Behavior Of Nonlinear Differential Equations Research Articles

New Computation of Unified Bounds via a More General Fractional Operator Using Generalized Mittag–Leffler Function in the Kernel

In the present case, we propose the novel generalized fractional integral operator describing Mittag–Leffler function in their kernel with respect to another function Ф. The proposed technique is to use graceful amalgamations of the Riemann–Liouville (RL) fractional integral operator and several other fractional operators. Meanwhile, several generalizations are considered in order to demonstrate the novel variants involving a family of positive functions n (n ∈ N) for the proposed fractional operator. In order to confirm and demonstrate the proficiency of the characterized strategy, we analyze existing fractional integral operators in terms of classical fractional order. Meanwhile, some special cases are apprehended and the new outcomes are also illustrated. The obtained consequences illuminate that future research is easy to implement, profoundly efficient, viable, and exceptionally precise in its investigation of the behavior of non-linear differential equations of fractional order that emerge in the associated areas of science and engineering.

A novel approach for nonlinear equations occurs in ion acoustic waves in plasma with Mittag-Leffler law

Purpose The purpose of this paper is to find the solution for special cases of regular-long wave equations with fractional order using q-homotopy analysis transform method (q-HATM). Design/methodology/approach The proposed technique (q-HATM) is the graceful amalgamations of Laplace transform technique with q-homotopy analysis scheme and fractional derivative defined with Atangana-Baleanu (AB) operator. Findings The fixed point hypothesis considered to demonstrate the existence and uniqueness of the obtained solution for the proposed fractional-order model. To illustrate and validate the efficiency of the future technique, the authors analysed the projected nonlinear equations in terms of fractional order. Moreover, the physical behaviour of the obtained solution has been captured in terms of plots for diverse fractional order. Originality/value To illustrate and validate the efficiency of the future technique, we analysed the projected nonlinear equations in terms of fractional order. Moreover, the physical behaviour of the obtained solution has been captured in terms of plots for diverse fractional order. The obtained results elucidate that, the proposed algorithm is easy to implement, highly methodical, as well as accurate and very effective to analyse the behaviour of nonlinear differential equations of fractional order arisen in the connected areas of science and engineering.

Fractional Approach for Equation Describing the Water Transport in Unsaturated Porous Media With Mittag-Leffler Kernel

In this paper, we find the solution for fractional Richards equation describing the water transport in unsaturated porous media using q-homotopy analysis transform method (q-HATM).The proposed technique is graceful amalgamations of Laplace transform technique with q-homotopy analysis scheme, and fractional derivative defined with Atangana-Baleanu (AB) operator. The fixed point hypothesis considered in order to demonstrate the existence and uniqueness of the obtained solution for the proposed fractional order model. In order to validate and illustrate the efficiency of the future technique, we analysed the projected model in terms of fractional order. Meanwhile, the physical behaviour of the q-HATM solutions have been captured in terms of plots for diverse fractional order and the numerical simulation is also demonstrated. The achieved results illuminate that, the future algorithm is easy to implement, highly methodical as well as effective and very accurate to analyse the behaviour of nonlinear differential equations of fractional order arisen in the connected areas of science and engineering.

A topological approach to the existence of solutions for nonlinear differential equations with piecewise constant argument

In this paper, we investigate qualitative behavior of nonlinear differential equations with piecewise constant argument (PCA). A topological approach of Ważewski-type which gives sufficient conditions to guarantee that the graph of at least one solution stays in a given domain is formulated. Moreover, our results are also suitable for a class of general system of discrete equations. By using a regular polyfacial set, we apply our developed topological approach to cellular neural networks (CNNs) with PCA. Some new results are attained to reveal dynamic behavior of CNNs with PCA and discrete-time CNNs. Finally, an illustrative example of CNNs with PCA shows usefulness and effectiveness of our results.

Global asymptotic behavior of nonlinear difference equations

In this paper, the improved 3/2-conditions for the attractivity of the zero solution of nonlinear delay deference equation x ( n + 1 ) − x ( n ) = F ( n , x ( g ( n ) ) ) , n = 0 , 1 , 2 … ( 1 , 1 ) are obtained. Some applications are presented to illustrate the advantage of our results.

The Fibonacci sequence modulo π, chaos and some rational recursive equations

In this paper, we provide the closed form solution to the inter-related equations y n + 2 = y n y n + 1 − 1 y n + y n + 1 and z n + 2 = ( z n + 1 + z n ) mod π . Both of these equations were suggested as open problems in the book by Kocic and Ladas [V.L. Kocic, G. Ladas, Global Behavior of Nonlinear Difference Equations of High Order with Applications, Kluwer Academic, Dordrecht, 1993]. We also give the closed form solution to the equations x n + 1 = x n x n − 2 + a x n + x n − 2 and x n + 1 = x n − 1 x n − 2 + a x n − 1 + x n − 2 , studied by X. Li and D. Zhu [X. Li, D. Zhu, Two rational recursive sequences, Comput. Math. Appl. 47 (2004) 1487–1494].

Sharp Boundedness Conditions for a Difference Equation via the Chebyshev Polynomials

Sufficient conditions are provided for the boundedness of all positive solutions of the nonlinear difference equation where and is a given function. In case the classical Chebyshev polynomials of the second kind are used to obtain such sharp sufficient conditions. Also some convergence results are given. The results extend those given in [E. Camuzis, G. Ladas, I.W. Rodrigues and S. Northshield, The rational recursive sequence Advances in difference equations, Comp. Math. Appl. 28 (1994), 37–43; E. Camuzis, E.A. Grove, G. Ladas and V.L. Kosić, Monotone unstable solutions of difference equations and conditions for boundedness, J. Differ. Equations Appl. 1 (1995), 17–44; George L. Karakostas, Asymptotic behavior of the solutions of the difference equation J. Differ. Equations Appl. 9(6) (2001), 599–602; V.L. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order and Applications, Kluwer Academic Publishers, Dordrecht, 1993; Wan-Tong Li, Hong-Rui Sun and Xing-Xue Yan, The asymptotic behavior of a higher order delay nonlinear difference equations, Indian J. Pure Appl. Math. 34(10) (2003), 1431–1441; D.C. Zhang, B. Shi and M.J. Gai, On the rational recursive sequence , Indian J. Pure Appl. Math. (2) 32(5) (2001), 657–663] concerning boundedness of the solutions. toufik@math.haifa.ac.il

Discrete Halanay-type inequalities and applications

In this paper we derive new discrete Halanay-type inequalities and show some applications in the investigation of the asymptotic behavior of nonlinear difference equations. In particular, difference equations involving the “maximum” functional are considered.

Asymptotic behavior of nonlinear difference equations with delays

This paper deals with asymptotic behavior of nonlinear difference equations with delays. Sufficient conditions for attracting set and basin of attraction of the equations are obtained. Examples are given to show the applicability of the proposed approach.

GLOBAL BEHAVIOR OF NONLINEAR DIFFERENCE EQUATIONS OF HIGHER ORDER WITH APPLICATIONS (Mathematics and its Applications 256)

GLOBAL BEHAVIOR OF NONLINEAR DIFFERENCE EQUATIONS OF HIGHER ORDER WITH APPLICATIONS (Mathematics and its Applications 256)

Stability in variation for nonlinear integro-differential equations

One of the important techniques for investigating the qualitative behavior of nonlinear differential equations is to utilize the corresponding linear variational equations. In this paper, exployting this approach, we investigate the stability properties of nonlinear integro-differential equations by means of corresponding variational systems.

Unscrambling the Concept of Chaos through Thick and Thin: Reply

Melese and Transue rightly distinguish two types of chaos: that in which the scrambled set has positive measure (observable chaos); and that when it has zero measure (unobservable chaos). In a series of studies that began with the reference they cite, Wayne Shafer and I have explored the existence of observable chaos for a Keynesian business cycle model [Day and Shafer, 1985a,b; Day, 1985]. On the basis of this work it is clear that observable chaos can exist and be robust for a continuum of parameter values for a large class of difference equations that arise quite naturally in economics. The functional forms for this class involve piece-wise smooth functions-in effect, switching regimes. Readers concerned with these issues may also be interested to learn of extensive computational studies, carried out by Pianigiani, of the classic quadratic difference equation that I used as a special case of a more general example and to which Melese and Transue confine their attention. These simulations, which were shown to me in Siena and which augment similar studies by James Yorke (both of these scholars are among the pioneers in the field), show that throughout the range of values 3.57 < m < 4.00, behavior switches from stable, relatively loworder cycles (with thin chaos) to very high-order cycles or thick chaos. The high-order cycles appear to be nonperiodic for finite series shorter than their periodicity. The latter could be in the hundreds, thousands, millions, or higher! One should also be reminded that Lorenz [1963], in one of his seminal papers, provided evidence that the asymptotic behavior of solutions changed erratically within what is now known to be the chaos zone. Putting all this together, we see that (1) thick chaos can be robust in important classes of models; (2) thin chaos may frequently be associated with stable, but high-order cycles that are practically indistinguishable from chaos; and (3) the long-run behavior of nonlinear difference equations typically switches frequently and sometimes erratically with parameter shifts. In a nutshell, complex, essentially unpredictable behavior is a structurally stable feature of nonlinear difference equations through

Oscillatory behavior of nonlinear differential equations with deviating arguments

New oscillation criteria for nonlinear differential equations with deviating arguments of the formn even, are established.

Asymptotic behavior of non-linear differential equations via non-standard analysis. Part III. Boundedness and monotone behavior of the equation (a(t)φ(x)x')' + c(t)f(x) = q(t)

Asymptotic behavior of non-linear differential equations via non-standard analysis. Part III. Boundedness and monotone behavior of the equation (a(t)φ(x)x')' + c(t)f(x) = q(t)

Qualitative behavior of nonlinear differential equations describing adaptive filters using nonideal multipliers

We analyze the behavior of a well-known class of analog adaptive filters in which all the multipliers are replaced by nonideal multipliers for which the linearity requirement with respect to one of the inputs is relaxed. Applications of these filters have been proposed in many different engineering contexts which have in common the following idealized problem: a system has a vector input x <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> and a scalar output <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z_{t} = h' x_{t}</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</tex> is an unknown lime-invariant coefficient vector. From a knowledge of x <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> and z <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> it is required to estimate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</tex> . The estimate vector is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(\hat{h_{t}})</tex> where h <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> is generated in the filter as the solution of a differential equation and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(\cdot)</tex> is a nonlinear map defined by the characteristics of the nonideal multipliers. The effectiveness of the filter is determined by the convergence properties of the misalignment vector, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r=h-f(\hat{h})</tex> . With a weak nondegeneracy requirement on x <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> , the "mixing condition," we prove the exponential convergence of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\parallel r_{t} \parallel</tex> . Considerable emphasis is placed on the analysis of the effects of various important departures from the idealized problem as when noise is present, the coefficient vector <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</tex> is time varying, etc. The results show that in every important aspect the qualitative behavior is similar to that of the conventional filter in which ideal multipliers are used. However, new techniques are required for the analysis of this inherently nonlinear system.

On Stochastic Processes Defined by Differential Equations with a Small Parameter

On Stochastic Processes Defined by Differential Equations with a Small Parameter