On the Behavior of the Positive Solutions of the System of Two Higher-Order Rational Difference Equations

We study the convergence of the positive solutions of the system of the following two difference equations: where K is a positive integer, the parameters?A,B,?α, β? and the initial conditions are positive real numbers. Our results generalize well known results in [1,2].

Fuzzy Difference Equations in Diagnoses of Glaucoma from Retinal Images Using Deep Learning

The intuitive fuzzy set has found important application in decision-making and machine learning.To enrich and utilize the intuitive fuzzy set,this study designed and developed a deep neural network-based glaucoma eye detection using fuzzy difference equations in the domain where the retinal images converge.Retinal image detections are categorized as normal eye recognition,suspected glaucomatous eye recognition,and glaucomatous eye recognition.Fuzzy degrees associated with weighted values are calculated to determine the level of concentration between the fuzzy partition and the retinal images.The proposed model was used to diagnose glaucoma using retinal images and involved utilizing the Convolutional Neural Network(CNN)and deep learning to identify the fuzzy weighted regularization between images.This methodology was used to clarify the input images and make them adequate for the process of glaucoma detection.The objective of this study was to propose a novel approach to the early diagnosis of glaucoma using the Fuzzy Expert System(FES)and Fuzzy differential equation(FDE).The intensities of the different regions in the images and their respective peak levels were determined.Once the peak regions were identified,the recurrence relationships among those peaks were then measured.Image partitioning was done due to varying degrees of similar and dissimilar concentrations in the image.Similar and dissimilar concentration levels and spatial frequency generated a threshold image from the combined fuzzy matrix and FDE.This distinguished between a normal and abnormal eye condition,thus detecting patients with glaucomatous eyes.

GLOBAL ATTRACTIVITY IN A CLASS OF HIGHER-ORDER NONLINEAR DIFFERENCE EQUATION

In this paper the global attractivity of the nonlinear difference equationis investigated, where a,b, A ∈ (0,∞), k is an positive integer and the initial conditions x- k, …, x-1 and x0 are arbitrary positive numbers. It is shown that the unique positive equilibrium of the equation is global attractive. As a corollary, the result gives a positive confirmation on the conjecture presented by Kocic and Ladas [1,p154].

Sub-ODE's New Solutions and Their Applications to Two Nonlinear Partial Differential Equations with Higher-Order Nonlinear Terms

In the present paper, with the aid of symbolic computation, families of new nontrivial solutions of the first-order sub-ODE F12 = AF2 ＋ BF2＋p ＋ CF2＋2p （where F1= dF/dε, p 〉 0） are obtained. To our best knowledge, these nontrivial solutions have not been found in [X.Z. Li and M.L. Wang, Phys. Lett. A 361 （2007） 115] and IS. Zhang, W. Wang, and J.L. Tong, Phys. Lett. A 372 （2008） 3808] and other existent papers until now. Using these nontrivial solutions, the sub-ODE method is described to construct several kinds of exact travelling wave solutions for the generalized KdV-mKdV equation with higher-order nonlinear terms and the generalized ZK equation with higher-order nonlinear terms. By means of this method, many other physically important nonlinear partial differential equations with nonlinear terms of any order can be investigated and new nontrivial solutions can be explicitly obtained with the help of symbolic computation system Maple or Mathematics.

Higher-order differential variational principle and differential equations of motion for mechanical systems in event space

In this paper we study the higher-order differential variational principle and differential equations of motion for mechanical systems in event space. Based on the higher-order d＇Alembert principle of the system, the higher-order velocity energy and the higher-order acceleration energy of the system in event space are defined, the higher-order d＇Alembert- Lagrange principle of the system in event space is established, and the parametric forms of Euler-Lagrange, Nielsen and Appell for this principle are given. Finally, the higher-order differential equations of motion for holonomic systems in event space are obtained.

Solution of a One-Dimension Heat Equation Using Higher-Order Finite Difference Methods and Their Stability

One-dimensional heat equation was solved for different higher-order finite difference schemes, namely, forward time and fourth-order centered space explicit method, backward time and fourth-order centered space implicit method, and fourth-order implicit Crank-Nicolson finite difference method. Higher-order schemes have complexity in computing values at the neighboring points to the boundaries. It is required there a specification of the values of field variables at some points exterior to the domain. The complexity was incorporated using Hicks approximation. The convergence and stability analysis was also computed for those higher-order finite difference explicit and implicit methods in case of solving a one dimensional heat equation. The obtained numerical results were compared with exact solutions. It is found that backward time and fourth-order centered space implicit scheme along with Hicks approximation performed well over the other mentioned higher-order approaches.

HIGH RESOLUTION POSITIVITY-PRESERVING DIFFERENCE SCHEMES FOR TWO DIMENSIONAL EULER EQUATIONS

A class of high resolution positivity preserving Boltzmann type difference schemes for one and two dimensional Euler equations is studied. First, the relation between Boltzmann and Euler equations is analyzed. By using a kind of special interpolation, the high resolution Boltzmann type difference scheme is constructed. Finally, numerical tests show that the schemes are effective and useful.

Global Attractivity of a Kind of Delay Difference Equations

Several new sufficient conditions are given for the global attractivity of solutions of a kind of delay difference equations. They either include or improve some known results and put the study of Ladas' conjecture forward.

Asymptotic Behavior and Nonexistence of Positive Solutions of Delay Difference Equations *L

Aim To obtain new criteria for asymptotic behavior and nonexistence of positive solutions of nonlinear neutral delay difference equations. Methods By means of Hlder inequality and a method of direct analysis, some interesting Lemmas were offered. Results and Conclusion New criteria for asymptotic behavior and nonexistence of positive solutions of nonlinear neutral delay difference equations are established, which extend and improve the results obtained in the literature. Some interesting examples illustrating the importance of our results are also included.

Analysis of Extended Fisher-Kolmogorov Equation in 2D Utilizing the Generalized Finite Difference Method with Supplementary Nodes

In this study,we propose an efficient numerical framework to attain the solution of the extended Fisher-Kolmogorov(EFK)problem.The temporal derivative in the EFK equation is approximated by utilizing the Crank-Nicolson scheme.Following temporal discretization,the generalized finite difference method(GFDM)with supplementary nodes is utilized to address the nonlinear boundary value problems at each time node.These supplementary nodes are distributed along the boundary to match the number of boundary nodes.By incorporating supplementary nodes,the resulting nonlinear algebraic equations can effectively satisfy the governing equation and boundary conditions of the EFK equation.To demonstrate the efficacy of our approach,we present three numerical examples showcasing its performance in solving this nonlinear problem.

New exact solutions to some difference differential equations

In this paper, we use our method to solve the extended Lotka-Volterra equation and discrete KdV equation. With the help of Maple, we obtain a number of exact solutions to the two equations including soliton solutions presented by hyperbolic functions of sinh and cosh, periodic solutions presented by trigonometric functions of sin and cos, and rational solutions. This method can be used to solve some other nonlinear difference-differential equations.

THE GROWTH ORDER OF SOLUTIONS OF SYSTEMS OF COMPLEX DIFFERENCE EQUATIONS

Using Nevanlinna theory of the value distribution of meromorphic functions and Wiman-Valiron theory of entire functions, we investigate the problem of growth order of solutions of a type of systems of difference equations, and extend some results of the growth order of solutions of systems of differential equations to systems of difference equations.

KAMENEV-TYPE OSCILLATION CRITERIA FOR DELAY DIFFERENCE EQUATIONS

Some oscillation criteria are established by Raccati transformation techniques for the following second-order nonlinear neutral difference equation △（pn（△（Xn ＋ CnXn-τ））^γ） ＋ qnX^Bn-σ = 0, n ：0, 1, 2...which extend and include several oscillation criteria in [11], and also correct a theorem and its proof in [10].

MEROMORPHIC SOLUTIONS OF A TYPE OF SYSTEM OF COMPLEX DIFFERENTIAL-DIFFERENCE EQUATIONS

Applying Nevanlinna theory of the value distribution of meromorphic functions, we mainly study the growth and some other properties of meromorphic solutions of the type of system of complex differential and difference equations of the following form {j=1∑nαj（z）f1（λj1）（z＋cj）=R2（z,f2（z））,j=1∑nβj（z）f2（λj2）（z＋cj）=R1（z,f1（z））. where λij （j = 1, 2,…, n; i = 1, 2） are finite non-negative integers, and cj （j = 1, 2,… , n） are distinct, nonzero complex numbers, αj（z）, βj（z） （j = 1,2,… ,n） are small functions relative to fi（z） （i =1, 2） respectively, Ri（z, f（z）） （i = 1, 2） are rational in fi（z） （i =1, 2） with coefficients which are small functions of fi（z） （i = 1, 2） respectively.

ON DIFFERENCE EQUATIONS RELATING TO GAMMA FUNCTION

We consider the existence, the growth, poles, zeros, fixed points and the Borel exceptional value of solutions for the following difference equations relating to Gamma function y（z ＋ 1） -y（z） = R（z） and y（z ＋ 1） = P （z）y（z）.

ZEROS OF ENTIRE SOLUTIONS TO COMPLEX LINEAR DIFFERENCE EQUATIONS

In this article, we study the complex oscillation problems of entire solutions to homogeneous and nonhomogeneous linear difference equations, and obtain some relations of the exponent of convergence of zeros and the order of growth of entire solutions to complex linear difference equations.

ON MEROMORPHIC SOLUTIONS OF RICCATI AND LINEAR DIFFERENCE EQUATIONS

In this paper, meromorphic solutions of Riccati and linear difference equations are investigated. The growth and Borel exceptional values of these solutions are discussed, and the growth, zeros and poles of differences of these solutions are also investigated. Furthermore, several examples are given showing that our results are best possible in certain senses.

ON ENTIRE SOLUTIONS OF SOME TYPE OF NONLINEAR DIFFERENCE EQUATIONS

In this article, the existence of finite order entire solutions of nonlinear difference equations fn＋ Pd（z, f） = p1 eα1 z＋ p2 eα2 z are studied, where n ≥ 2 is an integer, Pd（z, f） is a difference polynomial in f of degree d（≤ n-2）, p1, p2 are small meromorphic functions of ez, and α1, α2 are nonzero constants. Some necessary conditions are given to guarantee that the above equation has an entire solution of finite order. As its applications, we also find some type of nonlinear difference equations having no finite order entire solutions.

An Algebraic Method for Constructing Exact Solutions to Difference-Differential Equations

In this paper, we present a method to solve difference differential equation（s）. As an example, we apply this method to discrete KdV equation and Ablowitz-Ladik lattice equation. As a result, many exact solutions are obtained with the help of Maple including soliton solutions presented by hyperbolic functions sinh and cosh, periodic solutions presented by sin and cos and rational solutions. This method can also be used to other nonlinear difference-differential equation（s）.

SOME DISCRETE NONLINEAR INEQUALITIES AND APPLICATIONS TO DIFFERENCE EQUATIONS

In this article, the authors establish some new nonlinear difference inequalities in two independent variables, which generalize some existing results and can be used as handy tools in the study of qualitative as well as quantitative properties of solutions of certain classes of difference equations.