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The study of the dynamics of the size of a population via mathematical modelling is a problem of interest and widely studied. Traditionally, continuous deterministic methods based on differential equations have been used to deal with this problem. However, discrete versions of some models are also available and sometimes more adequate. In this paper, we randomize the Pielou logistic equation in order to include the inherent uncertainty in modelling. Taking advantage of the method of transformation of random variables, we provide a full probabilistic description to the randomized Pielou logistic model via the computation of the probability density functions of the solution stochastic process, the steady state, and the time until a certain level of population is reached. The theoretical results are illustrated by means of two examples: The first one consists of a numerical experiment and the second one shows an application to study the diffusion of a technology using real data. [ABSTRACT FROM AUTHOR]

In this paper, the characteristics of a gravastar in f(Q)$f(\mathcal {Q})$ gravity, which is upheld by Krori–Barua (KB) metric are explored. The authors have used KB metric for the interior and shell regions of the gravastar. The field equations are deduced by using KB metric. In the outside regions of the gravastar, two regular black hole metrics are taken. Additionally, the Israel junction condition to calculate the potential difference across the thin shell concerning different types of regular black holes, such as Bardeen and Hayward is applied. The physical properties like proper length, entropy, energy, equation of state, and stability are also discussed. In this paper, the characteristics of a gravastar in f(Q)$f(Q)$ gravity, which is upheld by Krori–Barua (KB) metric are explored. The authors have used KB metric for the interior and shell regions of the gravastar. The field equations are deduced by using KB metric. In the outside regions of the gravastar, two regular black hole metrics are taken. Additionally, the Israel junction condition to calculate the potential difference across the thin shell concerning different types of regular black holes, such as Bardeen and Hayward is applied. The physical properties like proper length, entropy, energy, equation of state, and stability are also discussed. [ABSTRACT FROM AUTHOR]

It is well known that investigation on exact solutions of nonlinear fractional partial differential equations (PDEs) is a very difficult work compared with integer‐order nonlinear PDEs. In this paper, based on the separation method of semifixed variables and integral bifurcation method, a combinational method is proposed. By using this new method, a class of generalized time‐fractional thin‐film equations are studied. Under two kinds of definitions of fractional derivatives, exact solutions of two generalized time‐fractional thin‐film equations are investigated respectively. Different kinds of exact solutions are obtained and their dynamic properties are discussed. Compared to the results in the existing references, the types of solutions obtained in this paper are abundant and very different from those in the existing references. Investigation shows that the solutions of the model defined by Riemann–Liouville differential operator converge faster than those defined by Caputo differential operator. It is also found that the profiles of some solutions are very similar to solitons, but they are not true soliton solutions. In order to visually show the dynamic properties of these solutions, the profiles of some representative exact solutions are illustrated by 3D graphs. [ABSTRACT FROM AUTHOR]

In this paper, an improved zeroing neural network (ZNN) model is proposed to obtain the positive definite solutions of the continuous coupled Lyapunov matrix equations (CLMEs) associated with continuous‐time Markovian jump (CMJ) systems. To achieve this, a general ZNN model is established by constructing a matrix‐valued error function. Then, to accelerate the convergence rate of the proposed ZNN model, the latest estimation is introduced to obtain an improved ZNN model. Some convergence conditions have been derived for the presented improved ZNN model through Lyapunov theory. Comparisons among the improved ZNN model and the existing results are conducted to illustrate the advantages of the proposed improved ZNN model in numerical examples. [ABSTRACT FROM AUTHOR]

The KP‐I equation has family of solutions which decay to zero at space infinity. One of these solutions is the classical lump solution, which is a traveling wave, and the KP‐I equation in this case reduces to the Boussinesq equation. In this paper we classify all the 'lump‐type' solutions of the Boussinesq equation. Using a robust inverse scattering transform developed by Bilman–Miller for the Schrödinger equation, we show that the lump‐type solutions are rational and their τ$\tau $ functions have to be polynomials of degree k(k+1)$k(k+1)$ for some integer k$k$. In particular, this implies that the lump solution is the unique ground state of the KP‐I equation (as conjectured by Klein–Saut). The problem studied in this paper was mentioned in Airault–McKean–Moser, our result can be regarded as a two‐dimensional analogy of their theorem on the classification of rational solutions for the KdV equation. [ABSTRACT FROM AUTHOR]

This paper investigates the stability problem and large time behavior of solutions to the three‐dimensional magnetohydrodynamic equations with horizontal velocity dissipation and magnetic diffusion only in the x2$x_2$ direction. By applying the structure of the system, time‐weighted methods, and the method of bootstrapping argument, we prove that any perturbation near the background magnetic field (1, 0, 0) is globally stable in the Sobolev space H3(R3)$H^3(\mathbb {R}^3)$. Furthermore, explicit decay rates in H2(R3)$H^2(\mathbb {R}^3)$ are obtained. Motivated by the stability of the three‐dimensional Navier–Stokes equations with horizontal dissipation, this paper aims to understand the stability of perturbations near a magnetic background field and reveal the mechanism of how the magnetic field generates enhanced dissipation and helps stabilize the fluid. [ABSTRACT FROM AUTHOR]

In this paper, we develop a framework to analyze stochastic dynamic optimization problems in discrete time. We obtain new results about the existence and uniqueness of solutions to the Bellman equation through a notion of Banach contractions that generalizes known results for Banach and local contractions. We apply the results obtained to an endogenous growth model and compare our approach with other well‐known methods, such as the weighted contraction method, countable local contractions, and the Q‐transform. [ABSTRACT FROM AUTHOR]

This paper explores general coupled dynamic equations on time scales with multiple delays and investigates the existence of periodic solutions using the coincidence degree theory approach. The model studied includes the mathematical models of Nicholson, Mackey–Glass, and Lasota–Wazewska as special cases. Furthermore, we demonstrate that the solutions are not only asymptotically stable but also exponentially synchronized. The presented results significantly extend and complement existing findings in the field. The paper concludes with illustrative examples that highlight the practical implications of our analytical discoveries. [ABSTRACT FROM AUTHOR]

Summary: In this paper, for the first time, a new method with closed‐form analytical equations is presented to calculate the oscillation amplitude of fourth‐order oscillators, such as quadrature oscillators. This method is actually based on the general form of the differential equations describing the structure of the fourth‐order oscillators and finding a solution for the nonlinear differential equations governing this type of oscillator. The introduced method is a general method that is valid for all fourth‐order oscillators and is also independent of the oscillation frequency. Using the proposed method, complex and time‐consuming simulation tools will no longer be needed to calculate the oscillation amplitude in a steady state. Moreover, the presented closed‐form equations help the designers to understand the design compromises and design the oscillator for their specific and desired conditions. In addition, to evaluate the correctness of the presented equations, a comprehensive analysis of the oscillation amplitude of the quadrature oscillator in the steady state is performed, and a closed‐form equation is presented for the oscillation amplitude of the oscillator in the steady state, which is proposed for the first time in this paper. A comparison between the simulation results and theoretical computations confirms the validity of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we would like to consider the Cauchy problem for semilinear σ$$ \sigma $$‐evolution equations with time‐dependent damping for any σ≥1$$ \sigma \ge 1 $$. Motivated strongly by the classification of damping terms in some previous papers, the first main goal of the present work is to make some generalizations from σ=1$$ \sigma =1 $$ to σ>1$$ \sigma >1 $$ and simultaneously to investigate decay estimates for solutions to the corresponding linear equations in the so‐called effective damping cases. For the next main goals, we are going not only to prove the global well‐posedness property of small data solutions but also to indicate blow‐up results for solutions to the semilinear problem. In this concern, the novelty which should be recognized is that the application of a modified test function combined with a judicious choice of test functions gives blow‐up phenomena and upper bound estimates for lifespan in both the subcritical case and the critical case, where σ$$ \sigma $$ is assumed to be any fractional number. Finally, lower bound estimates for lifespan in some spatial dimensions are also established to find out their sharp results. [ABSTRACT FROM AUTHOR]

An improved filtering algorithm, called the EttAF adaptive filtering, is proposed in this paper. Compared with the available algorithm, the equivalent observation error in the fusion centre from the external sensor is considered, which makes the observation noise matrix of the filter dynamically adjust according to the relative relationship between the fusion centre and the external sensor. The strong manoeuvring target is modelled based on the CS‐jerk model, and the error adaptive variation equation is derived for the external data. Simulation results show that compared to traditional algorithm the EttAF algorithm can effectively improve the accuracy of position and velocity by 10.8% and 22.3% respectively. [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]

Mathematical techniques have a long and rich history in ecology, often serving as a virtual laboratory to test hypotheses, generate novel predictions, and investigate underlying ecological mechanisms. Recently, novel simulation techniques, advances in computing power, and numerical methods for implementing statistical models have significantly advanced our ability to integrate empirical and theoretical ecology. However, a divide still remains between mathematical and empirical studies, their readership, and integration into the broader literature. Because insights from mathematical ecology are far more general than the techniques employed, limitations in communicating mathematical advances to a broad spectrum of ecologists have arguably hindered ecology's progress, particularly in confronting theoretical predictions with empirical experiments and data. Here, we present a guide for both authors and readers of mathematical ecology, with the aim of increasing the accessibility of mathematical ecology for a broad group of ecologists. We provide a list of best practices when both writing and reading mathematical ecology, incorporating examples from this Special Feature of Ecosphere. This guide complements current guides for writing science, focusing specifically on effective communication of mathematics. [ABSTRACT FROM AUTHOR]

Flat TFT‐LCD with curved back light units(BLU) can obtain curved display effect. However, when the Flat TFT‐LCD bent with external force, there will be blocky black mura on both left and right sides, which makes the display effect worse. After analyzing a large number of blocky black mura phenomena and corresponding pre‐tilt angle data of curved TFT‐LCD, we summarized the quantitative relationship between the blocky black mura and pre‐tilt angle, and established a theoretical equation, which can be used to prejudge whether there will be blocky black mura after the TFT‐LCD bent, and to guide the improvement of blocky black mura. [ABSTRACT FROM AUTHOR]

This correction notice addresses errors in the equations presented in the original paper titled "MRI‐based transfer function determination through the transfer matrix by jointly fitting the incident and scattered B1+ field." The paper discusses the use of different time conventions to describe the background and scattered B1+ field, resulting in different expressions for the field distributions. The errors in the equations are corrected in this erratum. The authors apologize for any confusion caused and assure readers that the results and conclusions of the paper remain valid. [Extracted from the article]

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]

In this paper, we consider an explicit fully‐discrete approximation of the Cahn–Hilliard–Cook (CHC) equation with additive noise, performed by a standard finite element method in space and a kind of nonlinearity‐tamed Euler scheme in time. The main result in this paper establishes strong convergence rates of the proposed scheme. The key ingredient in the proof of our main result is to employ uniform moment bounds for the numerical approximations. To the best of our knowledge, the main contribution of this work is the first result in the literature which establishes strong convergence for an explicit fully‐discrete finite element approximation of the CHC equation. Finally, numerical results are finally reported to confirm the previous theoretical findings. [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]

The hypercircle theorem leads to a posteriori error estimates for the three‐field variational formulation of the Biot problem involving displacements, total pressure and fluid pressure. Based thereon, adaptive strategies can be constructed. In particular, the error estimator is derived on the basis of Hdiv‐conforming reconstructions of stress and flux approximations while the symmetry of the reconstructed stress is allowed to be satisfied only weakly. The reconstructions can therefore be performed locally on a set of vertex patches. The local nature of the reconstruction leads to two different possible adaptive strategies. In this paper, we review these two different adaptive strategies for the Biot problem. Numerical experiments underline the differences between the two strategies. [ABSTRACT FROM AUTHOR]

The numerical solution of dynamic problems for porous fluid‐saturated solids is often performed with the use of simplified equations known as the u‐p approximation. The simplification of the equations consists in neglecting some acceleration terms, which is justified for a certain class of problems related, in particular, to geomechanics and earthquake engineering. There exist two u‐p approximations depending on how many acceleration terms are neglected. All comparative studies of the exact and u‐p formulations are focused on the question of how well the u‐p solutions approximate those obtained with the exact equations. In this paper, the equations are compared from a different point of view, addressing the question of well‐posedness of boundary value problems. The exact equations must be hyperbolic and satisfy the corresponding hyperbolicity conditions for the boundary value problems to be well posed. The u‐p equations are not of the form to which the conventional definition of hyperbolicity applies. A slight extension of the approach makes it possible to derive hyperbolicity conditions as necessary conditions for well‐posedness for the u‐p approximations. The hyperbolicity conditions derived in this paper for the u‐p approximations are formulated in terms of the acoustic tensor of the skeleton. They differ essentially from the hyperbolicity conditions for the exact equations. [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]

In this paper we generalize an equation studied by Mossino and Temam in [7], to the fully nonlinear case. This equation arises in plasma physics as an approximation to Grad equations, which were introduced by Harold Grad in [4], to model the behavior of plasma confined in a toroidal vessel called TOKAMAK. We prove existence of a W2,p‐viscosity solution and regularity up to C1,αΩ¯ for any α<1 (we improve this regularity near the boundary). The difficulty of this problem lies in the right‐hand side which involves the measure of the superlevel sets, making the problem nonlocal. © 2021 Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

Let Hn=Cn×R$\mathbb {H}^{n}=\mathbb {C}^{n}\times \mathbb {R}$ be the n$n$‐dimensional Heisenberg group, Q=2n+2$Q=2n+2$ be the homogeneous dimension of Hn$\mathbb {H}^{n}$. In this paper, we investigate the existence of a least energy solution to the Q$Q$‐subLaplacian Schrödinger equation with either a constant V=γ$V=\gamma$ or a degenerate potential V$V$ vanishing on a bounded open subset of Hn$\mathbb {H}^n$: 0.1−divH∇HuQ−2∇Hu+V(ξ)uQ−2u=fu$$\begin{equation} -\mathrm{div}_{\mathbb {H}}{\left({\left|\nabla _{\mathbb {H}}u\right|}^{Q-2} \nabla _{\mathbb {H}}u\right)} +V(\xi) {\left|u\right|}^{Q-2}u=f{\left(u\right)} \end{equation}$$with the non‐linear term f$f$ of maximal exponential growth exp(αtQQ−1)$\exp (\alpha t^{\frac{Q}{Q-1}})$ as t→+∞$t\rightarrow +\infty$. Since the Pólya–Szegö‐type inequality fails on Hn$\mathbb {H}^n$, the coercivity of the potential has been a standard assumption in the literature for subelliptic equations to exclude the vanishing phenomena of Palais–Smale sequence on the entire space Hn$\mathbb {H}^n$. Our aim in this paper is to remove this strong assumption. To this end, we first establish a sharp critical Trudinger–Moser inequality involving a degenerate potential on Hn$\mathbb {H}^n$. Second, we prove the existence of a least energy solution to the above equation with the constant potential V(ξ)=γ>0$V(\xi)=\gamma >0$. Third, we establish the existence of a least energy solution to the Q$Q$‐subelliptic equation (0.1) involving the degenerate potential which vanishes on some open bounded set of Hn$\mathbb {H}^{n}$. We develop arguments that avoid using any symmetrization on Hn$\mathbb {H}^n$ where the Pólya–Szegö inequality fails. Fourth, we also establish the existence of a least energy solution to (0.1) when the potential is a non‐degenerate Rabinowitz type potential but still fails to be coercive. Our results in this paper improve significantly on the earlier ones on quasilinear Schrödinger equations on the Heisenberg group in the literature. We note that all the main results and their proofs in this paper hold on stratified groups with the same proofs. [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]

Summary: Operational amplifier (Op Amp) is a special class of analog integrated circuits. Multiple‐stage Op Amp design is of particular interest recently. However, analysis of this class of circuits has to be done manually and requires advanced skills. Automatic analysis of this class of circuits is studied in this paper, which takes a novel approach by circuit recognition and symbolic generation. Circuit recognition is deterministic, implemented by deterministic circuit partitioning and functional block extraction methods. Symbolic analysis of extracted circuit blocks can subsequently be performed with greatly reduced complexity while generated models and equations bear more useful information for circuit design. Symbolic results so generated can be applied in design space exploration, including tasks like finding the limitation in circuit topology, conducting initial sizing, and determining performance tradeoffs. This paper proposes the key algorithms for realizing the recognition of a few well‐known functional blocks frequently used in CMOS Op Amp circuits and further explores the possibility of using the recognized circuit cells in combination with the gm/ID method for circuit sizing. Preliminary tests show that this approach is a potential candidate for locating the performance boundaries of multiple‐stage Op Amps. [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]

In this paper, we present an adaptive framework for constructing a pseudo‐optimal reduced model for the frequency‐limited H2‐optimal model order reduction problem. We show that the frequency‐limited pseudo‐optimal reduced‐order model has an inherent property of monotonic decay in error if the interpolation points and tangential directions are selected appropriately. We also show that this property can be used to make an automatic selection of the order of the reduced model for an allowable tolerance in error. The proposed algorithm adaptively increases the order of the reduced model such that the frequency‐limited H2‐norm error decays monotonically irrespective of the choice of interpolation points and tangential directions. The stability of the reduced‐order model is also guaranteed. Additionally, it also generates the approximations of the frequency‐limited system Gramians that monotonically approach the original solution. Further, we show that the low‐rank alternating direction implicit iteration method for solving large‐scale frequency‐limited Lyapunov equations implicitly performs frequency‐limited pseudo‐optimal model order reduction. We consider two numerical examples to validate the theory presented in the paper. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to discuss the global well‐posedness, asymptotic behavior, and blow‐up of solutions in finite time for a parabolic system of equations driven by the (p(x), q(x))‐Laplacian operator with variable exponents and sign‐changing nonlinearity. More precisely, we consider ut−Δp(x)u=|v|s(x)|u|s(x)−2ulog(|uv|)inΩ×ℝ+,vt−Δq(x)v=|u|s(x)|v|s(x)−2vlog(|uv|)inΩ×ℝ+,u=v=0in∂Ω×ℝ+,u(x,0)=u0,v(x,0)=v0inΩ,where Ω⊂ℝN is a smooth bounded domain, N ≥ 1, p,q,s:Ω¯→ℝ+ are continuous functions and satisfy some conditions. The main characteristic of this paper is to study the global existence, asymptotic behavior, and blow‐up of solutions to the above system where the reaction terms are sign‐changing and satisfy neither the monotonicity condition nor the Ambrosetti–Rabinowitz condition. [ABSTRACT FROM AUTHOR]

The article titled "Corrigendum to: A Hill type equation can predict target gene expression driven by p53 pulsing" contains an error in Equation 18. The corrected equation is provided in the document. It is important to note that this change does not impact the results of the paper. [Extracted from the article]

We consider a class of diffusion equations with the Caputo time‐fractional derivative ∂tαu=Lu$$ {\partial}_t&#x0005E;{\alpha }u&#x0003D; Lu $$ subject to the homogeneous Dirichlet boundary conditions. Here, we consider a fractional order 0<α<1$$ 0<\alpha <1 $$ and a second‐order operator L$$ L $$ which is elliptic and non‐symmetric. In this paper, we show that the logarithmic convexity extends to this non‐symmetric case provided that the drift coefficient is given by a gradient vector field. Next, we perform some numerical experiments to validate the theoretical results in both symmetric and non‐symmetric cases. Finally, some conclusions and open problems will be mentioned. [ABSTRACT FROM AUTHOR]

This quest paper covers the most important topic in the field of mechanics. It is the nonlinear viscoelastic wave equations with variable degenerate damping and sources acting in both equations. We seek to prove that this system has no solution with positive initial energy. So, to establish our goal, we utilize the methods and techniques used by Said‐Houari in (J. Nonlinear. Differ. Equ. Appl, vol. 18, 2011, pp. 659‐684.). [ABSTRACT FROM AUTHOR]

In this paper, the Riemann‐Hilbert (RH) approach for the integrable discrete Hirota equation with bounded boundary conditions is presented. In the direct scattering problem, we study the analyticity, asymptotics, symmetries of the eigenfunctions, and scattering coefficients and analyze the distribution of discrete eigenvalues. In the inverse scattering problem, the RH problem is constructed and solved as well as the reconstruction formula of potential is derived based on asymptotics. Finally, combining the time evolution, we solve the first‐ and second‐order dark soliton solutions on the nonzero background under the reflectionless condition. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence of multiplicity of positive solutions to a new generalized Riemann–Liouville type fractional Fisher‐like equation on a semi‐infinite interval equipped with nonlocal multipoint boundary conditions involving Riemann–Liouville fractional derivative and integral operators. The existence of at least two positive solutions for the given problem is established by using the concept of complete continuity and iterative positive solutions. We show the existence of at least three positive solutions to the problem at hand by applying the generalized Leggett–Williams fixed‐point theorem due to Bai and Ge [Z. Bai, B. Ge, Existence of three positive solutions for some second‐order boundary value problems, Comput. Math. Appl. 48 (2014) 699‐70]. Illustrative examples are constructed to demonstrate the effectiveness of the main results. It has also been indicated in Section 5 that some new results appear as special cases by choosing the parameters involved in the given problem appropriately. [ABSTRACT FROM AUTHOR]

The paper considers the initial‐boundary value problem for equation Dtρu(x,t)+(−Δ)σu(x,t)=0,ρ∈(0,1),σ>0$$ {D}_t&#x0005E;{\rho }u\left(x,t\right)&#x0002B;{\left(-\Delta \right)}&#x0005E;{\sigma }u\left(x,t\right)&#x0003D;0,\rho \in \left(0,1\right),\sigma >0 $$, in an N‐dimensional domain Ω$$ \Omega $$ with a homogeneous Dirichlet condition. The fractional derivative is taken in the sense of Caputo. The main goal of the work is to solve the inverse problem of simultaneously determining two parameters: the order of the fractional derivative ρ$$ \rho $$ and the degree of the Laplace operator σ$$ \sigma $$. A new formulation and solution method for this inverse problem are proposed. It is proved that in the new formulation the solution to the inverse problem exists and is unique for an arbitrary initial function from the class L2(Ω)$$ {L}_2\left(\Omega \right) $$. Note that in previously known works, only the uniqueness of the solution to the inverse problem was proved and the initial function was required to be sufficiently smooth and non‐negative. [ABSTRACT FROM AUTHOR]

In this paper, we establish the well‐posedness of the Cauchy problem for incompressible magneto‐hydrodynamic equations by using parametric Meyer wavelets. The velocity field and magnetic field are expanded according to parametric wavelets given by a discrete set of thresholds. The evolution of these two types of fields is described by the relationship of parameterized flag microlocal quantities over binary time intervals. The corresponding iteration space is defined by the decay of the single norm, and the single norm is defined by the parametric flag microlocal quantities at a set of binary time interval. [ABSTRACT FROM AUTHOR]

In this paper, we consider the parabolic–elliptic Keller–Segel system, which is coupled to the incompressible Navier–Stokes equations through transportation and friction. It is shown that when the system is diffused by Lévy motion, the well‐posedness of the mild solution to the corresponding Cauchy problem in homogeneous Besov spaces is established by means of the Banach fixed point theorem. Furthermore, we prove the Lorentz regularity in time direction and the maximal regularity of solutions. In addition, we obtain the additional regularity and explore the time decay property of global mild solutions. [ABSTRACT FROM AUTHOR]

This paper studies the existence and asymptotic properties of solutions for a semilinear measure‐driven evolution equation with nonlocal conditions on an infinite interval. The existence result of the solutions for the considered equation is established by Schauder's fixed point theorem. Then, the asymptotic stability of solutions is further proved to show that all the solutions may converge to the unique solution of the corresponding Cauchy problem. In addition, under some conditions the existence of global attracting sets and quasi‐invariant sets of mild solutions is investigated as well. Finally, an example is provided to illustrate the applications of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, the author investigates particular disconjugate even‐order linear difference equations with two terms and classify them based on the properties of their recessive solutions at plus and minus infinity. The main theorem described states that the studied equation is (k−p+1)$(k-p+1)$‐critical whenever a specific second‐order linear difference equation is p$p$‐critical. In the proof, the author derived closed‐form solutions for the studied equation wherein the solutions of the said second‐order equation appear. Furthermore, the solutions were organized, in order to determine recessive solutions, into a linear chain by sequence ordering that compares the solutions at ±∞$\pm \infty$. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to characterize stable sets based on the asymptotic behavior of solutions as t$t$ goes to infinity for the following class of parabolic Kirchhoff equations: ut+∥u∥(θ−1)Ns(−Δ)N/ssu=λ|u|q−2uexpα0|u|NN−s|x|γinΩ,t>0,u=0inRN∖Ω,t>0,u(x,0)=u0(x)inΩ,$$\begin{eqnarray*} \hspace*{13pc}{\left\lbrace \def\eqcellsep{&}\begin{array}{llc}u_{t}+\Vert u\Vert ^{\frac{(\theta -1)N}{s}}(-\Delta)^{s}_{N/s}u=\frac{\lambda |u|^{q-2}u\exp {\left(\alpha _{0}|u|^{\frac{N}{N-s}}\right)}}{|x|^{\gamma }} &\text{in}\ &\Omega,\;t>0, \\ u =0 &\text{in} & \mathbb {R}^{N}\backslash \Omega,\;t > 0, \\ u(x,0)=u_{0}(x)& \text{in} &\Omega, \end{array} \right.} \end{eqnarray*}$$where ∥u∥Ns=∫R2N|u(x,t)−u(y,t)|N/s|x−y|2Ndxdy,$$\begin{equation*} \hspace*{7pc}\Vert u\Vert ^{\frac{N}{s}}=\int _{\mathbb {R}^{2N}}\frac{|u(x,t)-u(y,t)|^{N/s}}{|x-y|^{2N}}\,dxdy, \end{equation*}$$Ω⊂RN(N≥2)$\Omega \subset \mathbb {R}^N \, (N\ge 2)$ is a bounded domain with a Lipschitz boundary, 0∈Ω$0\in \Omega$, α0,λ>0$\alpha _{0},\lambda >0$, θ≥1$\theta \ge 1$, γ∈[0,N)$\gamma \in [0, N)$, q>Nθ/s$q>N\theta /s$, and (−Δ)N/ss$(-\Delta)_{N/s}^{s}$ is the fractional N/s$N/s$‐Laplacian operator, s∈(0,1)$s\in (0,1)$. [ABSTRACT FROM AUTHOR]

Small noise problems are quite important for all types of stochastic differential equations. In this paper, we focus on rough differential equations driven by scaled fractional Brownian rough path with Hurst parameter H∈(1/4,1/2]$H\in (1/4, 1/2]$. We prove a moderate deviation principle for this equation as the scale parameter tends to zero. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with ground state solutions to fractional equations with Coulomb potential and critical exponent. The existence of ground state solutions is established under certain conditions by proposing a new analytical method. [ABSTRACT FROM AUTHOR]

The linear weakly singular fractional Volterra integro‐differential equations involving the Caputo derivative have solutions whose derivatives are unbounded at the left endpoint of the integration interval. In this paper, we use suitable transformations to prevail on this nonsmooth behavior. We used the product integration method based on the new fractional basis function to solve these equations, which led to the production of the new interpolation formula and weights for the method. We investigate the convergence of the presented method. The proposed scheme is employed to solve some numerical examples to test its efficiency and accuracy. [ABSTRACT FROM AUTHOR]

This paper establishes the global well‐posedness of strong solutions for the 3D compressible magnetohydrodynamics (MHD) with non‐flat doping profile and Coulomb force, which is of hyperbolic–parabolic–elliptic mixed type. It is essentially shown that for the initial value problem with initial density allowed to contain vacuum states, the strong solution exists globally under small energy but possible large oscillations. [ABSTRACT FROM AUTHOR]

In this paper, we propose integrating factor (IF)‐based methods for solving the Cahn–Hilliard (CH) equation, which was originally proposed to describe the phase separation phenomenon for binary melted alloy below the critical temperature, and since then, it has appeared in many fields. Being a nonlinear equation, it is of great importance to develop efficient numerical methods for investigating the solution behavior of the CH equation. We present a few important linear and nonlinear schemes for the CH equation and obtain error estimates for those schemes. The integrators that we present here are of first and second order in time. We show numerical results to illustrate our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, the three‐dimensional hydrostatic magnetohydrodynamic (HMHD) equations are considered on a thin domain. We showed the global existence and uniqueness (regularity) of strong solutions to the three‐dimensional incompressible HMHD equations without any small assumption on the initial data. More precisely, there exists a unique strong solution globally in time for any given H2$$ {H}&#x0005E;2 $$‐smooth initial data. [ABSTRACT FROM AUTHOR]

In this paper, the initial value problem of the semilinear σ$$ \sigma $$‐evolution equations with a memory term is concerned. Firstly, using the energy method in the Fourier space, the decay estimates for the solutions to the corresponding linear problem are established. Additionally, assuming small initial data in suitable time‐weighted Sobolev spaces, the global‐in‐time existence of the solutions to the semilinear issue is proved by contraction mapping. Finally, the decay estimates of solutions are obtained under the additional regularity assumption on the initial data. [ABSTRACT FROM AUTHOR]

In this paper, a second‐order splitting method is applied to the Kuramoto–Sivashinsky equation, and then a finite Pointset method (FPM) is employed to solve the resulting coupled system. FPM is a meshfree particle method which is a local iterative procedure based on the weighted least square approximation. The numerical example confirms that the FPM is effective and accurate for this kind of problem. [ABSTRACT FROM AUTHOR]

This paper considers the following fractional Kirchhoff–Schrödinger–Poisson equation: a+b∫ℝ3(−Δ)α2u2dx(−Δ)αu+ϕu=h(u)+λuinℝ3,(−Δ)βϕ=u2inℝ3,$$ \left\{\begin{array}{lll}& \left(a&#x0002B;b{\int}_{{\mathrm{\mathbb{R}}}&#x0005E;3}{\left&#x0007C;{\left(-\Delta \right)}&#x0005E;{\frac{\alpha }{2}}u\right&#x0007C;}&#x0005E;2\mathrm{d}x\right){\left(-\Delta \right)}&#x0005E;{\alpha }u&#x0002B;\phi u&#x0003D;h(u)&#x0002B;\lambda u& \mathrm{in}\kern0.4em {\mathrm{\mathbb{R}}}&#x0005E;3,\\ {}& {\left(-\Delta \right)}&#x0005E;{\beta}\phi &#x0003D;{u}&#x0005E;2& \mathrm{in}\kern0.4em {\mathrm{\mathbb{R}}}&#x0005E;3,\end{array}\right. $$where a,b>0,0<α,β<1,2α+2β>3,λ∈ℝ,h∈C(ℝ,ℝ)$$ a,b>0,0<\alpha, \beta <1,2\alpha &#x0002B;2\beta >3,\lambda \in \mathrm{\mathbb{R}},h\in C\left(\mathrm{\mathbb{R}},\mathrm{\mathbb{R}}\right) $$. For the case of h(u)∼|u|p−2u$$ h(u)\sim {\left&#x0007C;u\right&#x0007C;}&#x0005E;{p-2}u $$ with p∈(8α+4β−3α+β,8α3+2)∪(8α3+2,2α∗),2α∗=63−2α$$ p\in \left(\frac{8\alpha &#x0002B;4\beta -3}{\alpha &#x0002B;\beta },\frac{8\alpha }{3}&#x0002B;2\right)\cup \left(\frac{8\alpha }{3}&#x0002B;2,{2}_{\alpha}&#x0005E;{\ast}\right),{2}_{\alpha}&#x0005E;{\ast }&#x0003D;\frac{6}{3-2\alpha } $$, the nonlinear term h$$ h $$ satisfies more general conditions, we obtain the existence of normalized solutions for the above system by using Pohozaev manifold and variational method. [ABSTRACT FROM AUTHOR]

In this paper, we consider two string–beam–string boundary transmission problems with localized frictional damping and delay. The first transmission problem is that the frictional damping and delay are in the left string equation and the right string equation is thermoelastic. The second transmission problem is that the damping and the delay are both only acting on the middle beam equation. Using the semigroup theory, we get the existence and the uniqueness of solution under suitable assumptions on the weight of the damping and the weight of the delay. Furthermore, we prove that the first transmission problem is exponentially stable and the second transmission problem is polynomially stable using the frequency domain method and the multiplier techniques. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the nonlocal‐in‐time Allen‐Cahn equation (NiTACE), which incorporates a nonlocal operator in time with a finite nonlocal memory. Our objective is to examine the well‐posedness of the NiTACE by establishing the maximal Lp$$ {L}^p $$ regularity for the nonlocal‐in‐time parabolic equations with fractional power kernels. Furthermore, we derive a uniform energy bound by leveraging the positive definite property of kernel functions. We also develop an energy‐stable time stepping scheme specifically designed for the NiTACE. Additionally, we analyze the discrete maximum principle and energy dissipation law, which hold significant importance for phase field models. To ensure convergence, we verify the asymptotic compatibility of the proposed stable scheme. Lastly, we provide several numerical examples to illustrate the accuracy and effectiveness of our method. [ABSTRACT FROM AUTHOR]