requirements of KCC theory. By means of the Euler‐Lagrange extension of a flow on a Riemannian manifold, this paper gives five geometric invariants of some three‐dimensional systems with great convenience, and focus on the analysis of two of them. The results show that the hyperbolic equilibria corresponding to the seven standard forms of three‐dimensional linear systems are Jacobi unstable. This is completely different from what we got before in two‐dimensional systems, where Jacobi stable and Jacobi unstable correspond to focus and node, respectively. All equilibria of classical Lü chaotic system and Yang‐Chen chaotic system are Jacobi unstable. Meanwhile, in three‐dimensional linear case, the torsion tensors at any point of the trajectory are identically equal to zero, but the two nonlinear systems have nonzero torsion tensors components. [ABSTRACT FROM AUTHOR]

In this paper, approximate solutions for stochastic Fitzhugh–Nagumo partial differential equations are obtained using two‐dimensional shifted Legendre polynomial (2DSLP) approximation. The problem's suitability and solvability are confirmed. The convergence analysis for the proposed methodology and the error analysis in the L2$$ {L}^2 $$ norm are carried out. Using Maple software, an algorithm is created and implemented to arrive at the numerical solution. The solution thus obtained is compared with the exact solution and the solution obtained using the explicit order RK1.5 method. [ABSTRACT FROM AUTHOR]

Recently, certain biological field experiments and research findings have revealed that predators not only reduce prey populations through direct predation (lethal effect) but also indirectly affect the growth rate of prey by inducing fear; these nonlethal effects can be carried over seasons or generations. In this paper, we proposed an autonomous and a nonautonomous Leslie–Gower model incorporating some biological factors such as fear and its carry‐over effects, prey refuge, and nonlinear predator harvesting. In the autonomous model, first, we examined the well‐posedness, positivity solutions, and their boundedness. Also, it is shown that new equilibrium points emerge and disappear with the change in intrinsic growth rate of predator. Further, we compute and analyzed the local and global stability at the interior equilibrium points. Furthermore, Hopf bifurcation and direction and stability of limit cycle at interior equilibrium points are established. Moreover, the sensitivity analysis of the biological parameters is carried out numerically with two statistical methods via Latin hypercube sampling and partial rank correlation coefficients. It was found that at the small values of fear factor, carry‐over effect, and refuge behavior of prey parameters, the autonomous system exhibits oscillatory behavior, whereas for small values of prey birth rate and harvesting effort, the system remains stable. However, when intrinsic growth rate of predator increases from a low value to a higher value, the system shows transition from stability to instability and back to stability. We also explore the effects of seasonal variations in biological parameters in the nonautonomous model. Additionally, we investigate the theoretical existence of positive periodic solutions using the continuation theorem. The influences of seasonal variations in some parameters such as prey's birth rate, fear, and its carry‐over effects; refuge behavior of prey; intrinsic growth rate of predator; and harvesting effort are then numerically investigated. The results reveal the emergence of positive periodic solutions and bursting patterns in the seasonally forced model. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate similarity and consimilarity of hyper‐dual generalized quaternions and their matrices. For this purpose, we give different conjugates according to the generalized quaternionic units i,j,k$$ i,j,k $$. We present i$$ i $$‐consimilarity of hyper‐dual generalized quaternions and their matrices except hyper‐dual 14$$ \frac{1}{4} $$‐quaternions. For the generalization consisting of hyper‐dual coefficients quaternion and split quaternion, we search j$$ j $$‐consimilarity and k$$ k $$‐consimilarity with the help of j$$ j $$‐conjugate and k$$ k $$‐conjugate. We also give i,j,k$$ i,j,k $$‐coneigenvalues and i,j,k$$ i,j,k $$‐coneigenvectors of the matrices of these generalizations. In addition, we examine right coneigenvalue problem in generalized quaternion matrices for real and split quaternions. The complex matrix representation obtained through the complex adjoint matrix representation of this generalization is introduced, and its properties are presented. Besides, we give algebraic methods for the concept of right coneigenvalues and coneigenvectors for matrices, which are the generalization of real quaternion and split quaternion. [ABSTRACT FROM AUTHOR]

In this paper, we have formulated and analyzed a mathematical model with multiple time delays of the genetic regulatory network of the microRNA‐17‐92 (miR‐17‐92) cluster. We have focused on the effect of the associated transcriptional and translational time delays of both positive and negative feedback loops on the miR‐17‐92 regulatory network. To gain better insight, we have explored the role of individual time delay in controlling the transition of the G1/S cell cycle process. We pay particular attention to the concentration of E2F and Myc, which trigger the protein ON and OFF switches. To analyze the situation mathematically, we employ delay‐dependent stability analysis and bifurcation analysis using delay differential equations and a biological study of the time‐delayed random transitions between the ON and OFF states for protein synthesis. Based on our study, we can conclude that the translational time delay (τ2)$$ \left({\tau}_2\right) $$ can halt the healthy transition of cells in the G1/S cell cycle phase as the small increase in (τ2)$$ \left({\tau}_2\right) $$ can destroy bistability of the system. Simulations are carried out to numerically show the solution trajectories under the combined effect of delays. Finally, we summarize the simulation results and the impact of delays on the dual behavior of miR‐17‐92 as it can act as an oncogene or as a tumor suppressor gene. [ABSTRACT FROM AUTHOR]

In this paper, we focus on designing a well‐conditioned Galerkin spectral methods for solving a two‐sided fractional diffusion equations with drift in which the fractional operators are defined neither in Riemann–Liouville nor Caputo sense, and its physical meaning is clear. Based on the image spaces of Riemann–Liouville fractional integral operators on Lp([a, b]) space discussed in our previous work, after a step by step deduction, three kinds of Galerkin spectral formulations are proposed, the final obtained corresponding scheme of which shows to be well‐conditioned—the condition number of the stiff matrix can be reduced from O(N2α) to O(Nα), where N is the degree of the polynomials used in the approximation. Another point is that the obtained schemes can also be applied successfully to approximate fractional Laplacian with generalized homogeneous boundary conditions, whose fractional order α ∈ (0, 2), not only having to be limited to α ∈ (1, 2). Several numerical experiments demonstrate the effectiveness of the derived schemes. Besides, based on the numerical results, we can observe the behavior of mean first exit time, an interesting quantity that can provide us with a further understanding about the mechanism of abnormal diffusion. [ABSTRACT FROM AUTHOR]

In that paper, firstly, we get two additional different non‐null space curve evolutions in Minkowski 3‐space by considering Landau–Lifshitz (LL) equation where we identify the spin vector with the binormal vector and the normal vector of these curves, respectively. Then, we obtain some links for constructing the moving non‐null space curves by using the integrable LL equation. Finally, we give as an application the exact solution of the moving non‐null curve evolutions obtained by taking the spin vector, the normal vector of the curve, and we showed graphically that these solutions are wave solutions. [ABSTRACT FROM AUTHOR]

In this paper, based on the differential quadrature method (DQM), matrix operators are derived for fractional integration and Caputo differentiation. These operators generalize the efficient DQM to fractional calculus. The proposed fractional differential/integral quadrature method (FDIQM) is used to solve various types of fractional ordinary and partial differential equations. FDIQM unifies the solution of multi‐integer fractional‐order differential equations leading to significant simplification in the implementation. Numerous examples are presented to demonstrate the accuracy of the operators. Other examples are presented to solve various fractional differential equations including time‐fractional sub‐diffusion equation, linear/nonlinear, and multiorder fractional differential equations. In addition, numerous boundary conditions are considered including mixed fractional derivatives. Further, a nonlinear fractional eigenvalue problem is solved efficiently, and its bifurcation diagrams are obtained. Comparisons between the proposed method and the existing ones are included, showing the ease of implementation, efficiency, and applicability of FDIQM. [ABSTRACT FROM AUTHOR]