Second-order Differential Systems Research Articles

Canonical equations of Hamilton for the nonlinear Schrödinger equation

We define two different systems of mathematical physics: the second order differential system (SODS) and the first order differential system (FODS). The Newton's second law of motion and the nonlinear Schrödinger equation (NLSE) are the exemplary SODS and FODS, respectively. We obtain a new kind of canonical equations of Hamilton (CEH), which exhibit some kind of symmetry in form and are formally different from the conventional CEH without symmetry [H. Goldstein, C. Poole, J. Safko, Classical Mechanics, third ed., Addison- Wesley, 2001]. We also prove that the number of the CEHs is equal to the number of the generalized coordinates for the FODS, but twice the number of the generalized coordinates for the SODS. We show that the FODS can only be expressed by the new CEH, but not introduced by the conventional CEH, while the SODS can be done by both the new and the conventional CEHs. As an example, we prove that the nonlinear Schrödinger equation can be expressed with the new CEH in a consistent way.

Periodic Solutions of the Second-Order Neutral Functional Differential Systems with Operator Varying in Time and Delays

UDC 517.9 We study the existence of periodic solutions for second-order neutral functional differential systems with operator varying in time and delays. An interesting issue is that the coefficient matrix of the neutral difference operator is not a constant matrix, which contradicts the available literature data. Some properties of the neutral difference operator variable as a function of time and the results on the existence of periodic solutions of the second-order neutral functional differential systems are obtained. Moreover, numerical simulations are performed to illustrate the theoretical results.

Existence and localization of solutions for operatorial systems defined on Cartesian product of Fréchet spaces using a new vector version of Krasnoselskii’s cone compression–expansion theorem

A generalization of Krasnoselskii’s compression–expansion fixed point theorem is presented for treating nonlinear systems defined on the Cartesian product of Fréchet spaces. The compression–expansion conditions are given componentwise, and therefore each component can separately behave in its own way. Applications to differential systems of second order on the half line are presented, with existence, localization and multiplicity results.

Erratum to: Periodic solution of second-order impulsive delay differential system via generalized mountain pass theorem

Erratum to: Periodic solution of second-order impulsive delay differential system via generalized mountain pass theorem

Variational Properties of the Solutions for Second-Order Differential Equations and Systems on Semi-Line

In this article, an abstract theory regarding variational properties of the fixed points of contractions and Perov contractions is applied to boundary value problems on semi-line for second-order differential equations and systems. The main result states that under suitable conditions the unique solution of such a system is a Nash-type equilibrium of the corresponding energy functionals.

Existence of unbounded solutions of boundary value problems for singular differential systems on whole line

Motivated by (Kyoung and Yong-Hoon in Sci. China Math. 53:967-984, 2010) and (Chen and Zhang in Sci. China Math. 54:959-972, 2011), this paper is concerned with a boundary value problem of singular second-order differential systems with quasi-Laplacian operators on the whole line. By constructing a completely continuous nonlinear operator and using a fixed point theorem, sufficient conditions guaranteeing the existence of at least one unbounded solution are established. The methods used are standard, however, their exposition in the framework of such a kind of problems is new and skillful. Three concrete examples are given to illustrate the main theorem.

Convergence analysis of multidimensional parametric deformable models

Deformable models are mathematical tools, used in image processing to analyze the shape and movement of real objects due to their ability to emulate physical features such as elasticity, stiffness, mass and damping. In the original approach, parametric models are obtained from the minimization of an energy functional by means of the Euler–Lagrange equation. Finite element method is used for spatial discretization. The shape and position of the model is governed by a second-order partial differential equation system, which is obtained by applying the calculus of variations. Subsequent work propose a model formulation defined completely in the frequency domain, by translating the PDE system into the Fourier domain. This new approach offers important computational efficiency and an easier generalization to multidimensional models, since each spectral component of the model is ruled by an independent PDE. This paper reviews the frequency based formulation and analyzes the convergence and stability of these multidimensional parametric deformable models. Results show that the accuracy and speed of convergence depend on the dynamic parameters of the system and the spectrum of the data to be characterized, providing a procedure to speed-up the convergence by an appropriate choice of these parameters.

A weighted $L_p$-theory for second-order parabolic and elliptic partial differential systems on a half space

In this article we consider parabolic systems and $L_p$ regularityof the solutions. With zero boundary condition the solutionsexperience bad regularity near the boundary. This article addressesa possible way of describing the regularity nature. Our space domainis a half space and we adapt an appropriate weight into our functionspaces. In this weighted Sobolev space setting we develop aFefferman-Stein theorem, a Hardy-Littlewood theorem and sharpfunction estimations. Using these, we prove uniqueness and existenceresults for second-order elliptic and parabolic partial differentialsystems in weighed Sobolev spaces.

Optimal Urban Development Density along A Multi-Modal Linear Travel Corridor with Time-distance Toll Scheme

Consider a travel corridor with a multi-modal transport system (i.e., highway and railway) that connects a continuum of residential locations to a point of CBD. Both highway and railway are subject to congestion effects. All commuters travel along the corridor from home to work in the morning peak hour. The travel costs include travel time, schedule delay and monetary cost. The spatial dynamics of the traffic congestion on both transportation systems are determined by the trip-timing condition, that no traveler will experience a lower travel cost by departing at a different time or switching to a different mode. The flow dynamics on the highway will be considered by applying basic LWR model, while crowdedness (i.e., passenger density on the train) is used to describe the congestion on the railway. The simultaneous temporal and spatial dynamics of commute traffic pattern will be modeled by applying a second-order partial differential complementarity system approach. A time-distance road pricing scheme is applied to achieve the system optimal condition. The urban population is assumed to be located continuously along the corridor. However, the spatial population density distribution is regarded as variant. As is well known that the urban planning issue of population density distribution affects the transportation system significantly, this study aims to find the optimal urban population density distribution in a linear continuous travel corridor leading to optimal transportation system performance, with basic assumptions that it follows some given distribution pattern like negative exponential distribution. The problem is eventually formulated into a mathematical program with complementarity constraints and efficient solution algorithm is developed. Finally, numerical examples are conducted to test the model formulation validity and efficiency.

Symmetric positive solutions for second-order singular differential systems with multi-point coupled integral boundary conditions

Symmetric positive solutions for second-order singular differential systems with multi-point coupled integral boundary conditions

Existence and stability solutions of second-order partial neutral stochastic nonlinear differential systems with infinite delay

In this paper we consider a class of second-order neutral stochastic differential equations with infinite delay. First we establish a new set of sufficient conditions for the existence and uniqueness results for the system of equations which are discussed under non-Lipschitz with Lipschitz condition by the means of the successive approximation and theory of cosine families. Moreover, we give the continuous dependence of solutions on the initial data by the way of corollary of the Bihari's inequality. Finally, an example is provided to illustrate our results.

Periodic solution of second-order impulsive delay differential system via generalized mountain pass theorem

In this paper we use variational methods and generalized mountain pass theorem to investigate the existence of periodic solutions for some second-order delay differential systems with impulsive effects. To the authors’ knowledge, there is no paper about periodic solution of impulses delay differential systems via critical point theory. Our results are completely new.

Accurate simulations of pure quasi-P-waves in complex anisotropic media

Reverse time migration (RTM) in complex anisotropic media requires calculation of the propagation of a single-mode wave, the quasi-P-wave. This was conventionally realized by solving a [Formula: see text] system of second-order partial differential equations. The implementation of this [Formula: see text] system required at least twice the computational resources as compared with the acoustic wave equation. The S-waves, an introduced auxiliary function in this system, were treated as artifacts in the RTM. Furthermore, the [Formula: see text] system suffered numerical stability problems at the places in which abrupt changes of symmetric axis of anisotropy exist, which brings more challenges to real data implementation. On the other hand, the Alkhalifah’s equation, which governs the pure quasi-P-wave propagation, was hard to solve because it was a pseudodifferential equation. We proposed a pure quasi-P-wave equation that can be easily implemented within current imaging framework. Our new equation was obtained by decomposing the original pseudodifferential operator into two numerical solvable operators: a differential operator and a scalar operator. The combination of these two operators yielded an accurate phase of quasi-P-wave propagation. Our solution was S-wave free and numerically stable for very complicated models. Because only one equation was required to resolve numerically, the new proposed scheme was more efficient than those conventional schemes that solve the [Formula: see text] second-order differential equations system. For tilted transverse isotropy (TTI) RTM implementation, the required increase of numerical cost was minimal, and we could expect at least a factor of two of improvement of efficiency. We showed the effectiveness and robustness of our method with numerical examples with simple and very complicated TTI models, the SEG Advanced Modeling (SEAM) model. Further extension of our approach to orthorhombic anisotropy or tilted orthorhombic anisotropy was straightforward.

Homoclinic solutions for a second-order p-Laplacian functional differential system with local condition

By means of critical point theory and some analysis methods, the existence of homoclinic solutions for the p-Laplacian system with delay, d d t [ | u ′ ( t ) | p − 2 u ′ (t)]= ∇ x G(t,u(t),u(t+τ))+ ∇ y G(t−τ,u(t−τ),u(t))+e(t), is investigated. Some new results are obtained. The interesting thing is that the function G(t,x,y) is only required to satisfy a local condition. Furthermore, the results are all explicitly related to the value of delay τ.MSC:34C37, 58E05, 70H05.

Existence of Solutions of Boundary Value Problems for Coupled Singular Differential Equations on Whole Lines with Impulses

This paper is concerned with boundary value problems of second-order impulsive differential systems on whole lines. By using Banach spaces and nonlinear operators, together with the Schauder fixed point theorem, sufficient conditions to guarantee the existence of at least one solution are established.

New approach in dynamics of regenerative chatter research of turning

In this paper, regenerative chatter phenomena in a turning process is discussed from impulsive dynamical point of view. By introducing the instantaneous pulse when vibration occurs and the vibratory condition set, we optimize the models and present a certain kind of second-order impulsive differential systems, which is a specific discontinuous dynamical system. Then we search for the general results of the nonoccurrence of chatter phenomena by discussing the number of the vibration pulse times, utilizing the method of flow theory in discontinuous systems and transversal property at the boundary. Our results give a convenient way to estimate the available parameters to keep the turning process stable.

Controllability of Second-Order Systems with Nonlocal Conditions in Banach Spaces

In this article, exact controllability of second-order semilinear differential system with nonlocal conditions in Banach spaces is studied. To establish sufficient conditions for exact controllability, Sadovskii's fixed point theorem together with the theory of strongly continuous cosine family is been used. The cosine family and the nonlinear function associated with the system are assumed to be non-compact. An example is given to illustrate the developed theory.

A Dynamical Approach to an Inertial Forward-Backward Algorithm for Convex Minimization

We introduce a new class of forward-backward algorithms for structured convex minimization problems in Hilbert spaces. Our approach relies on the time discretization of a second-order differential system with two potentials and Hessian-driven damping, recently introduced in [H. Attouch, P.-E. Mainge, and P. Redont, Differ. Equ. Appl., 4 (2012), pp. 27--65]. This system can be equivalently written as a first-order system in time and space, each of the two constitutive equations involving one (and only one) of the two potentials. Its time dicretization naturally leads to the introduction of forward-backward splitting algorithms with inertial features. Using a Liapunov analysis, we show the convergence of the algorithm under conditions enlarging the classical step size limitation. Then, we specialize our results to gradient-projection algorithms and give some illustrations of sparse signal recovery and feasibility problems.

Symplectic explicit methods of Runge–Kutta–Nyström type for solving perturbed oscillators

The construction of symplectic methods of Runge–Kutta–Nyström type (RKN-type) specially adapted to the numerical solution of perturbed oscillators is analyzed. Based on the symplecticity conditions for this class of methods, new fourth-order explicit methods of RKN type for solving perturbed oscillators are constructed. The derivation of the new symplectic methods is carried out paying special attention to the minimization of the principal term of the local truncation error. The numerical experiments carried out show the qualitative behavior and the efficiency of the new methods when they are compared with some standard and specially adapted symplectic methods proposed in the scientific literature for solving second-order oscillatory differential systems.

On one special case of the existence of solutions of quasilinear differential systems represented by Fourier series with slowly varying parameters

For a quasilinear differential system of second order, whose coefficients have the form of Fourier series with slowly varying coefficients and frequency, we obtain conditions for the existence of a particular solution of analogous structure in one special case.