Second-order Dynamic Equations Research Articles

Multiple Positive Solutions in the Sense of Distributions of Singular BVPs on Time Scales and an Application to Emden-Fowler Equations

This paper is devoted to using perturbation and variational techniques to derive some sufficient conditions for the existence of multiple positive solutions in the sense of distributions to a singular second-order dynamic equation with homogeneous Dirichlet boundary conditions, which includes those problems related to the negative exponent Emden-Fowler equation.

Interval criteria for oscillation of nonlinear second-order dynamic equations on time scales

Interval oscillation criteria are established for a second-order nonlinear dynamic equation on time scales by utilizing a generalized Riccati technique and an integral averaging technique. The theory can be applied to second-order dynamic equations regardless of the choice of delta or nabla derivatives.

Existence of weak solutions of two-point boundary value problems for second-order dynamic equations on time scales

In this paper, by using critical point theory, we establish the existence of weak solutions of the two-point boundary value problem for a second-order dynamic equation on an arbitrary time scale T , so that the well known case of differential dynamic systems ( T = R ) and the recently developed case of discrete dynamic systems ( T = Z ) are unified. To the best of our knowledge, this is the first time that boundary value problems of dynamic equations on time scales have been dealt with by using critical point theory.

Oscillation of second-order forced functional dynamic equations with oscillatory potentials

Oscillation criteria are established for a second-order forced dynamic equation on time scales containing both delay and advance arguments. Moreover, the potentials are allowed to change sign. Several nontrivial examples from difference equations are provided to illustrate the easy application of the results. The theory can be applied to second-order dynamic equations regardless of the choice of delta or nabla derivatives.

Oscillation criteria for a class of second-order Emden–Fowler delay dynamic equations on time scales

By means of Riccati transformation technique, we establish some new oscillation criteria for the second-order Emden–Fowler delay dynamic equations x Δ Δ ( t ) + p ( t ) x γ ( τ ( t ) ) = 0 on a time scale T ; here γ is a quotient of odd positive integers with p ( t ) real-valued positive rd-continuous functions defined on T . To the best of our knowledge nothing is known regarding the qualitative behavior of these equations on time scales. Our results in this paper not only extend the results given in [R.P. Agarwal, M. Bohner, S.H. Saker, Oscillation of second-order delay dynamic equations, Can. Appl. Math. Q. 13 (1) (2005) 1–18] but also unify the oscillation of the second-order Emden–Fowler delay differential equation and the second-order Emden–Fowler delay difference equation.

Oscillation results for second-order nonlinear neutral delay dynamic equations on time scales

In this article, we consider the second-order nonlinear neutral delay dynamic equation on a time scale and establish some new oscillation and nonoscillation criteria. Also from these we deduce the Leighton–Wintner, Hille–Kneser, Kamenev, and Philos types oscillation criteria. Our results are different and complement the existence oscillation results for neutral delay dynamic equations on time scales in (Agarwal et al. 2004, Oscillation criteria for second-order nonlinear neutral dynamic equations. Journal of Mathematical Analysis and Applications, 300, 203–217) and (S.H. Saker, 2006, Oscillation of second-order nonlinear neutral delay dynamic equations on time scales. Journal of Computational and Applied Mathematics, 187, 123–141). Our results can be applied on the time scales , , , for h > 0, , , , and when where {t n } is the set of harmonic numbers, etc. Some examples are considered to illustrate the main results.

Oscillation Criteria for Second-Order Delay Dynamic Equations on Time Scales

By means of Riccati transformation technique, we establish some new oscillation criteria for the second-order nonlinear delay dynamic equations on a time scale , here is a quotient of odd positive integers with p and q real-valued positive rd-continuous functions defined on .

Trench's Perturbation Theorem for Dynamic Equations

We consider a nonoscillatory second-order linear dynamic equation on a time scale together with a linear perturbation of this equation and give conditions on the perturbation that guarantee that the perturbed equation is also nonoscillatory and has solutions that behave asymptotically like a recessive and dominant solutions of the unperturbed equation. As the theory of time scales unifies continuous and discrete analysis, our results contain as special cases results for corresponding differential and difference equations by William F. Trench.

A Reduced Model for Flame-Flow Interaction

The paper is concerned with an extension of the classical relation between the ∞ame speed and the curvature-∞ow stretch, valid only for high Lewis numbers (difiusively stable ∞ames). At low Lewis numbers the corresponding ∞ame-∞ow system sufiers short- wavelength instability, making the associated initial value problem ill-posed. In this study the di-culty is resolved by incorporation of higher-order efiects. As a result one ends up with a reduced model based on a coupled system of second-order dynamic equations for the ∞ame interface and its temperature. As an illustration the new model is applied for description of difiusively unstable stagnation-point ∞ow ∞ames.

Asymptotic behavior of second-order dynamic equations

We prove several growth theorems for second-order dynamic equations on time scales. These theorems contain as special cases results for second-order differential equations, difference equations, and q-difference equations.

Oscillation criteria for second-order nonlinear delay dynamic equations

In this paper, we consider the second-order nonlinear delay dynamic equation ( r ( t ) x Δ ( t ) ) Δ + p ( t ) f ( x ( τ ( t ) ) ) = 0 , on a time scale T . By employing the generalized Riccati technique we will establish some new sufficient conditions which ensure that every solution oscillates or converges to zero. The obtained results improve the well-known oscillation results for dynamic equations and include as special cases the oscillation results for differential equations. Some applications and examples are considered to illustrate the main results.

A stretch-temperature model for flame-flow interaction

The classical relation between the flame speed and the stretch, employed in modeling flame-flow interaction, is valid only for positive Markstein lengths (high Lewis numbers). At negative Markstein lengths (low Lewis numbers) the corresponding dynamical system suffers short-wavelength instability, making the associated initial value problem ill-posed. In this study the difficulty is resolved by incorporation of higher-order effects using a geometrically-invariant extrapolation from the linear analysis data. As a result one ends up with a reduced model based on a coupled system of second-order dynamic equations for the flame interface and its temperature. As an illustration the new model is applied for description of diffusively unstable stagnation-point flow flames.

Oscillation of second-order damped dynamic equations on time scales

The study of dynamic equations on time scales has been created in order to unify the study of differential and difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so-called time scale, which may be an arbitrary closed subset of the reals. This way results not only related to the set of real numbers or set of integers but those pertaining to more general time scales are obtained. In this paper, by employing the Riccati transformation technique we will establish some oscillation criteria for second-order linear and nonlinear dynamic equations with damping terms on a time scale T . Our results in the special case when T = R and T = N extend and improve some well-known oscillation results for second-order linear and nonlinear differential and difference equations and are essentially new on the time scales T = h N , h > 0 , T = q N for q > 1 , T = N 2 , etc. Some examples are considered to illustrate our main results.

Force-feedback control of a spherical haptic device featuring an electrorheologicalfluid

This paper presents the force-feedback performance of an active type of haptic devicefeaturing an electrorheological (ER) fluid via a sliding mode controller (SMC), which isvery robust to external disturbances and parameter uncertainties. In order to generate a2-degrees-of-freedom (2-DOF) active force, a spherical ER joint is designed based on themathematical torque model. The ER joint is then manufactured by adopting twobi-directional clutch actuators and the spherical geometry of the electrode, andits field-dependent generating force is experimentally evaluated. Subsequently,the 2-DOF force display is manufactured by integrating the ER joint with ACmotors. The force response of the manufactured haptic device is experimentallyevaluated, and expressed by a second-order dynamic equation. In order to achieve thedesired force trajectories, the SMC is formulated and experimentally realized byimposing an Euler angle relationship. Tracking control performances for various forcetrajectories are presented in the time domain, and their tracking errors are evaluated.

Existence of Positive Solutions for Second-Order m-Point Boundary Value Problems on Time Scales

This paper investigates the existence of positive solutions of the m-point boundary value problem for second-order dynamic equations on time scales, and obtain the result that the problem has at least one positive solution by using functional-type cone expansion-compression fixed point theorem.

A new existence result for impulsive dynamic equations on timescales

The paper is concerned with the boundary value problem for second-order impulsive dynamic equations on timescales. A new existence result is acquired by using a fixed point theorem due to Krasnoselskii and Zabreiko. An example is also included to illustrate our main result.

Oscillation criteria for second-order nonlinear neutral variable delay dynamic equations

This paper is concerned with oscillation of second-order nonlinear neutral dynamic equations on time scales with a variable delay. By using the generalized Riccati technique and integral averaging technique, new oscillation criteria are obtained for all solutions of the equation. Some results extend known results for difference equations when the time scale is the set Z + of positive integers and for differential equations when the time scale is R . Several examples are given to illustrate the results of the paper.

Asymptotic stability for dynamic equations on time scales

We examine the conditions of asymptotic stability of second-order linear dynamic equations on time scales. To establish asymptotic stability we prove the stability estimates by using integral representations of the solutions via asymptotic solutions, error estimates, and calculus on time scales.

Hille-Kneser-type criteria for second-order dynamic equations on time scales

We consider the pair of second-order dynamic equations, (r(t)(xΔ) γ )Δ + p(t)x γ (t) = 0 and (r(t)(xΔ) γ )Δ + p(t)x γσ (t) = 0, on a time scale , where γ > 0 is a quotient of odd positive integers. We establish some necessary and sufficient conditions for nonoscillation of Hille-Kneser type. Our results in the special case when involve the well-known Hille-Kneser-type criteria of second-order linear differential equations established by Hille. For the case of the second-order half-linear differential equation, our results extend and improve some earlier results of Li and Yeh and are related to some work of Doslý and Řehak and some results of Řehak for half-linear equations on time scales. Several examples are considered to illustrate the main results.

An existence result for a multipoint boundary value problem on a time scale

We will expand the scope of application of a fixed point theorem due to Krasnosel'skiĭ and Zabreiko to the family of second-order dynamic equations described by uΔΔ(t) = f(u σ (t)), , with multipoint boundary conditions u(0) = 0, , and for the purpose of establishing existence results. We will determine sufficient conditions on our function f such that the assumptions of the fixed point theorem are satisfied, which in return gives us the existence of solutions.