Second-order Differential Systems Research Articles

Generalized inverse-Cowling approximation for polar w-mode oscillations of neutron stars

Adopting the Lindblom-Detweiler formalism for polar oscillations of neutron stars, we study the w-mode oscillation and find that the Lagrangian change in pressure, measured by the physical quantity X, is negligibly small. Based on this observation, we develop the generalized inverse-Cowling approximation (GICA) with the approximation X = X' = 0, where X' is the derivative of X with respect to the circumferential radius for w-mode oscillations of neutron stars. Under the GICA, w-mode oscillations are described by a second-order differential system, which can yield accurate frequencies and damping rates of quasi-normal modes.

Existence of positive solutions for a class of second-order ordinary differential systems

In this paper, we present some new concepts of superlinearity and sublinearity for nonlinear terms of second-order ordinary differential systems, and then consider the existence of positive solutions for systems with such superlinear and sublinear nonlinear terms, especially proving the existence of positive solutions for systems in which one nonlinear term is superlinear and the other is sublinear.

Positive solutions of singular sub-linear boundary value problems for fourth-order and second-order differential equation systems

This paper investigates the existence of positive solutions of singular sub-linear boundary value problems for fourth-order and second-order differential equation systems. First of all, we establish some important Lemmas. Then, we define a partial ordering in C 2[ a, b] ∩ C 4( a, b) × C[ a, b] ∩ C 2( a, b) and construct lower and upper solutions to give a necessary and sufficient condition for the existence of C 2[0, 1] × C[0, 1] positive solutions as well as C 3[0, 1] × C 1[0, 1] positive solutions. Our nonlinearity f( t, x 1, x 2, x 3), g( t, x 1, x 2) may be singular at x 1 = 0, x 2 = 0, x 3 = 0, t = 0 and or t = 1.

Subharmonic solutions for nonautonomous sublinear second-order differential inclusions systems with [formula omitted]-Laplacian

An existence result is obtained for infinite subharmonic solutions of nonautonomous second-order sublinear differential inclusions systems with p -Laplacian by the minimax methods in the critical point theory.

Drainage after total knee replacement

AbstractA simple lumped hydraulic model of knee drainage following arthroplasty is developed incorporating a pressure-volume equation of state for the knee capsule and a wound healing rate dynamically retarded by the blood flow-induced shear stress. The resulting second-order nonlinear ordinary differential system is examined numerically and qualitatively to map the parameter space. In the model, moderate suction or a slight back-pressure promotes gradual drainage and healing whereas excessive suction can lead to a bifurcation in which healing is retarded or even prevented. Guided, then, by the model, the literature, and experience, continuous drainage with a small constant back-pressure appeared beneficial so we prospectively evaluated a series of ten patients. The results are consistent with the model and promising.

New oscillation criteria for second-order nonlinear matrix differential equations

Some new oscillation criteria are established for the second-order matrix differential system (r(t)Z′(t))′+p(t)Z′(t)+Q(t)F(Z′(t))G(Z(t))=0, t≥0>0, are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of [t0, ∞), rather than on the whole half-line. The results weaken the condition of Q(t) and generalize some well-known results of Wong (1999) to nonlinear matrix differential equation.

Dynamics of the center of mass in rotating Bose–Einstein condensates

In this paper, we derive the analytical solution for a second-order ordinary differential system which governs the motion of the center of mass in the dynamics of a stationary state with its center shifted. A leap-frog Fourier pseudospectral (LFFP) method is presented for efficient and accurate numerical simulations of the Gross–Pitaevskii equation (GPE) with an angular momentum rotation term. Different motion patterns for the center of mass are observed and classified from the analytical solution and confirmed by directly simulating the GPE. To show the effectiveness of the LFFP method, the dynamics of vortex lattices are studied, and the numerical results demonstrate the efficiency and extremely high resolution of our method.

Riesz Basis Generation of Abstract Second-Order Partial Differential Equation Systems with General Non-Separated Boundary Conditions

Riesz basis analysis for a class of general second-order partial differential equation systems with nonseparated boundary conditions is conducted. Using the modern spectral analysis approach for parameterized ordinary differential operators, it is shown that the Riesz basis property holds for the general system if its associated characteristic equation is strongly regular. The Riesz basis property can then be readily established in a unified manner for many one-dimensional second-order systems such as linear string and beam equations with collocated or noncollocated boundary feedbacks and tip mass attached systems. Three demonstrative examples are presented.

Periodic solutions of second-order differential inclusions systems with p-Laplacian

Some existence results are obtained for periodic solutions of nonautonomous second-order differential inclusions systems with p-Laplacian.

Efficient iterations for Gauss methods on second-order problems

We consider some important aspects about the implementation of high order implicit formulas (specially the Gauss methods) for solving second-order differential systems having high frequencies and small amplitudes superimposed. The choice of an appropriate iterative scheme is discussed in detail. Important topics about the predictors (initial guesses) are analyzed and a variable order strategy to select the best predictor at each integration step is supplied. A few numerical experiments on some standard test problems confirm the theory presented.

Positive solutions of second-order two-delay differential systems with twin-parameter

The paper is concerned with the existence of positive solutions for a second order two-delay differential systems with twin-parameter. Firstly, we present some sufficient conditions with the twin-parameter belonging to an open rectangular region (eigenvalue region) to ensure the existence of positive solutions for the differential systems with singularity. Secondly, we deal with the superlinear semipositone problem of the differential systems and establish the existence of positive solutions for the systems. The proofs are based on a fixed-point theorem in cones.

Oscillation, stability, and boundedness of second-order differential systems with random impulses

The model of second-order linear differential systems with random impulses is brought forward in this paper. Then, necessary and sufficient conditions for oscillation in mean, p-moment stability and p-moment boundedness are obtained by several theorems that compare solutions of this system with the corresponding nonimpulsive differential system. At last, an example is presented to show the application of obtained results.

Existence of positive solutions for a second-order ordinary differential system

In this paper we consider the existence of positive solutions for a second-order ordinary differential system, where the nonlinear term is superlinear in one equation and sublinear in the other equation. By constructing a cone K 1 × K 2 which is the Cartesian product of two cones in space C [ 0 , 1 ] and computing the fixed point index in K 1 × K 2 , we establish the existence of positive solutions for the system. We remark that, differently from the literature, we deal with our problem on the Cartesian product of two cones, in which the feature of two equations can be exploited better.

Nonlinear functionals and a theorem of Sun

We use a recently developed theory of nonlinear functionals in the study of oscillations of second-order symmetric vector differential systems to extend a number of theorems of Sun [New Kamenev type theorems for second order linear matrix differential systems, Appl. Math. Lett., 2004, in press] under a common theme. The criteria presented here are of the form: the integral of the coefficient matrix is bounded at infinity (in a sense to be made explicit in the paper) and bounded away from a positive absolute constant implies oscillation at infinity.

Necessary and sufficient conditions for existence and uniqueness of bounded or almost-periodic solutions for differential systems with convex potential

We give necessary and sufficient conditions for the existence and uniqueness of bounded or almost-periodic solutions of the first-order differential system: $u' + \nabla \Phi (u) = e(t)$, when $\nabla \Phi$ denotes the gradient of a convex function on $\mathbb R^N$. We also study the relations of continuity between the forcing term $e$ and the solution $u$. Then we give similar results for the second-order differential system: $u'' = \nabla \Phi (u) + e(t)$.

Multiple positive solutions of a singular gelfand type problem for second-order impulsive differential systems

We study the existence of multiple positive solutions of a singular boundary valueproblem for second-order impulsive differential systems with real multiparameters. The existence phenomenon of positive solutions depends on the parameters and we give a full analysis about multiplicity, existence, and nonexistence with respect to the parameters. Proofs are mainly employed by the upper and lower solutions method for systems and the fixed-point index argument.

New Kamenev type theorems for second-order linear matrix differential systems

Some new Kamenev type oscillation criteria are established for the second-order matrix differential system ( P(t)Y′)′ + Q(t)Y = 0, where Y, P, and Q are n × n real continuous matrix functions with P(t), Q(t) symmetric and P(t) > 0 for t ε t 0, ∞) . Our results are sharper than some previous results, and can be applied to the cases which are not covered by known criteria.

Oscillation of self-adjoint linear matrix Hamiltonian systems

By means of monotone functionals defined on suitable matrix spaces and new methods, oscillation criteria for self-adjoint linear Hamiltonian matrix system of the form X′(x)=A(x)X(x)+B(x)Y(x), Y′(x)=C(x)X(x)−A ∗(x)Y(x), are obtained. There results are sharper than some previous results even for self-adjoint second-order matrix differential systems. In particular, our theorems are generalizations and improvements of many existing results including those obtained by Sun [J. Math. Anal. Appl. 279 (2003) 651–658].

Maximum principle and existence of almost-periodic solutions of second-order differential systems

We give sufficient conditions for the existence of almost-periodic solutions of the second-order differential system $u'' = f(t,u,u')$, which satisfies the maximum principle. Our hypotheses do not impose the uniqueness of bounded solutions.

Periodic solutions for a kind of second-order neutral differential systems with deviating arguments

By means of the generalized continuation theorem of coincidence degree theory, a kind of second-order neutral differential systems with deviating arguments as follows d 2 dt 2 (x(t)+Cx(t−r))+ d dt grad F(x(t))+ grad G(x(t−τ(t)))=p(t) is studied. A new result on the existence of periodic solutions is obtained, which relates to the deviating arguments τ( t) and r.