Order Nonlinear Dynamic Equations Research Articles

New Oscillation Criteria for Higher Order Nonlinear Dynamic Equations

AbstractThis paper deals with some new criteria for the oscillation of higher-order nonlinear dynamic equations with mixed deviating arguments. The purpose of the present paper is the linearization of the equation under consideration. Specifically, we will infer the oscillation of the studied equation from its linear form and establish new oscillation criteria by comparing it with first-order equations whose oscillatory behaviour is known. The obtained results are new, improve, and correlate many of the known oscillation criteria appearing in the literature. The results are illustrated by two examples.

Oscillation Properties of Third Order Nonlinear Delay Dynamic Equations on Time Scales

In this paper, we shall investigate the oscillatory properties of third order nonlinear delay dynamic equations. Applying suitable comparison theorems and by a Riccati transformation technique, we establish some new sufficient conditions which insure that every solution of this equation either oscillates or converges to zero. Our results not only unify the oscillation of third order nonlinear differential and difference equations but also can be applied to different types of time scales with

On the oscillations of the solutions for a class of second order nonlinear functional dynamic equations on time scales with unbounded neutral coefficients

On the oscillations of the solutions for a class of second order nonlinear functional dynamic equations on time scales with unbounded neutral coefficients

Oscillation of higher order nonlinear dynamic equations with a nonlinear neutral term

This study introduces the oscillation of higher order nonlinear dynamic equations with a nonlinear neutral term. The findings are obtained via utilising an integral criterion as well as a comparison theorem with the oscillatory properties of a first‐order dynamic equation. We propose novel oscillation criteria that improve, extend and simplify existing ones in the literature. The results are associated with several numerical examples.

Oscillation of even order nonlinear dynamic equations on time-scales

In this paper, the authors discuss the oscillatory behavior of solutions to a class of even order nonlinear dynamic equations on time scales. The results are established by a comparison with n-th order delay dynamic inequalities or first-order delay dynamic equations whose oscillatory characters are known. Several corollaries are obtained for special cases.

The approximate analytical solutions of nonlinear gas dynamics equation within caputo-fabrizo fractional operator

The adomian decomposition method (ADM) is applied to solve fractional order nonlinear gas dynamic equation with Caputo-Fabrizo fractional operator in this paper. In this method, illustration example evaluate the accuracy and applicability of the mentioned procedure. The outcomes got by ADM are in acceptable concurrence with the special arrangement of the problem.

Oscillation of second-order nonlinear delay dynamic equations on time scales with a nonpositive neutral term

In this paper, we establish some new oscillation criteria for second- order nonlinear neutral delay dynamic equation of the form (r(t)((m(t)y(t) ? p(t)y(τ(t)))∆)γ )∆ + ƒ (t, y(?(t))) = 0 on a time scale

Oscillation of nonlinear neutral dynamic equations on time scales

The authors present necessary and sufficient conditions for the oscillation of a class of second order non-linear neutral dynamic equations with non-positive neutral coefficients by using Krasnosel’skii’s fixed point theorem on time scales. The nonlinear function may be strongly sublinear or strongly superlinear.

Adomian polynomials method for dynamic equations on time scales

In a recent paper, a series solution method based on combining the Laplace transform and Adomian polynomial expansion was proposed to 
 find an approximate solution of nonlinear differential equations \cite{FA2016}. It uses the expansion in Adomian polynomials defined in \cite {A1,A2}. An important drawback of the Laplace transform method is the fact that it cannot be applied in the case of nonlinear differential equation in general. In order to cope with this problem, the authors of \cite{FA2016} suggested the use of Adomian polynomial expansion of the nonlinear function of the dependent variable involved in the differential equation. 
 
 In this work, we propose a counterpart of this method on an arbitrary time scale and derive its general formulation for a dynamic equation of any order. 
 We confirm that when the time scale is the set of real numbers, our method reduces to that in \cite{FA2016}. 
 
 Our presentation is organized as follows. First, we recollect some preliminary information on time scales in Secton 2. In Section 3, we derive the method for an $n$-th order nonlinear dynamic equation. The next section contains the application of the method to specific examples of first order nonlinear dynamic equations. The last section is devoted to conclusion and some further directions for study.

New oscillatory results for non-linear delay dynamic equations with super-linear neutral term

A new approach is introduced for the oscillatory solutions of the second order non-linear delay dynamic equations on time scale with a super-linear neutral term. The result are obtained through various integral and differential relations. The Kamenev and Philos-type criteria for oscillation are also discussed. The results improve and extend the ones reported in the literature in the sense that the inequalities are easy to verify and results cover various kinds of equations by choosing different time scales. Examples for particular time scales are given for illustration.

New oscillation results for non-canonical higher order nonlinear neutral dynamic equations

New oscillation results for non-canonical higher order nonlinear neutral dynamic equations

Some Oscillation Criteria for a Class of Higher Order Nonlinear Dynamic Equations with a Delay Argument on Time Scales

Some Oscillation Criteria for a Class of Higher Order Nonlinear Dynamic Equations with a Delay Argument on Time Scales

On Hyers–Ulam and Hyers–Ulam–Rassias Stability of a Nonlinear Second-Order Dynamic Equation on Time Scales

In this paper, we obtain sufficient conditions for Hyers–Ulam and Hyers–Ulam–Rassias stability of an abstract second–order nonlinear dynamic equation on bounded time scales. An illustrative example is given to show the applicability of the theoretical results.

Oscillation of second-order nonlinear neutral dynamic equations with “Maxima” on time scales

This paper is concerned with the oscillation of solutions of the second order nonlinear neutral delay dynamic equation with “Maxima” of the formon a time scale 𝕋. The obtained results improve and generalize some recent published results. Examples are given to illustrate the main results.

Oscillation Behavior of Second Order Nonlinear Dynamic Equation with Damping on Time Scales

Oscillation Behavior of Second Order Nonlinear Dynamic Equation with Damping on Time Scales

Oscillatory behavior of higher order nonlinear homogeneous neutral delay dynamic equations with positive and negative coefficients

Abstract In this paper, we derive some sufficient conditions for the oscillatory and asymptotic behavior of solutions of the higher order nonlinear neutral delay dynamic equation with positive and negative coefficients. The results of this paper extend and generalize the results of [S. Panigrahi and P. Rami Reddy, Oscillatory and asymptotic behavior of fourth order non-linear neutral delay dynamic equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 20 2013, 143–163] and [S. Panigrahi, J. R. Graef and P. Rami Reddy, Oscillation results for fourth order nonlinear neutral dynamic equations, Commun. Math. Anal. 15 2013, 11–28]. Examples are included to illustrate the validation of the results.

Oscillation Criteria of Singular Initial-Value Problem for Second Order Nonlinear Dynamic Equation on Time Scales

AbstractBy using of generalized Opial’s type inequality on time scales, a new oscillation criterion is given for a singular initial-value problem of second-order dynamic equation on time scales. Some oscillatory results of its generalizations are also presented. Example with various time scales is given to illustrate the analytical findings.

Oscillation of third order nonlinear damped dynamic equation with mixed arguments on time scales

The objective of this paper is to offer sufficient conditions for the oscillation of all solutions of the third order nonlinear damped dynamic equation with mixed arguments of the form \t\t\t(r2(r1(yΔ)α)Δ)Δ(t)+p(t)ψ(t,yΔ(a(t)))+q(t)f(t,y(g(t)))=0\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\bigl(r_{2}\\bigl(r_{1}\\bigl(y^{\\Delta}\\bigr)^{\\alpha}\\bigr)^{\\Delta}\\bigr)^{\\Delta}(t)+p(t)\\psi \\bigl(t,y^{\\Delta}\\bigl(a(t)\\bigr)\\bigr)+q(t)f\\bigl(t,y\\bigl(g(t)\\bigr) \\bigr)=0 $$\\end{document} on time scales, where a(t)geq t and g(t)leq t. Using Riccati transformation, integral averaging technique, and comparison theorem, we give some new criteria for the oscillation of the studied equation. Our results essentially improve and complement the earlier ones.

OSCILLATORY AND ASYMPTOTIC CRITERIA OF THIRD ORDER NONLINEAR DELAY DYNAMIC EQUATIONS WITH DAMPING TERM ON TIME SCALES

In this paper, we are concerned with oscillatory and asymptotic behavior of third order nonlinear delay dynamic equations with damping term on time scales. By using a generalized Riccati function and inequality technique, we establish some new oscillatory and asymptotic criteria. The established results on one hand extend some known results in the literature, on the other hand unify continuous and discrete analysis as two special cases of an arbitrary time scale. We also present some applications for the established results.

On the oscillation of certain third order nonlinear dynamic equations with a nonlinear damping term

Abstract New oscillation criteria for certain third order nonlinear dynamic equations with a nonlinear damping term are established. Examples to illustrate the results are included.