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The precise determination of earthquake location is the fundamental basis in seismological community, and is crucial for analyzing seismic activity and performing seismic tomography. First arrivals are generally used to practically determine earthquake locations. However, first-arrival traveltimes are not sensitive to focal depths. Moreover, they cannot accurately constrain focal depths. To improve the accuracy, researchers have analyzed the depth phases of earthquake locations. The traveltimes of depth phases are sensitive to focal depths, and the joint inversion of depth phases and direct phases can be implemented to potentially obtain accurate earthquake locations. Generally, researchers can determine earthquake locations in layered models. Because layered models can only represent the first-order feature of subsurface structures, the advantages of joint inversion are not fully explored if layered models are used. To resolve the issue of current joint inversions, we use the traveltimes of three seismic phases to determine earthquake locations in heterogeneous models. The three seismic phases used in this study are the first P-, sPg- and PmP-waves. We calculate the traveltimes of the three seismic phases by solving an eikonal equation with an upwind difference scheme and use the traveltimes to determine earthquake locations. To verify the accuracy of the earthquake location method by the inversion of three seismic phases, we take the 2021 MS6.4 Yangbi, Yunnan earthquake as an example and locate this earthquake using synthetic and real seismic data. Numerical tests demonstrate that the eikonal equation-based earthquake location method, which involves the inversion of multiple phase arrivals, can effectively improve earthquake location accuracy. [ABSTRACT FROM AUTHOR]

This work investigates the dynamics of discrete reaction-diffusion Gierer-Meinhardt system as mathematical models of biological pattern formation. We study the system's local asymptotic behaviour with and without the diffusion once developing the discrete integer variant of the well-known Gierer-Meinhardt model and proving that the model has a unique equilibrium. The requirements for the steady-state solution's local and global stability are found with the help of relevant approaches and the Lyapunov technique. Two large biological models and simulations are used throughout the work to validate the utility of the suggested technique. [ABSTRACT FROM AUTHOR]

The logistic map is a nonlinear difference equation well studied in the literature, used to model self-limiting growth in certain populations. It is known that, under certain regularity conditions, the stochastic logistic map, where the parameter is varied according to a specified distribution, has a unique invariant distribution. In these cases we can compare the long-term behaviour of the stochastic system with that of the deterministic system evaluated at the average parameter value. Here we examine the relationship between the mean of the stochastic logistic equation and the mean of orbits of the deterministic logistic equation at the expected value of the parameter. We formally prove that, in some cases, the addition of noise is beneficial to the populations, in the sense that it increases the mean, while for other ranges of parameters it is detrimental. A conjecture based on numerical evidence is presented at the end. [ABSTRACT FROM AUTHOR]

In this paper, the open-loop and closed-loop local and remote stochastic nonzero-sum game (LRSNG) problem is investigated. Different from previous works, the stochastic nonzero-sum game problem under consideration is essentially a special class of two-person nonzero-sum game problem, in which the information sets accessed by the two players are inconsistent. More specifically, both the local player and the remote player are involved in the system dynamics, and the information sets obtained by the two players are different, and each player is designed to minimise its own cost function. For the considered LRSNG problem, both the open-loop and closed-loop Nash equilibrium are derived. The contributions of this paper are given as follows. Firstly, the open-loop optimal Nash equilibrium is derived, which is determined in terms of the solution to the forward and backward stochastic difference equations (FBSDEs). Furthermore, by using the orthogonal decomposition method and the completing square method, the feedback representation of the optimal Nash equilibrium is derived for the first time. Finally, the effectiveness of our results is verified by a numerical example. [ABSTRACT FROM AUTHOR]

In this paper, we investigate several new classes of multi-dimensional almost automorphic type sequences and focus on their applications to various difference equations involving Volterra difference equations. We provide many structural results, illustrative examples and open problems about the notion under consideration. [ABSTRACT FROM AUTHOR]

It is shown that if the equation f (z + 1) n = R (z , f) , where R(z, f) is rational in both arguments and deg f (R (z , f)) ≠ n , has a transcendental meromorphic solution, then the equation above reduces into one out of several types of difference equations where the rational term R(z, f) takes particular forms. Solutions of these equations are presented in terms of Weierstrass or Jacobian elliptic functions, exponential type functions or functions which are solutions to a certain autonomous first-order difference equation having meromorphic solutions with preassigned asymptotic behavior. These results complement our previous work on the case deg f (R (z , f)) = n of the equation above and thus provide a complete difference analogue of Steinmetz' generalization of Malmquist's theorem. [ABSTRACT FROM AUTHOR]

Let p be a prime and let r , s be positive integers. In this paper, we prove that the Goormaghtigh equation $(x^m-1)/(x-1)=(y^n-1)/(y-1)$ , $x,y,m,n \in {\mathbb {N}}$ , $\min \{x,y\}>1$ , $\min \{m,n\}>2$ with $(x,y)=(p^r,p^s+1)$ has only one solution $(x,y,m,n)=(2,5,5,3)$. This result is related to the existence of some partial difference sets in combinatorics. [ABSTRACT FROM AUTHOR]

On the occasion of the centennial of his mentor Alvin Hansen, Paul Samuelson published in 1988 a modified version of his seminal 1939 multiplier-accelerator model to address aspects of Hansen's secular stagnation hypothesis. The "Keynes-Hansen-Samuelson" model (or KHS, as he called it) was built to analyse the effects of population growth on the economy's trajectory. Several changes were then made. Instead of difference equations and a tight accelerator, as in his 1939 model, Samuelson deployed differential equations and a flexible accelerator to produce a nonlinear limit cycle. Despite Samuelson's strong claims for the analytical contributions of his 1988 paper, it has – in contrast with the 1939 model – received only scant attention by macroeconomists and historians of economics alike. Samuelson's 1988 paper was his last published macroeconomic model, based on his long-established tradition of non-optimising macro-dynamics. Our paper provides a close reading of that article and some analytical results that shed new light on the formal aspects of Samuelson's 1988 model. We also discuss how it historically links up with business cycle models advanced by John Hicks, Nicholas Kaldor, Roy Harrod and Richard Goodwin and examine how far Samuelson's use of the term secular stagnation differs from Larry Summers's recent reconstruction of it. [ABSTRACT FROM AUTHOR]

In this paper, we provide sharp criteria of global attraction for a class of non-autonomous reaction–diffusion equations with delay and Neumann conditions. Our methodology is based on a subtle combination of some dynamical system tools and the maximum principle for parabolic equations. It is worth mentioning that our results are achieved under very weak and verifiable conditions. We apply our results to a wide variety of classical models, including the non-autonomous variants of Nicholson's equation or the Mackey–Glass model. In some cases, our technique gives the optimal conditions for the global attraction. [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]

Purpose: This work examines the returns to education for workers who pursue additional education after time out of the labor force. It compares those who remain in the labor force during additional education with those who drop out of the labor force during additional education. It compares two cohorts of the National Longitudinal Survey of Youth (NLSY). Design/methodology/approach: This work utilizes a difference equation to estimate the returns to education for workers who pursue additional education after time spent out of school and in the labor force. Findings: The results indicate a sheepskin return of approximately 14% for those who remain in the labor force and a return of approximately 9% to years of additional education for those who drop out of the labor force. This contrasting pattern of returns is robust to sample selection correction and a variety of checks. Research limitations/implications: This work does not fully account for all threats to causation. Further research could pursue these and make use of data from more clearly defined periods of education. Practical implications: This work finds key differences between the internal labor market faced by those remaining in the labor force and the external labor market faced by those dropping out of the labor force. A policy focused on re-training workers should account for these differences. Originality/value: This is the first work to compare workers who pursue additional education while remaining in the labor force to workers who pursue additional education and drop out of the labor force. [ABSTRACT FROM AUTHOR]

A singularly perturbed stationary problem for a one-dimensional reaction-diffusion equation in the case when the degenerate equation has multiple roots is studied. This is a new class of problems with discontinuous reactive terms along some curve that is independent of the small parameter. The existence of a smooth solution with the transition from the triple root of one degenerate equation to the double root of the other degenerate equation in the neighborhood of some point on the discontinuous curve is studied. Based on the existence theorem of classical boundary value problems and the technique of matching asymptotic expansion, the existence of a smooth solution is proved. And the point itself and the asymptotic representation of this solution are constructed by the matching technique and modified boundary layer function method. [ABSTRACT FROM AUTHOR]

Closed-form solutions to some boundary-value problems of fractional differential geofiltration dynamics in a fractured porous medium are obtained for a model with weakly permeable porous blocks. In particular, the direct and inverse boundary-value problems of filtration for the finite thickness layer are solved, the conditions for the existence of their regular solutions are given, and a solution to the problem of filtration dynamics with nonlocal boundary conditions is found. For a particular case of the filtration model, the problem of modeling the anomalous dynamics of filtration pressure fields on a star-shaped graph is considered. [ABSTRACT FROM AUTHOR]

A singularly perturbed transport equation is considered. A variable two-step backward differentiation formulas (BDF2) on a Shishkin-type mesh is used to discrete the first-order derivatives of the singularly perturbed transport equation. The stability and error analysis are derived by using the discrete orthogonal convolution kernels. It is proved that the scheme is second-order uniformly convergent with respect to the small parameter, which improves previous results. Numerical experiments are presented to support the theoretical result. [ABSTRACT FROM AUTHOR]

Long-term memory is a feature observed in systems ranging from neural networks to epidemiological models. The memory in such systems is usually modeled by the time delay. Furthermore, the nonlocal operators, such as the "fractional order difference," can also have a long-time memory. Therefore, the fractional difference equations with delay are an appropriate model in a range of systems. Even so, there are not many detailed studies available related to the stability analysis of fractional order systems with delay. In this work, we derive the stability conditions for linear fractional difference equations with an arbitrary delay τ and even for systems with distributed delay. We carry out a detailed stability analysis for the cases of single delay with τ = 1 and τ = 2. The results are extended to nonlinear maps. The formalism can be easily extended to multiple time delays. [ABSTRACT FROM AUTHOR]

Let $ f_1,\ldots,f_n $ f 1 , ... , f n be n ( $ n\geq ~2 $ n ≥ 2) continuous real-valued functions on $ \mathbf{R} $ R such that \[ \lim_{|t|\to +\infty}\frac{\int_0^tf_k(s)\,{\rm d}s}{t^2}=-\infty \] lim | t | → + ∞ ∫ 0 t f k (s) d s t 2 = − ∞ for all $ k=1,\ldots,n $ k = 1 , ... , n. This sole condition is far from ensuring the existence of multiple solutions for the classical problem \[ \begin{cases} -(x_{k+1}-2x_k+x_{k-1})=f_k(x_k)\quad k=1,\ldots,n, \\ x_0=x_{n+1}=0. \end{cases} \] { − (x k + 1 − 2 x k + x k − 1) = f k (x k) k = 1 , ... , n , x 0 = x n + 1 = 0. However, as a by-product of a much more general result, we get the following: for each $ \rho \in \mathbf{R} $ ρ ∈ R and for each $ i=1,\ldots,n $ i = 1 , ... , n , there exists $ (\lambda,\mu)\in \mathbf{R}^2 $ (λ , μ) ∈ R 2 such that the problem \[ \begin{cases} -\rho(x_{k+1}-2x_k+x_{k-1})=f_k(x_k)\quad k=1,\ldots,n,\quad k\neq i \\ -\rho(x_{i+1}-2x_i+x_{i-1})=f_i(x_i)+\lambda x_i+\mu \\ x_0=x_{n+1}=0 \end{cases} \] { − ρ (x k + 1 − 2 x k + x k − 1) = f k (x k) k = 1 , ... , n , k ≠ i − ρ (x i + 1 − 2 x i + x i − 1) = f i (x i) + λ x i + μ x 0 = x n + 1 = 0 has at least three solutions. [ABSTRACT FROM AUTHOR]

In A. Czornik et al. [Spectra based on Bohl exponents and Bohl dichotomy for non-autonomous difference equations, J. Dynam. Differ. Equ. (2023)] the concept of Bohl dichotomy is introduced which is a notion of hyperbolicity for linear non-autonomous difference equations that is weaker than the classical concept of exponential dichotomy. In the class of systems with bounded invertible coefficient matrices which have bounded inverses, we study the relation between the set $ \mathrm {BD} $ BD of systems with Bohl dichotomy and the set $ \mathrm {ED} $ ED of systems with exponential dichotomy. It can be easily seen from the definition of Bohl dichotomy that $ \mathrm {ED} \subseteq \mathrm {BD} $ ED ⊆ BD . Using a counterexample we show that the closure of $ \mathrm {ED} $ ED is not contained in $ \mathrm {BD} $ BD . The main result of this paper is the characterization $ \operatorname {int}\mathrm {BD} = \mathrm {ED} $ int ⁡ BD = ED . The proof uses upper triangular normal forms of systems which are dynamically equivalent and utilizes a diagonal argument to choose subsequences of perturbations each of which is constructed with the Millionshikov Rotation Method. An Appendix describes the Millionshikov Rotation Method in the context of non-autonomous difference equations as a universal tool. [ABSTRACT FROM AUTHOR]

In this paper, we present a new method to the state bounding problem for a class of coupled differential-difference equations (CDDEs) with bounded disturbances and time-varying delays. Instead of using the state transformation to reformulate the considered problem into the one for non-perturbed CDDEs, our novel idea is to construct suitable comparison systems, from which the state vector can be estimated directly and the information on the initial value is exploited more effectively in order to derive a sharper state bound. For computing and minimising the state bound, a numerical linear programming-based algorithm is also presented. The effectiveness of the presented method is verified via two numerical examples. [ABSTRACT FROM AUTHOR]

We investigate the following two types of nonlinear differential-difference equations L (z , f) + H (z , f) = ∑ k = 1 r α k (z) e β k z ; L (z , f) + H (z , f) = ∑ k = 1 r F k (z) , where α 1 , … , α r are meromorphic functions of order < 1 , and F 1 , … , F r are periodic transcendental entire functions, and L, H are defined by L (z , f) = ∑ k = 1 p a k (z) f (m k) (z + τ k) ≢ 0 , H (z , f) = ∑ k = 1 q b k (z) [ f (n k) (z + ζ k) ] s k with small meromorphic coefficients a i , b j. By introducing a new method, we obtain the exact forms of the solutions of these two equations under certain growth conditions. [ABSTRACT FROM AUTHOR]

We are investigating the existence of solutions to a system of two fractional q -difference equations containing fractional q -integral terms, subject to multi-point boundary conditions that encompass q -derivatives and fractional q -derivatives of different orders. In our main results, we rely on various fixed point theorems, such as the Leray–Schauder nonlinear alternative, the Schaefer fixed point theorem, the Krasnosel'skii fixed point theorem for the sum of two operators, and the Banach contraction mapping principle. Finally, several examples are provided to illustrate our findings. [ABSTRACT FROM AUTHOR]

The current manuscript proposes an innovative approach for obtaining exact solutions to the conformable time-fractional nonlinear differential-difference equations (NDDEs). The fundamental concept of this approach involves the generalized exponential rational function method (GERFM). In this method, the exact solutions include trigonometric and hyperbolic functions. In order to assess this method's efficacy, we consider its application to the conformable time-fractional Hybrid Lattice equation. After solving this model, we provide its dynamic behavior through the 3D, 2D, and Contour graphs. The soliton solutions obtained are used to describe any related physical phenomena. [ABSTRACT FROM AUTHOR]

Based on the Girsanov theorem for a kind of Volterra-Gaussian process, which are the generalization of fractional Brownian motion, Liouville fractional Brownian motion, and fractional Ornstein-Uhlenbeck process, we establish the Harnack inequalities for a class of stochastic functional differential equations driven by a kind of Volterra-Gaussian processes with a subordinator by an approximation technique. Some known results have been generalized and improved. [ABSTRACT FROM AUTHOR]

To accurately depict inventory management over time, this paper introduces a fractional inventory management model that builds upon the existing classical inventory management framework. According to the definition of fractional difference equation, the numerical solution and phase diagram of an inventory management system are obtained by MATLAB simulation. The influence of parameters on the nonlinear behavior of the system is analyzed by a bifurcation diagram and largest Lyapunov exponent (LLE). Combined with the related indexes of time series, the complex characteristics of a quantization system are analyzed using spectral entropy and C0. This study concluded that the changing law of system complexity is consistent with the LLE of the system. By analyzing the influence of order on the system, it is found that the inventory changes will be periodic in some areas when the system is fractional, which is close to the actual conditions of the company's inventory situation. The research results of this paper provide useful information for inventory managers to implement inventory and facility management strategies. [ABSTRACT FROM AUTHOR]

Summary: In this article, an event‐triggered recursive filtering problem is studied for a class of nonlinear multi‐rate systems (MRSs) with random transmission delays (RTDs). The RTDs are described by utilizing random variables with a known probability distribution and the Kronecker δ$$ \delta $$ function. To facilitate further study, the MRS is converted into a single‐rate one by adopting an iteration equation approach. To address the challenge of filter design caused by different measurement sampling periods, a modified prediction method of measurements is given. Moreover, an event‐triggered mechanism (ETM) is introduced to regulate the innovation transmission frequency. The objective of the addressed filtering problem is to design a recursive distributed filtering method for MRSs subject to ETM and RTDs, where a minimum upper bound on the filter error covariance is obtained. Moreover, the filter gain matrix is formulated by resorting to the solutions to matrix difference equations. Besides, the boundedness in the mean‐square sense of the filtering error is analyzed and a sufficient condition is provided. Finally, simulations with comparison experiments are presented to demonstrate the effectiveness of the newly proposed theoretical results. [ABSTRACT FROM AUTHOR]

We consider pairs of automorphisms acting on fields of Laurent or Puiseux series: pairs of shift operators .W x 7 x C h1; W x 7 x C h2/, of q-difference operators .W x 7 q1x, W x 7 q2x/, and of Mahler operators .W x 7 xp1 ; W x xp2 /. Given a solution f to a linear -equation and a solution g to an algebraic -equation, both transcendental, we show that f and g are algebraically independent over the field of rational functions, assuming that the corresponding parameters are sufficiently independent. As a consequence, we settle a conjecture about Mahler functions put forward by Loxton and van der Poorten in 1987. We also give an application to the algebraic independence of q-hypergeometric functions. Our approach provides a general strategy to study this kind of question and is based on a suitable Galois theory: the -Galois theory of linear -equations. [ABSTRACT FROM AUTHOR]

The performance of a narrowband active noise control (NANC) system can be significantly degraded due to the frequency mismatch (FM). In this paper, the statistical performance of a typical FxLMS-based NANC system in the presence of FM is analyzed in detail. Difference equations governing the system dynamics and closed-form steady-state mean-square error expressions are derived and discussed. The stochastic analysis results reveal that the FM introduces small troublesome sinusoids into the NANC system. The controller has to track these sinusoids to minimize the residual noise, which leads to a serious performance deterioration. As a by-product, the optimal step sizes that minimize the effect of a relatively small FM are also derived. The findings significantly enrich our understanding of the stochastic behavior of the FxLMS algorithm in the presence of FM and also provide some useful information for NANC system design. Extensive simulations are conducted to confirm the validity of the analytical findings. [ABSTRACT FROM AUTHOR]

This paper is concerned with approximations of invariant probability measures for stochastic functional differential equations (SFDEs) using a backward Euler-Maruyama (BEM) scheme under one-sided Lipschitz condition on the drift coefficient. Firstly, the strong convergence of the numerical "segment sequence" from the BEM scheme on finite time interval [ 0 , T ] is established. In addition, it is also demonstrated that the numerical segment sequence from the BEM scheme is a Markov process, and the corresponding discrete-time semigroup generated by this BEM scheme admits a unique numerical invariant probability measure. Finally, it is revealed that the numerical invariant probability measure converges to the underlying one in a Wasserstein distance. [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]

This study is devoted to the analysis of transport of chemical components of the Mars atmosphere. We investigate the turbulent diffusion of chemical components of the Mars atmosphere. To solve this problem in the approximation of diffusion of the minor component, we composed the continuity equation and the corresponding difference scheme. We formulated the boundary conditions in accordance with the available experimental and theoretical data and obtained the required temperature and pressure profiles. For simulation, we chose two models of turbulent diffusion, which were used in subsequent calculations. The simulation was performed using the modified Newton method. The models showed significant differences in the distribution of minor components of the atmosphere, in particular, hydrogen-containing molecules, which indicates the importance of choosing a model for describing turbulent diffusion when constructing a one-dimensional photochemical model of the atmosphere. [ABSTRACT FROM AUTHOR]

We give real representations of $ A^n $ A n (or $ e^{t A} $ e tA ) based on $ A^nP_{\mu } $ A n P μ for a real square matrix A, where $ P_{\mu } $ P μ is the projection to the generalized eigenspace associated with an imaginary eigenvalue μ of A. Our method is based on the spectral decomposition theorem. As applications, we can easily obtain realifications of representations of solutions of inhomogeneous linear difference equations with constant coefficients. [ABSTRACT FROM AUTHOR]

A weak Galerkin finite element method is applied to singularly perturbed delay reaction-diffusion problems. A robust uniform convergence has been proved both in the energy and balanced norms using higher-order piecewise discontinuous polynomials on Shishkin meshes. The error analysis for singularly perturbed reaction-diffusion problems with negative or positive shift in the balanced norm has appeared for the first time. The proposed method uses piecewise polynomials of order $ k\geq ~1 $ k ≥ 1 on interior of each element and piecewise constant polynomials on the end points of each element. By the Schur complement technique, the interior degrees of foredoom (DOF) can be eliminated from the discrete system resulting from the numerical scheme, and thus the degrees of freedom of the proposed method comparable with the classical finite element methods, and it is remarkably less than that of the discontinuous Galerkin method. Finally, we give various numerical experiments to verify the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, a discrete-time model of a plant–herbivore system is qualitatively analyzed using difference equations to describe population dynamics over time. The goal is to examine how the model behaves under varying parameter values and initial conditions. Results reveal that the model exhibits diverse dynamical behaviors such as stable equilibria, period-doubling cascade, and chaotic attractors. The analysis indicates that changes in crucial parameters greatly affect the system's dynamics. This study offers crucial insights into plant–herbivore systems and highlights the value of qualitative analysis in comprehending intricate ecological systems. [ABSTRACT FROM AUTHOR]

The present study investigates an algorithm numerically for finding the solution of partial differential equation with differences involved singular perturbation parameter(SPPDE) on non-uniform grid. Taylor series expansion provides a close approximation of the delay and advance terms in the convection-diffusion terms. After the approximations in shift containing terms, we applied the Crank-Nicolson application on uniform grid in the vertical direction. Subsequently, the resultant system is employed by the method of tension spline on a piece-wise uniform grid. Empirical evidence has shown that the suggested approach exhibits second-order characteristics in both the spatial and temporal dimensions. The effectiveness of derived scheme demonstrated through the solution of examples and the results are compared with existed methods. In the conclusion section, we will discuss the effect of shift parameters behavior for various singular perturbation parameter. [ABSTRACT FROM AUTHOR]

In this work we introduce discrete convolution operators and study their most basic properties. We then solve linear difference equations depending on such operators. The theory herein developed generalizes, in particular, the theory of discrete fractional calculus and fractional difference equations. To that matter we make use of the so-called Sonine pairs of kernels. [ABSTRACT FROM AUTHOR]

In this article, we focus on a specific version of the NDDEs which is the relativistic Toda lattice equation. We employ the generalized exponential rational function method on a nonlinear model of surface wave propagation to recognize their diverse singular soliton and multi-soliton wave structures. What is remarkable in this article is the use of graphic diagrams, which have diversified the solutions for solving such equations, leading to a greater understanding of the movements of particles and the strengthening of nonlinear lattice dynamics. The efficiency and strength of the employed method are illustrated, signifying its applicability to a wide spectrum of NDDEs in physical phenomena. [ABSTRACT FROM AUTHOR]

This article aims to study the generalized exponential rational function method to solve the differential-difference Burgers' equation. Our approach is an efficient method for solving nonlinear partial differential equations, and it can be used for a specific type of nonlinear differential-difference equations. To recognize diverse singular soliton and multi-soliton wave structures, we displayed the 3-D and contour graphs associated with the solutions. [ABSTRACT FROM AUTHOR]

The fractional Lindley distribution is used to model the distribution of perturbations in count data regressions, which allow for dealing with widely dispersed data. It is obtained from the non-fractional Lindley distribution by replacing the support $\mathbb{T} = {\mathbb{R}^ + }$ T = R + by ${\mathbb{T}} = {\mathbb{N}}\backslash \{ 0\} $ T = N ∖ { 0 } and applying time scale theory, whose ambition is to unify the theories of difference equations and differential equations, integral and differential calculus, and the calculus of finite differences. It thus provides a framework for the study of dynamical systems in discrete-continuous time. Delta moments are discrete-time Laplace transforms of the frequency function of the fractional Lindley distribution. The parameter of the fractional Lindley distribution is estimated by least squares, weighted least squares, maximum likelihood, moments, and proportions. The moment estimator always exists, so that delta moments result from the nabla Laplace transform of the frequency function of the fractional Lindley distribution. The maximum likelihood estimates have the least mean-square errors. The proportion method works satisfactorily only when the mode of the distribution is null and the proportion of zeros is high. A simulation allows for quantifying the mean-square errors associated with the estimators. A count regression based on the fractional Lindley distribution with data on the total number of stays after hospital admission among U.S. residents aged 65 and over shows that the Akaike information criteria is significantly lower than with the uniform Poisson and Poisson regressions. [ABSTRACT FROM AUTHOR]

Fractional calculus provides some fractional operators for us to model different real-world phenomena mathematically. One of these important study fields is the mathematical model of the elastic beam changes. More precisely, in this paper, based on the behavior patterns of an elastic beam, we consider the generalized sequential boundary value problems of the Navier difference equations by using the post-quantum fractional derivatives of the Caputo-like type. We discuss on the existence theory for solutions of the mentioned (p ; q) -difference Navier problems in two single-valued and set-valued versions. We use the main properties of the (p ; q) -operators in this regard. Application of the fixed points of the ρ - θ -contractions along with the endpoints of the multi-valued functions play a fundamental role to prove the existence results. Finally in two examples, we validate our models and theoretical results by giving numerical models of the generalized sequential (p ; q) -difference Navier problems. [ABSTRACT FROM AUTHOR]

This paper investigates an autonomous discrete-time glycolytic oscillator model with a unique positive equilibrium point which exhibits chaos in the sense of Li–Yorke in a certain region of the parameters. We use Marotto's theorem to prove the existence of chaos by finding a snap-back repeller. The illustration of the results is presented by using numerical simulations. [ABSTRACT FROM AUTHOR]

We consider general classes of Euler type linear and half‐linear difference equations, which are conditionally oscillatory. Applying the adapted Riccati technique, we improve known oscillation criteria for these equations. More precisely, our presented main criterion is the full oscillatory counterpart of a non‐oscillation criterion. Thus, in this paper, we enlarge the set of conditionally oscillatory Euler type difference equations. We highlight that our results are new even for linear equations with periodic coefficients. This fact is documented by simple examples of such equations at the end of this paper. [ABSTRACT FROM AUTHOR]

In this article, an explicit solution of the homogeneous fractional delay oscillation difference equation of order 1<ι<2$$ 1<\iota <2 $$ is given by constructing discrete sine‐ and cosine‐type delayed Mittag‐Leffler functions. Then, the discrete Laplace transform technique as an effective tool for solving the nonhomogeneous term, which is utilized to explore the solution of corresponding nonhomogeneous equation. Next, finite‐time stability of the homogeneous equation is studied based on the representation of the solution. Furthermore, we show a numerical example to elaborate the correctness of stability theory. Finally, an exact solution for the nonhomogeneous fractional difference equation with 0<ι<1$$ 0<\iota <1 $$ is further presented via using the discrete two‐parameter delayed Mittag‐Leffler function. [ABSTRACT FROM AUTHOR]

We study the qKZ difference equations with values in the n$n$th tensor power of the vector sl2${\mathfrak {sl}_2}$ representation V$V$, variables z1,⋯,zn$z_1,\dots,z_n$, and integer step κ$\kappa$. For any integer N$N$ relatively prime to the step κ$\kappa$, we construct a family of polynomials fr(z)$f_r(z)$ in variables z1,⋯,zn$z_1,\dots,z_n$ with values in V⊗n$V^{\otimes n}$ such that the coordinates of these polynomials with respect to the standard basis of V⊗n$V^{\otimes n}$ are polynomials with integer coefficients. We show that fr(z)$f_r(z)$ satisfy the qKZ equations modulo N$N$. Polynomials fr(z)$f_r(z)$ are modulo N$N$ analogs of the hypergeometric solutions of the qKZ given in the form of multidimensional Barnes integrals. [ABSTRACT FROM AUTHOR]

In this contribution we deal with sequences of polynomials orthogonal with respect to a Sobolev type inner product. A banded symmetric operator is associated with such a sequence of polynomials according to the higher order difference equation they satisfy. Taking into account the Darboux transformation of the corresponding matrix we deduce the connection with a sequence of orthogonal polynomials associated with a Christoffel perturbation of the measure involved in the standard part of the Sobolev inner product. A connection with matrix orthogonal polynomials is stated. The Laguerre-Sobolev type case is studied as an illustrative example. Finally, the bispectrality of such matrix orthogonal polynomials is pointed out. [ABSTRACT FROM AUTHOR]