Class Of Nonlinear Differential Equations Research Articles

Diagnostic Inferring on the Bases of Nonlinear Models

Physical phenomena accompanying destruction processes of technical systems are nonlinear and “low-energetic” by nature, while during wearing out mainly a nonlinear disturbance changes. Out of many inference techniques — on the observation basis of the state of the system — the best one is undoubtedly the well-defined mathematical model, allowing inferring ‘backwards’ and ‘forward’, which means finding the genesis and prognosis of the phenomenon. However, such model should be nonlinear. Problems related to the identification of nonlinear dynamic models in a frequency domain and a proposition of solving this problem for the needs of technical diagnostics, i.e. in situations when the observed wear out effects are significantly smaller than the dynamic effects — are discussed in the hereby paper. The bases of the proposed method constitute the discussion of possible solutions of a certain class of nonlinear differential equations and resulting from that statements on the possibility of nonlinear disturbance approximations by a series of the selected harmonic frequencies.

Approximate solutions to the nonlinear vibrations of multiwalled carbon nanotubes using Adomian decomposition method

This paper applies the Adomian decomposition method (ADM) to the search for the approximate solutions to the problem of the nonlinear vibrations of multiwalled carbon nanotubes embedded in an elastic medium. A multiple-beam model is utilized in which the governing equations of each layer are coupled with those of its adjacent ones via the van der Waals inter layer forces. The amplitude–frequency curves for large-amplitude vibrations of single-walled, double-walled and triple-walled carbon nanotubes are obtained. The influence of changes in material constants of the surrounding elastic medium and the effect of changes in nanotube geometrical parameters on the vibration characteristics are studied by comparing the results with those from the open literature. This method needs less work in comparison with the traditional methods and decreases considerable volume of calculation, and it’s powerful mathematical tool for solving wide class of nonlinear differential equations. Special attention is given to prove the convergence of the method. Some examples are given to illustrate the determination approximate solutions of the proposed problem.

Q-Analogues of Freud weights and nonlinear difference equations

In this paper we derive the nonlinear recurrence relation for the recursion coefficients β n of polynomials orthogonal with respect to q-analogues of Freud exponential weights. An asymptotic relation for β n is given under assuming a certain smoothing condition and the Plancherel–Rotach asymptotic for the corresponding orthogonal polynomials is derived. Special interest is paid to the case m = 2 . We prove that the nonlinear recurrence relation of β n in this case obeys the discrete Painlevé property. Motivated by Lew and Quarles, we study possible periodic solutions for a class of nonlinear difference equations of second order. Finally we prove that the Bernstein approximation problem associated to the weights under consideration has a positive solution.

Asymptotic Stability for a Class of Nonlinear Difference Equations

We study the global asymptotic stability of the equilibrium point for the fractional difference equation xn+1 = (axn−lxn−k)/(α + bxn−s + cxn−t), n = 0,1, …, where the initial conditions x−r, x−r+1, …, x1, x0 are arbitrary positive real numbers of the interval (0, α/2a), l, k, s, t are nonnegative integers, r = max {l, k, s, t} and α, a, b, c are positive constants. Moreover, some numerical simulations are given to illustrate our results.

Existence of Homoclinic Solutions for a Class of Nonlinear Difference Equations

Existence of Homoclinic Solutions for a Class of Nonlinear Difference Equations

Existence of Homoclinic Solutions for a Class of Nonlinear Difference Equations

By using the critical point theory, we establish some existence criteria to guarantee that the nonlinear difference equation has at least one homoclinic solution, where , and is non periodic in . Our conditions on the nonlinear term are rather relaxed, and we generalize some existing results in the literature.

Homogenization with uncertain input parameters

We homogenize a class of nonlinear differential equations set in highly heterogeneous media. Contrary to the usual approach, the coefficients in the equation characterizing the material properties are supposed to be uncertain functions from a given set of admissible data. The problem with uncertainties is treated by means of the worst scenario method, when we look for a solution which is critical in some sense.

A result on second order nonlinear operators arising in high-speed networking applications

We consider the behaviour of a class of nonlinear difference equations that arise in networks employing congestion control protocols. For a class of such systems we show that trajectories, experiencing the same set of congestion notifications, but starting at different initial conditions, converge exponentially to each other.

Asymptotic Behavior of Equilibrium Point for a Class of Nonlinear Difference Equation

We study the asymptotic behavior of the solutions for the following nonlinear difference equation where the initial conditions are arbitrary nonnegative real numbers, are nonnegative integers, , and are positive constants. Moreover, some numerical simulations to the equation are given to illustrate our results.

Solutions to a class of nonlinear differential equations of fractional order

In this paper we investigate the formulation of a class of boundary value problems of fractional order with the Riemann-Liouville fractional derivative and integral-type boundary conditions. The existence of solutions is established by applying a fixed point theorem of Krasnosel’skiuo and Zabreiko for asymptotically linear mappings.

Comparing numerical methods for Boussinesq equation model problem

AbstractThis article applies three methods to solve a class of nonlinear differential equations. We obtain the exact solution and numerical solution of the Boussinesq equation for certain initial condition. Comparsion of the results with those of other methods have led us to significant consequences. The numerical solutions are compared with the known analytical solutions. The numerical results demonstrate that those methods are quite accurate and readily implemented. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009

Global Asymptotic Stability for a Class of Nonlinear Chemical Equations

We consider a class of nonlinear differential equations that arises in the study of chemical reaction systems known to be locally asymptotically stable and prove that they are in fact globally asymptotically stable. More specifically, we will consider chemical reaction systems that are weakly reversible, have a deficiency of zero, and are equipped with mass action kinetics. We show that if for each $c \in \mathbb{R}_{>0}^m$ the intersection of the stoichiometric compatibility class $c + S$ with the subsets on the boundary that could potentially contain equilibria, $L_W$, are at most discrete, then global asymptotic stability follows. Previous global stability results for the systems considered in this paper required $(c + S) \cap L_W = \emptyset$ for each $c \in \mathbb{R}^m_{> 0}$.

Nonlinear differential equations of second Painlevé type with the quasi-Painlevé property

We present a class of nonlinear differential equations of second Painleve type. These equations, with a single exception, admit the quasi-Painleve property along a rectifiable curve, that is, for general solutions, every movable singularity defined by a rectifiable curve is at most an algebraic branch point. Moreover we discuss the global many-valuedness of their solutions. For the exceptional equation, by the method of successive approximation, we construct a general solution having a movable logarithmic branch point.

Oscillation criteria for second-order nonlinear differential equations with damping

For a second-order nonlinear differential equation with a damping term, we obtain new oscillation criteria without an assumption that has been required for related results reported in the literature over the last two decades. Two approaches refining the standard integral averaging technique suggested in the paper can be used to derive a variety of simpler oscillation theorems for different classes of nonlinear differential equations, and both sets of oscillation criteria established in the paper are of independent interest. Several examples are provided to illustrate the relevance of the new theorems.

Modified variational iteration method

In this Letter, we introduce a modified variational iteration method by inserting some unknown parameters into the correctional functional. The main advantage of this method is that one can avoid the uncontrollability problems, of the nonzero endpoint conditions, encountered in the traditional variational iteration method. Moreover, the method is applied to some nonlinear equations and the numerical solutions reveal that the modified method is accurate and efficient to solve a large class of nonlinear differential equations. Furthermore, the method does not share the drawbacks of the conventional variational iteration method, namely the restriction of the order of the nonlinearity term or even the form of the boundary conditions.

Approximate analytical solutions for a class of nonlinear differential equations

This paper presents a theoretical analysis for a class of generalized diffusion equations. An effective analytical decomposition and numerical technique is given by similarity transformations and the analytical decomposition technique. The approximate analytical solution may be expressed in terms of a rapid convergent power series with elegantly computable terms. The reliability and efficiency of approximate solutions are illustrated by good agreement with numerical results.

Periodic solutions of a class of nonlinear difference equations via critical point method

In this paper, we obtain some new sufficient conditions for the existence of nontrivial m-periodic solutions of the following nonlinear difference equation Δ ( p n ( Δ x n − 1 ) δ ) + f ( n , x n ) , n ∈ Z , by using the critical point method, where f: Z × R → R is continuous in the second variable, m ≥ 2 is a given positive integer, p n+ m = p n for any n ∈ Z and f( t + m, z) = f( t, z) for any ( t, z) ∈ Z × R, (−1) δ = −1 and δ > 0.

Frequency locking and complex dynamics near a periodically forced robust heteroclinic cycle

Robust heteroclinic cycles occur naturally in many classes of nonlinear differential equations with invariant hyperplanes. In particular they occur frequently in models for ecological dynamics and fluid mechanical instabilities. We consider the effect of small-amplitude time-periodic forcing and describe how to reduce the dynamics to a two-dimensional map. In the limit where the heteroclinic cycle loses asymptotic stability, intervals of frequency locking appear. In the opposite limit, where the heteroclinic cycle becomes strongly stable, the dynamics remains chaotic and no frequency locking is observed.

A qualitative mathematical analysis of a class of nonlinear differential equations: Applications in biochemistry

In this paper, we develop a mathematical approach which can be used to display, in a systematic, rigorous and efficient way, various qualitative properties of a large class of dynamical and quasi-steady biochemical models. The applicability of the methodology has been examined for various biochemical reactions, enzyme kinetics and multi enzyme systems.

Conditions for the existence of bounded solutions of one class of nonlinear differential equations

For systems of nonlinear differential equations (dx/dt) = A(x)x + f(t) in a Banach space, we establish sufficient conditions for the existence of their solutions bounded on the entire real axis R.