In this paper, we introduce some explicit formulas for the positive integer order q-derivative of a quotient of two functions. We establish the connections between linear algebra, the homomorphisms between the commutative algebras and the properties of q-operators. Especially, the third formula is a beautiful one involving the determinant of the functions and their q-derivatives. Then, we show that the formulas in the classical mathematical analysis are their special cases. Finally, the last formula is used for proving of some interesting identities. MSC CLASSIFICATION: 05A30, 26A24

In this article, the inverse problems for the wave equation in a medium in which multiple types of cavities and inclusion exist in a mixture are considered. From the point of view of the indicator function of the enclosure method, there are two types of heterogeneous parts:“minus group” and “plus group”. For example, cavities with the Dirichlet boundary condition belong to the minus group, while inclusions with smaller propagation velocity belong to the plus group. The heterogeneous part of the minus group gives a negative sign to the indicator function, and the heterogeneous part of the plus group gives a positive sign. In general, the presence of many types of heterogeneous parts causes cancellation of the sign of the indicator function. Such cases are referred to as “mixed cases”. Here we consider the case that the shortest length obtained from the indicator function is attained only by heterogeneous parts of the same group. This case is called the “mixed but separated case” and it is shown that the method of elliptic estimates developed by Ikehata works well. We also show that the case of a two-layered background medium with a flat layer can be considered in the same way as the case of a homogeneous background medium.

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.

In this paper, we use the modified potential well method to study the long time behaviors of solutions to the heat flow of H-system in a bounded smooth domain of $R^2$. Global existence and finite time blowup of solutions are proved when the initial energy is in three cases. When the initial energy is low or critical, we not only give a threshold result for the global existence and blowup of solutions, but also obtain the decay rate of the $L^2$ norm for global solutions. When the initial energy is high, sufficient conditions for the global existence and blowup of solutions are also provided. We extend the recent results which were obtained in \cite{r4}.

We proved the existence and uniqueness of comporessible subsonic impinging jet flow in the work \cite{WX}. As a continuation, in this paper, we investigate the shape of free boundary to the impinging jet flow established in \cite{WX}. More specifically, if the nozzle wall is concave to the fluid, then the free boundary of flow will be convex to the fluid. On another hand, the higher regularity of free boundary at separation point is obtained, provided that the nozzle satisfies corresponding hypotheses.

In this paper, we consider the global solvability and energy conservation for initial value problem of nonlinear semi-discrete wave equation of Kirchhoff type, which is a discretized model of Kirchhoff equation.

This paper investigates a stochastic two predators-one prey system with ratio-dependent functional response under regime switching. The stochastic extinction of species and the existence of ergodic stationary distribution for the system are established, and the transition probability of the solution converging to the stationary distribution also is obtained. To illustrate our theoretical results, the numerical examples and simulations are presented. Our findings also demonstrate that the stationary distribution and extinction of species for the stochastic two predators-one prey system are affected by random perturbations, leading to an imbalance in ecology.

In this paper, we propose a new nonuniform mesh method to simulate acoustic scattering problems in two dimensional periodic structures with non-periodic incident fields numerically. As existing methods are difficult to extend to higher dimensions, we have designed the new method with such extensions in mind. With the help of the Floquet-Bloch transform, the solution to the original scattering problem is written as an integral of a family of quasi-periodic problems. These are defined in bounded domains for each value of the Floquet parameter which varies in a bounded interval. The key step in our method is the numerical approximation of the integral by a quadrature rule adapted to the regularity of the family of quasi-periodic solutions. We design a nonuniform mesh with a Gaussian quadrature rule applied on each subinterval. We prove that the numerical method converges exponentially with respect to both the number of subintervals and the number of Gaussian quadrature points. Some numerical experiments are provided to illustrate the results.

The uniform exponential stabilities (UESs) of two hybrid control systems comprised of a wave equation and a second-order ordinary differential equation are investigated in this study. Linear feedback law and local viscosity are considered, as are nonlinear feedback law and internal anti-damping. The hybrid system is first reduced to a first order port-Hamiltonian system with dynamical boundary conditions, and the resulting system is discretized using the average central-difference scheme. Second, the UES of the discrete system is obtained without prior knowledge of the exponential stability of the continuous system. The frequency domain characterization of UES for a family of contractive semigroups and the discrete multiplier approach are used to validate the main conclusions. Finally, the Trotter-Kato Theorem is used to perform a convergence study on the numerical approximation approach. Most notably, the exponential stability of the continuous system is derived by the convergence of energy and UES, which is a novel approach to studying the exponential stability of some complex systems. Numerical simulation is used to validate the effectiveness of the numerical approximating strategy.

This paper deals with the existence of weak solutions of the system that describes the two-phase two-component fluid flow in porous media. Both two-phase and possible one-phase flow regions are taken into account. Our research is based on a global pressure, introduced in3, an artificial variable that allows us to decouple the original equations. We introduce another variable that does not have physical meaning in one-phase region and name it capillary pseudo-pressure. By using these two variables we rewrite system's equations in a fully equivalent form. This allows us to derive an existence result for weak solutions in more tractable way compared to15.

Cauchy matrix approach for the discrete Ablowitz-Kaup-Newell-Segur equations is reconsidered, where two ‘proper’ discrete Ablowitz-Kaup-Newell-Segur equations and two ‘unproper’ discrete Ablowitz-Kaup-Newell-Segur equations are derived. The ‘proper’ equations admit local reduction, while the ‘unproper’ equations admit nonlocal reduction. By imposing the local and nonlocal complex reductions on the obtained discrete Ablowitz-Kaup-Newell-Segur equations, two local and nonlocal discrete complex modified Korteweg-de Vries equations are constructed. For the obtained local and nonlocal discrete complex modified Korteweg-de Vries equations, soliton solutions and Jordan-block solutions are presented by solving the determining equation set. The dynamical behaviors of 1-soliton solution are analyzed and illustrated. Continuum limits of the resulting local and nonlocal discrete complex modified Korteweg-de Vries equations are discussed.

Based on partial Fourier series analysis, we adapt on a model case a new approach to classical results obtained in the literature describing the singularities of a family a solutions of a second order elliptic problems on open non-convex planar sectors. The method allows the exhibition of singular and regular frequencies, explicit decomposition and description of coefficients of singularities of the solution. As a main result, explicit and sharp estimates with respect to the opening angle parameter are obtained via this method. They are not uniform near π where corners have opening angle generating a jump of singularity in Sobolev exponent, contrarily to the results obtained in A. Tami (2016),(2019),(2021) for harmonic and/or biharmonic problems on a family of convex planar sectors.

This research deals with the Kaup-Newell model (KNM), a class of nonlinear Schrödinger equations with important applications in plasma physics and nonlinear optics. Soliton solutions are essential for analyzing nonlinear wave behaviors in different physical systems, and the KNM is also significant in this context. The model’s ability to represent sub-picosecond pulses makes it a significant tool for the research of nonlinear optics and plasma physics. Overall, the KNM is an important research domain in these areas, with ongoing efforts focused on understanding its various solutions and potential applications. A new version of the generalized exponential rational function method (nGERFM) and ( G ′ G 2 ) -expansion function methods are utilized to discover diverse soliton solutions. The nGERFM facilitates the generation of multiple solution types, including singular, shock, singular periodic, exponential, combo trigonometric, and hyperbolic solutions in mixed forms. Thanks to ( G ′ G 2 ) -expansion function method, we obtain trigonometric, hyperbolic, and rational solutions. The modulation instability (MI) of the proposed model is examined, with numerical simulations complementing the analytical results to provide a better understanding of the solutions’ dynamic behavior. These results offer a foundation for future research, making the solutions effective, manageable, and reliable for tackling complex nonlinear problems. The methodologies used in this study are robust, influential, and practicable for diverse nonlinear partial differential equations (NLPDEs); to our knowledge, for this equation, these methods of investigation have not been explored before. The accuracy of each solution has been verified using the Maple software program.

In this work, we study and solve the normalized Ricci flow equation for circle bundles over surfaces. Moreover, we study the asymptotic behavior of the solutions and their connections to some model geometries.

The aim of this paper is to derive many novel formulas involving the sum of powers of consecutive integers, the Bernoulli polynomials, the Stirling numbers and moments arise from conditional probability, moment generating functions and arithmetic functions by using the methods and techniques, which are used in discrete distributions in statistics such as uniform distribution, moment generating functions, and other probability distributions. Moreover, relations among the generalized Euler totient function, finite distributions containing special numbers and polynomials, discrete probability formula, and other special function are given. By using the Riemann zeta function and the Liouville's function, we derive a novel moment formula probability distribution on the set positive integers. Finally, by using approximation formulas for certain family of finite sums, approximation formulas for the sum of powers of consecutive integers involving the Bernoulli polynomials,and certain classes conditional probability involving the Laplace's rule of succession are derived.

Consider the nonlinear difference equations of the form L u = f m ( u ) , m ∈ Z , where L is a Jacobi operator given by L u m = a m u m + 1 + a m − 1 u m − 1 + b m u m for m∈Z, { a m } and { b m } are real valued T-periodic sequences, and f:Z×R→R. Applying critical point theory and a new analytical method, we obtain that the above problem has ground state solutions and infinitely many geometrically distinct solutions under the local superlinear condition lim | x | → ∞ ∫ 0 x f m ( t ) dt | x | 2 = ∞ uniformly in m∈ D for some set D⊂Z instead of the global superlinear condition lim | x | → ∞ ∫ 0 x f m ( t ) dt | x | 2 = ∞ uniformly in m∈Z.

Two types of short-time Fourier transforms involving respectively nonlinear modulation and nonlinear translation are designed in this note. Some important properties such as twin in time-frequency domain and orthogonality relations are investigated. Moreover, Lieb type inequalities and uncertainty principle are established.

We consider a degenerate/singular wave equation in one dimension, with drift and in presence of a leading operator which is not in divergence form. We impose a homogeneous Dirichlet boundary condition where the degeneracy occurs and a boundary damping at the other endpoint. We provide some conditions for the uniform exponential decay of solutions for the associated Cauchy problem.