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Статті в журналах з теми "Difference equation system"

Автор: Grafiati

Опубліковано: 11 березня 2023

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1

Pasáčková, Jana. "Neutral Difference System and its Nonoscillatory Solutions." Tatra Mountains Mathematical Publications 71, no. 1 (December 1, 2018): 139–48. http://dx.doi.org/10.2478/tmmp-2018-0012.

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Анотація:

Abstract The paper deals with a system of four nonlinear difference equations where the first equation is of a neutral type. We study nonoscillatory solutions of the system and we present sufficient conditions for the system to have weak property B.

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2

Pogrebkov, Andrei. "Hirota Difference Equation and Darboux System: Mutual Symmetry." Symmetry 11, no. 3 (March 25, 2019): 436. http://dx.doi.org/10.3390/sym11030436.

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We considered the relation between two famous integrable equations: The Hirota difference equation (HDE) and the Darboux system that describes conjugate curvilinear systems of coordinates in R 3 . We demonstrated that specific properties of solutions of the HDE with respect to independent variables enabled introduction of an infinite set of discrete symmetries. We showed that degeneracy of the HDE with respect to parameters of these discrete symmetries led to the introduction of continuous symmetries by means of a specific limiting procedure. This enabled consideration of these symmetries on equal terms with the original HDE independent variables. In particular, the Darboux system appeared as an integrable equation where continuous symmetries of the HDE served as independent variables. We considered some cases of intermediate choice of independent variables, as well as the relation of these results with direct and inverse problems.

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3

Khaliq, Abdul, Muhammad Adnan, and Abdul Qadeer Khan. "Global Dynamics of Sixth-Order Fuzzy Difference Equation." Mathematical Problems in Engineering 2021 (October 5, 2021): 1–16. http://dx.doi.org/10.1155/2021/9769093.

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Across many fields, such as engineering, ecology, and social science, fuzzy differences are becoming more widely used; there is a wide variety of applications for difference equations in real-life problems. Our study shows that the fuzzy difference equation of sixth order has a nonnegative solution, an equilibrium point and asymptotic behavior. y i + 1 = D y i − 1 y i − 2 / E + F y i − 3 + G y i − 4 + H y i − 5 , i = 0,1,2 , … , where y i is the sequence of fuzzy numbers and the parameter D , E , F , G , H and the initial condition y − 5 , y − 4 , y − 3 , y − 2 , y − 1 , y 0 are nonnegative fuzzy number. The characterization theorem is used to convert each single fuzzy difference equation into a set of two crisp difference equations in a fuzzy environment. So, a pair of crisp difference equations is formed by converting the difference equation. The stability of the equilibrium point of a fuzzy system has been evaluated. By using variational iteration techniques and inequality skills as well as a theory of comparison for fuzzy difference equations, we investigated the governing equation dynamics, such as its boundedness, existence, and local and global stability analysis. In addition, we provide some numerical solutions for the equation describing the system to verify our results.

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4

MIHAILOVIĆ, DRAGUTIN T., and GORDAN MIMIĆ. "KOLMOGOROV COMPLEXITY AND CHAOTIC PHENOMENON IN COMPUTING THE ENVIRONMENTAL INTERFACE TEMPERATURE." Modern Physics Letters B 26, no. 27 (September 24, 2012): 1250175. http://dx.doi.org/10.1142/s0217984912501758.

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In this paper, we consider the chaotic phenomenon and Kolomogorov complexity in computing the environmental interface temperature. First, the environmental interface is defined in the context of the complex system, in particular for autonomous dynamical systems. Then we consider the following issues in modeling procedure: (i) how to replace given differential equations by appropriate difference equations in modeling of phenomena in the environmental world? (ii) whether a mathematically correct solution to the corresponding differential equation or system of equations is always physically possible and (iii) phenomenon of chaos in autonomous dynamical systems in environmental problems, in particular in solving the energy balance equation to calculate environmental interface temperature. The difference form of this equation for computing the environmental interface temperature is discussed and analyzed depending on parameters of equation, using the Lyapunov exponent and sample entropy. Finally, the Kolmogorov complexity of time series obtained from this difference equation is analyzed.

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5

İnan, B., and A. R. Bahadir. "Numerical solutions of the generalized Burgers-Huxley equation by implicit exponential finite difference method." Journal of Applied Mathematics, Statistics and Informatics 11, no. 2 (December 1, 2015): 57–67. http://dx.doi.org/10.1515/jamsi-2015-0012.

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Abstract In this paper, numerical solutions of the generalized Burgers-Huxley equation are obtained using a new technique of forming improved exponential finite difference method. The technique is called implicit exponential finite difference method for the solution of the equation. Firstly, the implicit exponential finite difference method is applied to the generalized Burgers-Huxley equation. Since the generalized Burgers-Huxley equation is nonlinear the scheme leads to a system of nonlinear equations. Secondly, at each time-step Newton’s method is used to solve this nonlinear system then linear equations system is obtained. Finally, linear equations system is solved using Gauss elimination method at each time-step. The numerical solutions obtained by this way are compared with the exact solutions and obtained by other methods to show the efficiency of the method.

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6

Porter, D., and N. R. T. Biggs. "SYSTEMS OF INTEGRAL EQUATIONS WITH WEIGHTED DIFFERENCE KERNELS." Proceedings of the Edinburgh Mathematical Society 47, no. 1 (February 2004): 205–30. http://dx.doi.org/10.1017/s0013091503000269.

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AbstractExplicit expressions are derived for the inverses of operators of a particular class that includes the operator corresponding to a system of coupled integral equations having weighted difference kernels. The inverses are expressed in terms of a finite number of functions and a systematic way of generating different sets of these functions is devised. The theory generalizes those previously derived for a single integral equation and an integral-equation system with pure difference kernels. The connection is made between the finite generation of inverses and embedding.AMS 2000 Mathematics subject classification: Primary 45A05

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7

Potts, R. B., and X. H. Yu. "Difference equation modelling of a variable structure system." Computers & Mathematics with Applications 28, no. 1-3 (August 1994): 281–89. http://dx.doi.org/10.1016/0898-1221(94)00116-2.

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8

Uslu, K. "GENERALIZED PERIOD OF NON-LINEAR DIFFERENCE EQUATION SYSTEM." Far East Journal of Applied Mathematics 95, no. 6 (January 4, 2017): 451–57. http://dx.doi.org/10.17654/am095060451.

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9

Ma, Junxia, Qiuling Fei, Fan Guo, and Weili Xiong. "Variational Bayesian Iterative Estimation Algorithm for Linear Difference Equation Systems." Mathematics 7, no. 12 (November 22, 2019): 1143. http://dx.doi.org/10.3390/math7121143.

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Анотація:

Many basic laws of physics or chemistry can be written in the form of differential equations. With the development of digital signals and computer technology, the research on discrete models has received more and more attention. The estimates of the unknown coefficients in the discretized difference equation can be obtained by optimizing certain criterion functions. In modern control theory, the state-space model transforms high-order differential equations into first-order differential equations by introducing intermediate state variables. In this paper, the parameter estimation problem for linear difference equation systems with uncertain noise is developed. By transforming system equations into state-space models and on the basis of the considered priors of the noise and parameters, a variational Bayesian iterative estimation algorithm is derived from the observation data to obtain the parameter estimates. The unknown states involved in the variational Bayesian algorithm are updated by the Kalman filter. A numerical simulation example is given to validate the effectiveness of the proposed algorithm.

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10

Kodipaka, Mamatha, Siva Prasad Emineni, and Phaneendra Kolloju. "Difference Scheme for Differential-Difference Problems with Small Shifts Arising in Computational Model of Neuronal Variability." International Journal of Applied Mechanics and Engineering 27, no. 1 (March 1, 2022): 91–106. http://dx.doi.org/10.2478/ijame-2022-0007.

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Abstract The solution of differential-difference equations with small shifts having layer behaviour is the subject of this study. A difference scheme is proposed to solve this equation using a non-uniform grid. With the non-uniform grid, finite - difference estimates are derived for the first and second-order derivatives. Using these approximations, the given equation is discretized. The discretized equation is solved using the tridiagonal system algorithm. Convergence of the scheme is examined. Various numerical simulations are presented to demonstrate the validity of the scheme. In contrast to other techniques, maximum errors in the solution are organized to support the method. The layer behaviour in the solutions of the examples is depicted in graphs.

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11

Farrar, Robert M. "Schnur's Site-Index Curves Formulated for Computer Applications." Southern Journal of Applied Forestry 9, no. 1 (February 1, 1985): 3–5. http://dx.doi.org/10.1093/sjaf/9.1.3.

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Abstract Two systems of interpolation equations are offered for the graphic site-index curves of Schnur for even-aged upland oak stands. One is Wiant's equation system, adjusted to pass through site index at index age. The second is similar but some-what more precise. For both, the average absolute difference between equation system and table values is about 0.73 foot and the maximum absolute difference is about 1.9 feet.

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12

Alamin, Abdul, Sankar Prasad Mondal, Shariful Alam, Ali Ahmadian, Soheil Salahshour, and Mehdi Salimi. "Solution and Interpretation of Neutrosophic Homogeneous Difference Equation." Symmetry 12, no. 7 (July 1, 2020): 1091. http://dx.doi.org/10.3390/sym12071091.

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Анотація:

In this manuscript, we focus on the brief study of finding the solution to and analyzingthe homogeneous linear difference equation in a neutrosophic environment, i.e., we interpreted the solution of the homogeneous difference equation with initial information, coefficient and both as a neutrosophic number. The idea for solving and analyzing the above using the characterization theorem is demonstrated. The whole theoretical work is followed by numerical examples and an application in actuarial science, which shows the great impact of neutrosophic set theory in mathematical modeling in a discrete system for better understanding the behavior of the system in an elegant manner. It is worthy to mention that symmetry measure of the systems is employed here, which shows important results in neutrosophic arena application in a discrete system.

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13

Gepreel, Khaled A., Taher A. Nofal, and Fawziah M. Alotaibi. "Exact Solutions for Nonlinear Differential Difference Equations in Mathematical Physics." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/756896.

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Анотація:

We modified the truncated expansion method to construct the exact solutions for some nonlinear differential difference equations in mathematical physics via the general lattice equation, the discrete nonlinear Schrodinger with a saturable nonlinearity, the quintic discrete nonlinear Schrodinger equation, and the relativistic Toda lattice system. Also, we put a rational solitary wave function method to find the rational solitary wave solutions for some nonlinear differential difference equations. The proposed methods are more effective and powerful to obtain the exact solutions for nonlinear difference differential equations.

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14

Zuev, Sergei. "A finite difference approach to find exact solution of differential equations." International Journal of Modern Physics: Conference Series 38 (January 2015): 1560080. http://dx.doi.org/10.1142/s2010194515600800.

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Анотація:

This paper contains the background and samples of an approach to construct exact solutions of a wide range of differential equations (DEs). This approach is based on the finite difference equation which corresponds to the given DE. There are three cases considered: linear partial differential equation (PDE) with constant coefficients and at least one non-zero root of characteristic equation, linear PDE with constant coefficients and completely zero roots of the characteristic equation, and a case of nonlinear autonomous dynamical system of second order. Each of these cases is illustrated by an example.

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15

Wang, Changyou, and Jiahui Li. "Periodic Solution for a Max-Type Fuzzy Difference Equation." Journal of Mathematics 2020 (July 10, 2020): 1–12. http://dx.doi.org/10.1155/2020/3094391.

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Анотація:

The paper is concerned with the dynamics behavior of positive solutions for the following max-type fuzzy difference equation system: xn+1=maxA/xn, A/xn−1, xn−2, n=0,1,2,…, where xn is a sequence of positive fuzzy numbers, and the parameter A and the initial conditions x−2, x−1, x0 are also positive fuzzy numbers. Firstly, the fuzzy set theory is used to transform the fuzzy difference equation into the corresponding ordinary difference equations with parameters. Then, the expression for the periodic solution of the max-type ordinary difference equations is obtained by the iteration, the inequality technique, and the mathematical induction. Moreover, we can obtain the expression for the periodic solution of the max-type fuzzy difference equation. In addition, the boundedness and persistence of solutions for the fuzzy difference equation is proved. Finally, the results of this paper are simulated and verified by using MATLAB 2016 software package.

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16

Hu, Guozhuan. "On the System of Three Order Rational Difference Equation." British Journal of Applied Science & Technology 17, no. 3 (January 10, 2016): 1–7. http://dx.doi.org/10.9734/bjast/2016/27524.

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17

Zhang, Qianhong, Jianjun Jiao, Wenzhuan Zhang, and Yuanfu Shao. "Dynamical behaviour of high-order rational difference equation system." International Journal of Dynamical Systems and Differential Equations 6, no. 4 (2016): 335. http://dx.doi.org/10.1504/ijdsde.2016.081817.

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18

Shao, Yuanfu, Jianjun Jiao, Wenzhuan Zhang, and Qianhong Zhang. "Dynamical behaviour of high-order rational difference equation system." International Journal of Dynamical Systems and Differential Equations 6, no. 4 (2016): 335. http://dx.doi.org/10.1504/ijdsde.2016.10002713.

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19

Çinar, Cengiz. "On the positive solutions of the difference equation system ,." Applied Mathematics and Computation 158, no. 2 (November 2004): 303–5. http://dx.doi.org/10.1016/j.amc.2003.08.073.

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20

Ibrahim, Tarek F., Somayah Refaei, Abdul Khaliq, Mohamed Abd El-Moneam, Bakri A. Younis, Waleed M. Osman, and Bushra R. Al-Sinan. "Qualitative Behavior of an Exponential Symmetric Difference Equation System." Symmetry 14, no. 12 (November 22, 2022): 2474. http://dx.doi.org/10.3390/sym14122474.

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Анотація:

We examine the unboundedness, persistence, boundedness, uniqueness, and existence of non-negative equilibrium of an exponential symmetric difference equations system: Ωn+1=α1+β1Ωn+γ1Ωn−1e−(Ωn+ϖn), ϖn+1=α2+β2ϖn+γ2ϖn−1e−(Ωn+ϖn),n=0,1,⋯, whereby initial values Ω−1,ϖ−1,Ω0,ϖ0 and parameters α1,α2 are non-negative real numbers and β1,β2∈(0,1) and γ1,γ2≤0. We will discuss asymptotic global and local stability and the convergence rate of this system. Ultimately, to check our results, we set out some numerical explanations.

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21

Matus, P., and A. Kołodyńska. "Exact Difference Schemes for Hyperbolic Equations." Computational Methods in Applied Mathematics 7, no. 4 (2007): 341–64. http://dx.doi.org/10.2478/cmam-2007-0021.

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AbstractIn the present paper, an exact difference scheme for the initial boundary- value problem of the third kind for an inhomogeneous hyperbolic equation of the second order with constant coefficients has been constructed on ordinary rectangular grids with constant space and time steps, where the Courant number γ=1. Later we proved a priori estimates of the stability in energy norm. For a quasi-linear wave equation on the moving characteristic grid a difference scheme has been constructed, which has the second order of approximation for the initial boundary-value problem and is exact for the Cauchy problem. The computational results for smooth functions and for a weak solution confirm the high accuracy of the introduced algorithm. We have also constructed exact difference schemes for the Cauchy problem for a system of two hyperbolic equations of the first order with constant coefficients on grids with constant space and time steps. Stability in energy norm for one of the constructed schemes has been proved. Using a method analogous to that used for the nonlinear wave equation a difference scheme for a nonlinear gas dynamic system has been constructed.

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22

Alymbaev, Asangul. "A SUM-DIFFERENCE METHOD FOR CONSTRUCTING AN ASYMPTOTIC SOLUTION TO A BOUNDARY VALUE PROBLEM OF A NONLINEAR DIFFERENCE EQUATION WITH A SMALL PARAMETER." Alatoo Academic Studies 19, no. 4 (December 30, 2019): 255–59. http://dx.doi.org/10.17015/aas.2019.194.29.

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Анотація:

Finite-difference equations proved to be a convenient mathematical model on describing impulse systems, combinatorial analysis problems, discrete analogues of mathematical physics equations, financial analysis tasks, etc. Oneshould point out that difference equations are encountered in the numerical solution of various classes of differential and integro- differential ones using the finite difference method. The article deals with methods of constructing an asymptotic solution to the boundary value problem of a system of a nonlinear difference equation with a small parameter. The problem is solved by reducing the boundary-value problem to the Cauchy problem for a system of total-difference equations with a small parameter. The efficiency of the method algorithm for the asymptotic expansion of the task of a boundary value problem in a definite example is shown.

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23

FRICKE, J. ROBERT. "QUASI-LINEAR ELASTODYNAMIC EQUATIONS FOR FINITE DIFFERENCE SOLUTIONS IN DISCONTINUOUS MEDIA." Journal of Computational Acoustics 01, no. 03 (September 1993): 303–20. http://dx.doi.org/10.1142/s0218396x93000160.

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The linear elastodynamic equations are ill-posed for models which contain high contrast density discontinuities. This paper presents a quasi-linear superset of the linear equations that is well-posed for this situation. The extended system contains a conservation of mass equation and a quasi-linear convective term in the momentum equation. Density, momentum, and stress are the field variables in the quasi-linear system, which is cast in a first order form. Using a Lax–Wendroff finite difference approximation, the utility of the quasi-linear system is demonstrated by modeling underwater acoustic scattering from a truncated ice sheet. The model contains air, ice, and water with a density contrast between air and ice or water of O(103). Superlinear convergence of the Lax–Wendroff scheme is demonstrated for his heterogeneous medium problem.

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24

Tuz, Münevver. "On The Solutions of Some Difference Equations Systems and Analytical Properties." ITM Web of Conferences 22 (2018): 01050. http://dx.doi.org/10.1051/itmconf/20182201050.

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Анотація:

In this study, we investigated the global asymptotic behaviors of their solutions by taking the second-order difference equation system. According to the given conditions, we obtained some asymptotic results for the positive balance of the system. We have also worked on q-fast changing functions. Such functions form the class of q-Caramate functions. We have applied q-Caramate functions to linear q-difference equations and We have also learned about the asymptotic behavior of solutions. In addition, we have studied the problem of initial and boundary value for the q-difference equation

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25

YUAN, ZHIJIANG, LIANGAN JIN, WEI CHI, and HENGDOU TIAN. "FINITE DIFFERENCE METHOD FOR SOLVING THE NONLINEAR DYNAMIC EQUATION OF UNDERWATER TOWED SYSTEM." International Journal of Computational Methods 11, no. 04 (August 2014): 1350060. http://dx.doi.org/10.1142/s0219876213500606.

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Анотація:

A wide body of work exists that describes numerical solution for the nonlinear system of underwater towed system. Many researchers usually divide the tow cable with less number elements for the consideration of computational time. However, this type of installation affects the accuracy of the numerical solution. In this paper, a newly finite difference method for solving the nonlinear dynamic equations of the towed system is developed. The mathematical model of tow cable and towed body are both discretized to nonlinear algebraic equations by center finite difference method. A newly discipline for formulating the nonlinear equations and Jacobian matrix of towed system are proposed. We can solve the nonlinear dynamic equation of underwater towed system quickly by using this discipline, when the size of number elements is large.

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26

Halim, Yacine, Asma Allam, and Zineb Bengueraichi. "Dynamical behavior of a P-dimensional system of nonlinear difference equations." Mathematica Slovaca 71, no. 4 (August 1, 2021): 903–24. http://dx.doi.org/10.1515/ms-2021-0030.

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Анотація:

Abstract In this paper, we study the periodicity, the boundedness of the solutions, and the global asymptotic stability of the positive equilibrium of the system of p nonlinear difference equations x n + 1 ( 1 ) = A + x n − 1 ( 1 ) x n ( p ) , x n + 1 ( 2 ) = A + x n − 1 ( 2 ) x n ( p ) , … , x n + 1 ( p − 1 ) = A + x n − 1 ( p − 1 ) x n ( p ) , x n + 1 ( p ) = A + x n − 1 ( p ) x n ( p − 1 ) $$\begin{equation*}x^{(1)}_{n+1}=A+\dfrac{x^{(1)}_{n-1}}{x^{(p)}_{n}},\quad x^{(2)}_{n+1}=A+\dfrac{x^{(2)}_{n-1}}{x^{(p)}_{n}},\quad\ldots,\quad x^{(p-1)}_{n+1}=A+\dfrac{x^{(p-1)}_{n-1}}{x^{(p)}_{n}},\quad x^{(p)}_{n+1}=A+\dfrac{x^{(p)}_{n-1}}{x^{(p-1)}_{n}} \end{equation*} $$ where n ∈ ℕ0, p ≥ 3 is an integer, A ∈ (0, +∞) and the initial conditions x − 1 ( j ) $x_{-1}^{(j)}$ , x 0 ( j ) $x_{0}^{(j)}$ , j = 1, 2, …, p are positive numbers.

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27

Skovpen, Sergey Mikhailovich, and Albert Saitovich Iskhakov. "Exact Solution of a Linear Difference Equation in a Finite Number of Steps." JOURNAL OF ADVANCES IN MATHEMATICS 14, no. 1 (April 25, 2018): 7560–63. http://dx.doi.org/10.24297/jam.v14i1.7206.

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Анотація:

An exact solution of a linear difference equation in a finite number of steps has been obtained. This refutes the conventional wisdom that a simple iterative method for solving a system of linear algebraic equations is approximate. The nilpotency of the iteration matrix is the necessary and sufficient condition for getting an exact solution. The examples of iterative equations providing an exact solution to the simplest algebraic system are presented.

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28

Ghezal, Ahmed, and Imane Zemmouri. "Representation of Solutions of a Second-Order System of Two Difference Equations With Variable Coefficients." Pan-American Journal of Mathematics 2 (January 15, 2023): 2. http://dx.doi.org/10.28919/cpr-pajm/2-2.

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Анотація:

A definition of system of two nonlinear difference equations with variable coefficients is given. Our main result shows that the difference equation is solvable in closed form and thus for the constant coefficients. Some applications of the main result are also given.

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29

Wei, Zhijian. "Periodicity in a Class of Systems of Delay Difference Equations." Journal of Applied Mathematics 2011 (2011): 1–11. http://dx.doi.org/10.1155/2011/735825.

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Анотація:

We study a system of delay difference equations modeling four-dimensional discrete-time delayed neural networks with no internal decay. Such a discrete-time system can be regarded as the discrete analog of a differential equation with piecewise constant argument. By using semicycle analysis method, it is shown that every bounded solution of this discrete-time system is eventually periodic. The obtained results are new, and they complement previously known results.

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30

Zlatanovska, Biljana, and Donc̆o Dimovski. "A modified Lorenz system: Definition and solution." Asian-European Journal of Mathematics 13, no. 08 (May 20, 2020): 2050164. http://dx.doi.org/10.1142/s1793557120501648.

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Анотація:

Based on the approximations of the Lorenz system of differential equations from the papers [B. Zlatanovska and D. Dimovski, Systems of difference equations approximating the Lorentz system of differential equations, Contributions Sec. Math. Tech. Sci. Manu. XXXIII 1–2 (2012) 75–96, B. Zlatanovska and D. Dimovski, Systems of difference equations as a model for the Lorentz system, in Proc. 5th Int. Scientific Conf. FMNS, Vol. I (Blagoevgrad, Bulgaria, 2013), pp. 102–107, B. Zlatanovska, Approximation for the solutions of Lorenz system with systems of differential equations, Bull. Math. 41(1) (2017) 51–61], we define a Modified Lorenz system, that is a local approximation of the Lorenz system. It is a system of three differential equations, the first two are the same as the first two of the Lorenz system, and the third one is a homogeneous linear differential equation of fifth order with constant coefficients. The solution of this system is based on the results from [D. Dimitrovski and M. Mijatovic, A New Approach to the Theory of Ordinary Differential Equations (Numerus, Skopje, 1995), pp. 23–33].

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31

Jeon, Youngmok, Eun-Jae Park, and Dong-wook Shin. "Hybrid Spectral Difference Methods for an Elliptic Equation." Computational Methods in Applied Mathematics 17, no. 2 (April 1, 2017): 253–67. http://dx.doi.org/10.1515/cmam-2016-0043.

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Анотація:

AbstractA locally conservative, hybrid spectral difference method (HSD) is presented and analyzed for the Poisson equation. The HSD is composed of two types of finite difference approximations; the cell finite difference and the interface finite difference. Embedded static condensation on cell interior unknowns considerably reduces the global couplings, resulting in the system of equations in the cell interface unknowns only. A complete ellipticity analysis is provided. The optimal order of convergence in the semi-discrete energy norms is proved. Several numerical results are given to show the performance of the method, which confirm our theoretical findings.

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32

Kara, Merve, and Yasin Yazlik. "Solvability of a nonlinear three-dimensional system of difference equations with constant coefficients." Mathematica Slovaca 71, no. 5 (October 1, 2021): 1133–48. http://dx.doi.org/10.1515/ms-2021-0044.

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Abstract In this paper, we show that the following three-dimensional system of difference equations x n + 1 = y n x n − 2 a x n − 2 + b z n − 1 , y n + 1 = z n y n − 2 c y n − 2 + d x n − 1 , z n + 1 = x n z n − 2 e z n − 2 + f y n − 1 , n ∈ N 0 , $$\begin{equation*} x_{n+1}=\frac{y_{n}x_{n-2}}{ax_{n-2}+bz_{n-1}}, \quad y_{n+1}=\frac{z_{n}y_{n-2}}{cy_{n-2}+dx_{n-1}}, \quad z_{n+1}=\frac{x_{n}z_{n-2}}{ez_{n-2}+fy_{n-1}}, \quad n\in \mathbb{N}_{0}, \end{equation*}$$ where the parameters a, b, c, d, e, f and the initial values x −i , y −i , z −i , i ∈ {0, 1, 2}, are complex numbers, can be solved, extending further some results in the literature. Also, we determine the forbidden set of the initial values by using the obtained formulas. Finally, an application concerning a three-dimensional system of difference equations are given.

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33

Kara, Merve, and Yasin Yazlik. "Solvability of a nonlinear three-dimensional system of difference equations with constant coefficients." Mathematica Slovaca 71, no. 5 (October 1, 2021): 1133–48. http://dx.doi.org/10.1515/ms-2021-0044.

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Анотація:

Abstract In this paper, we show that the following three-dimensional system of difference equations x n + 1 = y n x n − 2 a x n − 2 + b z n − 1 , y n + 1 = z n y n − 2 c y n − 2 + d x n − 1 , z n + 1 = x n z n − 2 e z n − 2 + f y n − 1 , n ∈ N 0 , $$\begin{equation*} x_{n+1}=\frac{y_{n}x_{n-2}}{ax_{n-2}+bz_{n-1}}, \quad y_{n+1}=\frac{z_{n}y_{n-2}}{cy_{n-2}+dx_{n-1}}, \quad z_{n+1}=\frac{x_{n}z_{n-2}}{ez_{n-2}+fy_{n-1}}, \quad n\in \mathbb{N}_{0}, \end{equation*}$$ where the parameters a, b, c, d, e, f and the initial values x −i , y −i , z −i , i ∈ {0, 1, 2}, are complex numbers, can be solved, extending further some results in the literature. Also, we determine the forbidden set of the initial values by using the obtained formulas. Finally, an application concerning a three-dimensional system of difference equations are given.

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34

Khakimzyanov, Gayaz S., Zinaida I. Fedotova, Oleg I. Gusev, and Nina Yu Shokina. "Finite difference methods for 2D shallow water equations with dispersion." Russian Journal of Numerical Analysis and Mathematical Modelling 34, no. 2 (April 24, 2019): 105–17. http://dx.doi.org/10.1515/rnam-2019-0009.

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Анотація:

Abstract Basic properties of some finite difference schemes for two-dimensional nonlinear dispersive equations for hydrodynamics of surface waves are considered. It is shown that stability conditions for difference schemes of shallow water equations are qualitatively different in the cases the dispersion is taken into account, or not. The difference in the behavior of phase errors in one- and two-dimensional cases is pointed out. Special attention is paid to the numerical algorithm based on the splitting of the original system of equations into a nonlinear hyperbolic system and a scalar linear equation of elliptic type.

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35

Ekimov, Alexander V., Aleksei P. Zhabko, and Pavel V. Yakovlev. "The stability of differential-difference equations with proportional time delay. I. Linear controlled system." Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes 16, no. 3 (2020): 316–25. http://dx.doi.org/10.21638/11701/spbu10.2020.308.

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The article considers a controlled system of linear differential-difference equations with a linearly increasing delay. Sufficient conditions for the asymptotic stability of such systems are known; however, general conditions for the stabilizability of controlled systems and constructive algorithms for constructing stabilizing controls have not yet been obtained. For a linear differential-difference equation of delayed type with linearly increasing delay, the canonical Zubov transformation is applied and conditions for the stabilization of such systems by static control are derived. An algorithm for checking the conditions for the existence of a stabilizing control and for its constructing is formulated. New theorems on stability analysis of systems of linear differential-difference equations with linearly increasing delay are proven. The results obtained can be applied to the case of systems with several proportional delays.

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36

Dang, Guoqiang, and Jinhua Cai. "Entire Solutions of the Second-Order Fermat-Type Differential-Difference Equation." Journal of Mathematics 2020 (July 7, 2020): 1–8. http://dx.doi.org/10.1155/2020/4871812.

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In this paper, the entire solutions of finite order of the Fermat-type differential-difference equation f″z2+△ckfz2=1 and the system of equations f1″z2+△ckf2z2=1 and f2″z2+△ckf1z2=1 have been studied. We give the necessary and sufficient conditions of existence of the entire solutions of finite order.

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37

Sroysang, Banyat. "Dynamics of a System of Rational Higher-Order Difference Equation." Discrete Dynamics in Nature and Society 2013 (2013): 1–5. http://dx.doi.org/10.1155/2013/179401.

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38

Yazlik, Y., D. T. Tollu, and N. Taskara. "On the solutions of a max-type difference equation system." Mathematical Methods in the Applied Sciences 38, no. 17 (February 2, 2015): 4388–410. http://dx.doi.org/10.1002/mma.3377.

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39

Li, Chuanzhong, and Anni Meng. "On the Full-Discrete Extended Generalised q-Difference Toda System." Zeitschrift für Naturforschung A 72, no. 8 (August 28, 2017): 703–9. http://dx.doi.org/10.1515/zna-2017-0113.

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AbstractIn this paper, we construct a full-discrete integrable difference equation which is a full-discretisation of the generalised q-Toda equation. Meanwhile its soliton solutions are constructed to show its integrable property. Further the Lax pairs of an extended generalised full-discrete q-Toda hierarchy are also constructed. To show the integrability, the bi-Hamiltonian structure and tau symmetry of the extended full-discrete generalised q-Toda hierarchy are given.

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40

Khalaf Hussain, Ali. "Transient-False Method for Solving System of Nonlinear Partial Differential Equations." Journal of Education College Wasit University 1, no. 25 (January 14, 2018): 509–22. http://dx.doi.org/10.31185/eduj.vol1.iss25.137.

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In this paper we study the false transient method to solve and transform a system of non-linear partial differential equations which can be solved using finite-difference method and give some problems which have a good results compared with the exact solution, whereas this method was used to transform the nonlinear partial differential equation to a linear partial differential equation which can be solved by using the alternating-direction implicit method after using the ADI method. The system of linear algebraic equations could be obtained and can be solved by using MATLAB.

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41

Sunandar, Endang, and Indrianto Indrianto. "Perbandingan Metode Newton-Raphson & Metode Secant Untuk Mencari Akar Persamaan Dalam Sistem Persamaan Non-Linier." PETIR 13, no. 1 (March 21, 2020): 72–79. http://dx.doi.org/10.33322/petir.v13i1.893.

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The numerical method is a technique used to formulate mathematical problems so that it can be solved using ordinary arithmetic operations. In general, numerical methods are used to solve mathematical problems that cannot be solved by ordinary analytic methods. In the Numerical Method, we recognize two types of systems of equations, namely the Linear Equation System and the Non-Linear Equation System. Each system of equations has several methods. In the Linear Equation System between methods is the Gauss Elimination method, the Gauss-Jordan Elimination method, the LU (Lower-Upper) Decomposition method. And for Non-Linear Equation Systems between the methods are the Bisection method, the Regula Falsi method, the Newton Raphson method, the Secant method, and the Fix Iteration method. In this study, researchers are interested in analyzing 2 methods in the Non-Linear Equation System, the Newton-Raphson method and the Secant method. And this analysis process uses the Java programming language tools, this is to facilitate the analysis of method completion algorithm, and monitoring in terms of execution time and analysis of output results. So we can clearly know the difference between what happens between the two methods.

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42

Witte, N. S. "Semiclassical orthogonal polynomial systems on nonuniform lattices, deformations of the Askey table, and analogues of isomonodromy." Nagoya Mathematical Journal 219 (September 2015): 127–234. http://dx.doi.org/10.1215/00277630-3140952.

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AbstractA 𝔻-semiclassical weight is one which satisfies a particular linear, first-order homogeneous equation in a divided-difference operator 𝔻. It is known that the system of polynomials, orthogonal with respect to this weight, and the associated functions satisfy a linear, first-order homogeneous matrix equation in the divided-difference operator termed the spectral equation. Attached to the spectral equation is a structure which constitutes a number of relations such as those arising from compatibility with the three-term recurrence relation. Here this structure is elucidated in the general case of quadratic lattices. The simplest examples of the 𝔻-semiclassical orthogonal polynomial systems are precisely those in the Askey table of hypergeometric and basic hypergeometric orthogonal polynomials. However within the 𝔻-semiclassical class it is entirely natural to define a generalization of the Askey table weights which involve a deformation with respect to new deformation variables. We completely construct the analogous structures arising from such deformations and their relations with the other elements of the theory. As an example we treat the first nontrivial deformation of the Askey–Wilson orthogonal polynomial system defined by the q-quadratic divided-difference operator, the Askey–Wilson operator, and derive the coupled first-order divided-difference equations characterizing its evolution in the deformation variable. We show that this system is a member of a sequence of classical solutions to the q-Painlevé system.

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43

Witte, N. S. "Semiclassical orthogonal polynomial systems on nonuniform lattices, deformations of the Askey table, and analogues of isomonodromy." Nagoya Mathematical Journal 219 (September 2015): 127–234. http://dx.doi.org/10.1017/s0027763000027124.

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Анотація:

AbstractA 𝔻-semiclassical weight is one which satisfies a particular linear, first-order homogeneous equation in a divided-difference operator 𝔻. It is known that the system of polynomials, orthogonal with respect to this weight, and the associated functions satisfy a linear, first-order homogeneous matrix equation in the divided-difference operator termed thespectral equation. Attached to the spectral equation is a structure which constitutes a number of relations such as those arising from compatibility with the three-term recurrence relation. Here this structure is elucidated in the general case of quadratic lattices. The simplest examples of the 𝔻-semiclassical orthogonal polynomial systems are precisely those in the Askey table of hypergeometric and basic hypergeometric orthogonal polynomials. However within the 𝔻-semiclassical class it is entirely natural to define a generalization of the Askey table weights which involve a deformation with respect to new deformation variables. We completely construct the analogous structures arising from such deformations and their relations with the other elements of the theory. As an example we treat the first nontrivial deformation of the Askey–Wilson orthogonal polynomial system defined by the q-quadratic divided-difference operator, the Askey–Wilson operator, and derive the coupled first-order divided-difference equations characterizing its evolution in the deformation variable. We show that this system is a member of a sequence of classical solutions to theq-Painlevé system.

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44

Ndayisenga, Serge, Leonid A. Sevastianov, and Konstantin P. Lovetskiy. "Finite-difference methods for solving 1D Poisson problem." Discrete and Continuous Models and Applied Computational Science 30, no. 1 (February 25, 2022): 62–78. http://dx.doi.org/10.22363/2658-4670-2022-30-1-62-78.

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The paper discusses the formulation and analysis of methods for solving the one-dimensional Poisson equation based on finite-difference approximations - an important and very useful tool for the numerical study of differential equations. In fact, this is a classical approximation method based on the expansion of the solution in a Taylor series, based on which the recent progress of theoretical and practical studies allowed increasing the accuracy, stability, and convergence of methods for solving differential equations. Some of the features of this analysis include interesting extensions to classical numerical analysis of initial and boundary value problems. In the first part, a numerical method for solving the one-dimensional Poisson equation is presented, which reduces to solving a system of linear algebraic equations (SLAE) with a banded symmetric positive definite matrix. The well-known tridiagonal matrix algorithm, also known as the Thomas algorithm, is used to solve the SLAEs. The second part presents a solution method based on an analytical representation of the exact inverse matrix of a discretized version of the Poisson equation. Expressions for inverse matrices essentially depend on the types of boundary conditions in the original setting. Variants of inverse matrices for the Poisson equation with different boundary conditions at the ends of the interval under study are presented - the Dirichlet conditions at both ends of the interval, the Dirichlet conditions at one of the ends and Neumann conditions at the other. In all three cases, the coefficients of the inverse matrices are easily found and the algorithm for solving the problem is practically reduced to multiplying the matrix by the vector of the right-hand side.

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45

Narita, K. "Solutions for a System of Difference-Differential Equations Related to the Self-Dual Network Equation." Progress of Theoretical Physics 106, no. 6 (December 1, 2001): 1079–96. http://dx.doi.org/10.1143/ptp.106.1079.

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46

Narita, K. "Solutions for a System of Difference-Differential Equations Related to the Self-Dual Network Equation." Progress of Theoretical Physics 108, no. 1 (July 1, 2002): 229. http://dx.doi.org/10.1143/ptp.108.229.

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47

Braverman, E., and S. H. Saker. "On a Difference Equation with Exponentially Decreasing Nonlinearity." Discrete Dynamics in Nature and Society 2011 (2011): 1–17. http://dx.doi.org/10.1155/2011/147926.

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We establish a necessary and sufficient condition for global stability of the nonlinear discrete red blood cells survival model and demonstrate that local asymptotic stability implies global stability. Oscillation and solution bounds are investigated. We also show that, for different values of the parameters, the solution exhibits some time-varying dynamics, that is, if the system is moved in a direction away from stability (by increasing the parameters), then it undergoes a series of bifurcations that leads to increasingly long periodic cycles and finally to deterministic chaos. We also study the chaotic behavior of the model with a constant positive perturbation and prove that, for large enough values of one of the parameters, the perturbed system is again stable.

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48

Yakar, Coşkun, and Mustafa Bayram Gücen. "Initial Time Difference Stability of Causal Differential Systems in terms of Lyapunov Functions and Lyapunov Functionals." Journal of Applied Mathematics 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/832015.

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We investigate the qualitative behavior of a perturbed causal differential equation that differs in initial position and initial time with respect to the unperturbed causal differential equations. We compare the classical notion of stability of the causal differential systems to the notion of initial time difference stability of causal differential systems and present a comparison result in terms of Lyapunov functions. We have utilized Lyapunov functions and Lyapunov functional in the study of stability theory of causal differential systems when establishing initial time difference stability of the perturbed causal differential system with respect to the unperturbed causal differential system.

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49

Jha, Navnit, Venu Gopal, and Bhagat Singh. "Geometric grid network and third-order compact scheme for solving nonlinear variable coefficients 3D elliptic PDEs." International Journal of Modeling, Simulation, and Scientific Computing 09, no. 06 (December 2018): 1850053. http://dx.doi.org/10.1142/s1793962318500538.

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By using nonuniform (geometric) grid network, a new high-order finite-difference compact scheme has been obtained for the numerical solution of three-space dimensions partial differential equations of elliptic type. Single cell discretization to the elliptic equation makes it easier to compute and exhibit stability of the numerical solutions. The monotone and irreducible property of the Jacobian matrix to the system of difference equations analyses the converging behavior of the numerical solution values. As an experiment, applications of the compact scheme to Schrödinger equations, sine-Gordon equations, elliptic Allen–Cahn equation and Poisson’s equation have been presented with root mean squared errors of exact and approximate solution values. The results corroborate the reliability and efficiency of the scheme.

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50

Wu, Kaining, Xiaohua Ding, and Liming Wang. "Stability and Stabilization of Impulsive Stochastic Delay Difference Equations." Discrete Dynamics in Nature and Society 2010 (2010): 1–15. http://dx.doi.org/10.1155/2010/592036.

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Анотація:

When an impulsive control is adopted for a stochastic delay difference system (SDDS), there are at least two situations that should be contemplated. If the SDDS is stable, then what kind of impulse can the original system tolerate to keep stable? If the SDDS is unstable, then what kind of impulsive strategy should be taken to make the system stable? Using the Lyapunov-Razumikhin technique, we establish criteria for the stability of impulsive stochastic delay difference equations and these criteria answer those questions. As for applications, we consider a kind of impulsive stochastic delay difference equation and present some corollaries to our main results.

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