Dynamics of a Higher‐Order Nonlinear Difference Equation

This is a continuation part of our investigation in which the second order nonlinear rational difference equation xn+1=(α+βxn+γxn-1)/(A+Bxn+Cxn-1), n=0,1,2,…, where the parameters A≥0 and B, C, α, β, γ are positive real numbers and the initial conditions x-1, x0 are nonnegative real numbers such that A+Bx0+Cx-1>0, is considered. The first part handled the global asymptotic stability of the hyperbolic equilibrium solution of the equation. Our concentration in this part is on the global asymptotic stability of the nonhyperbolic equilibrium solution of the equation.

In this article, we study the periodicity, the boundedness and the global stability of the positive solutions of the following nonlinear difference equation xn+1 = Axn +Bxn−k + Cxn−l +Dxn−σ + bxnxn−kxn−l dxn−k − exn−l , n = 0, 1, 2, ..... where the coefficients A,B,C,D, b, d, e ∈ (0,∞), while k, l and σ are positive integers. The initial conditions x−σ,..., x−l,..., x−k, ..., x−1, x0 are arbitrary positive real numbers such that k < l < σ. Some numerical examples will be given to illustrate our results.

Global asymptotic stability of equilibrium point for a family of rational difference equations

Solution of nonlinear partial differential equations from base equations

On a System of Two Nonlinear Difference Equations

Dynamics of a rational difference equation

In this paper, we investigate the global attractivity of the difference equation x n + 1 = A − B x n − 2 C + D x n − 1 , n = 0 , 1 , … , where A, B are nonnegative real numbers, C, D are positive real numbers and C + D x n − 1 ≠ 0 for all n ≥ 0.

We study the oscillatory behavior, the periodicity and the asymptotic behavior of the positive solutions of the system of two nonlinear difference equations xn+1 = A + xn−1/yn and yn+1 = A + yn−1/xn, where A is a positive constant, and n = 0, 1, ….

In this paper, we deal with the global behavior of the positive solutions of the system of k -difference equations u n + 1 1 = α 1 u n − 1 1 / β 1 + α 1 u n − 2 2 r 1 , u n + 1 2 = α 2 u n − 1 2 / β 2 + α 2 u n − 2 3 r 2 , … , u n + 1 k = α k u n − 1 k / β k + α k u n − 2 1 r k , n ∈ ℕ 0 , where the initial conditions u − l i l = 0,1,2 are nonnegative real numbers and the parameters α i , β i , γ i , and r i are positive real numbers for i = 1,2 , … , k , by extending some results in the literature. By the end of the paper, we give three numerical examples to support our theoretical results related to the system with some restrictions on the parameters.

Fortran programs for the time-dependent Gross–Pitaevskii equation in a fully anisotropic trap

In the paper, two new analytic methods using the decomposition and linearization technique on nonlinear differential/integral equations are developed, namely, the decomposition–linearization–sequential method (DLSM) and the linearized homotopy perturbation method (LHPM). The DLSM is realized by an integrating factor and the integral of certain function obtained at the previous step for obtaining a sequential analytic solution of nonlinear differential equation, which provides quite accurate analytic solution. Some first- and second-order nonlinear differential equations display the fast convergence and accuracy of the DLSM. An analytic approximation for the Volterra differential–integral equation model of the population growth of a species is obtained by using the LHPM. In addition, the LHPM is also applied to the first-, second-, and third-order nonlinear ordinary differential equations. To reduce the cost of computation of He’s homotopy perturbation method and enhance the accuracy for solving cubically nonlinear jerk equations, the LHPM is implemented by invoking a linearization technique in advance is developed. A generalization of the LHPM to the nth-order nonlinear differential equation is involved, which can greatly simplify the work to find an analytic solution by solving a set of second-order linear differential equations. A remarkable feature of those new analytic methods is that just a few steps and lower-order approximations are sufficient for producing reasonably accurate analytic solutions. For all examples, the second-order analytic solution x2(t) is found to be a good approximation of the real solution. The accuracy of the obtained approximate solutions are identified by the fourth-order Runge–Kutta method. The major objection is to unify the analytic solution methods of different nonlinear differential equations by simply solving a set of first-order or second-order linear differential equations. It is clear that the new technique considerably saves computational costs and converges faster than other analytical solution techniques existing in the literature, including the Picard iteration method. Moreover, the accuracy of the obtained analytic solution is raised.

A possibility for predicting the time-dependent pattern of sickness of human beings, i.e., biological catastrophe, has been shown by proposal of a nonlinear ordinary differential equation, describing temporal features of six macroscopic molecular groups chemically interacting in living beings, (Naitoh in Jpn J Ind Appl Math 28:15–26, 2011; Naitoh and Inoue in J Artif Life Robot 18:127–132, 2013). Logically, we also find that, six is minimum and essential as the number of macroscopic molecular groups for describing living systems. Then, along with the number theory applied for the differential equation, we derive critical mathematical conditions for predicting the premonition just before sickness (discrepancy from a healthy condition), which agree with an important knowledge revealed by the linear analysis proposed by Chen (Dynamical network biomarkers for identifying early-warning signals of complex diseases, Beppu, Oita Japan, 2015). In the present report, we first show that computational several time-histories of sickness obtained by solving the nonlinear differential equation with various parameters describing polymorphism agree well with actual time-dependent patterns of sickness for human beings, which is further evidence of usefulness of the nonlinear differential equation and its critical mathematical conditions. Thus, to examine the boundary between biological and abiological chemical reaction systems, which is related to the origin of living systems, we next check whether or not the nonlinear equation can also predict such abiological catastrophic phenomenon as misfire in artifacts including mechanical combustion engines.

In the current paper, a simplified model of Tower Cranes has been presented in order to investigate and analyze the nonlinear differential equation governing on the presented system in three different cases by Algebraic Method (AGM). Comparisons have been made between AGM and Numerical Solution, and these results have been indicated that this approach is very efficient and easy so it can be applied for other nonlinear equations. It is citable that there are some valuable advantages in this way of solving differential equations and also the answer of various sets of complicated differential equations can be achieved in this manner which in the other methods, so far, they have not had acceptable solutions. The simplification of the solution procedure in Algebraic Method and its application for solving a wide variety of differential equations not only in Vibrations but also in different fields of study such as fluid mechanics, chemical engineering, etc. make AGM be a powerful and useful role model for researchers in order to solve complicated nonlinear differential equations.

Improved algorithms for the continuous tree edge-partition problems and a note on ratio and sorted matrices searches

Nonlinear differential equations arise widely in applied mathematics, physics, and engineering, yet many conventional analytical and numerical methods remain limited in their ability to handle strong nonlinearities efficiently and accurately. This paper presents a novel computational framework based on the Modified Laplace–Adomian Polynomial Method (LAPM) for solving nonlinear differential equations. The proposed method integrates the Laplace transform with an enhanced form of the Adomian Decomposition Method, enabling complex nonlinear terms to be decomposed into rapidly convergent Adomian polynomials. This integration simplifies the solution procedure, reduces computational complexity, and preserves high accuracy. The performance of LAPM was validated using several benchmark nonlinear and linear differential equations, and the results demonstrated superior convergence speed, precision, and stability compared with traditional methods. The study concludes that the Modified Laplace–Adomian Polynomial Method is a reliable and efficient approach for solving a broad class of nonlinear differential equations. This work contributes to the advancement of computational methods by offering a robust alternative for the analysis of differential equation models encountered in mathematics, physics, and engineering.