System Of Second-order Ordinary Differential Equations Research Articles

Pole Placement for Delay Differential Equations With Time-Periodic Delays Using Galerkin Approximations

Abstract Many practical systems have inherent time delays that cannot be ignored; thus, their dynamics are described using delay differential equations (DDEs). The Galerkin approximation method is one strategy for studying the stability of time-delay systems (TDS). In this work, we consider delays that are time-varying and, specifically, time-periodic. The Galerkin method can be used to obtain a system of ordinary differential equations (ODEs) from a second-order time-periodic DDE in two ways: either by converting the DDE into a second-order time-periodic partial differential equation (PDE) and then into a system of second-order ODEs, or by first expressing the original DDE as two first-order time-periodic DDEs, then converting into a system of first-order time-periodic PDEs, and finally converting into a first-order time-periodic ODE system. The difference between these two formulations in the context of control is presented in this paper. Specifically, we show that the former produces spurious Floquet multipliers at a spectral radius of 1. We also propose an optimization-based framework to obtain feedback gains that stabilize closed-loop control systems with time-periodic delays. The proposed optimization-based framework employs the Galerkin method and Floquet theory and is shown to be capable of stabilizing systems considered in the literature. Finally, we present experimental validation of our theoretical results using a rotary inverted pendulum apparatus with inherent sensing delays as well as additional time-periodic state-feedback delays that are introduced deliberately.

An h-version adaptive FEM for eigenproblems in system of second order ODEs: vector Sturm-Liouville problems and free vibration of curved beams

Purpose This study aims to overcome the involved challenging issues and provide high-precision eigensolutions. General eigenproblems in the system of ordinary differential equations (ODEs) serve as mathematical models for vector Sturm-Liouville (SL) and free vibration problems. High-precision eigenvalue and eigenfunction solutions are crucial bases for the reliable dynamic analysis of structures. However, solutions that meet the error tolerances specified are difficult to obtain for issues such as coefficients of variable matrices, coincident and adjacent approximate eigenvalues, continuous orders of eigenpairs and varying boundary conditions. Design/methodology/approach This study presents an h-version adaptive finite element method based on the superconvergent patch recovery displacement method for eigenproblems in system of second-order ODEs. The high-order shape function interpolation technique is further introduced to acquire superconvergent solution of eigenfunction, and superconvergent solution of eigenvalue is obtained by computing the Rayleigh quotient. Superconvergent solution of eigenfunction is used to estimate the error of finite element solution in the energy norm. The mesh is then, subdivided to generate an improved mesh, based on the error. Findings Representative eigenproblems examples, containing typical vector SL and free vibration of beams problems involved the aforementioned challenging issues, are selected to evaluate the accuracy and reliability of the proposed method. Non-uniform refined meshes are established to suit eigenfunctions change, and numerical solutions satisfy the pre-specified error tolerance. Originality/value The proposed combination of methodologies described in the paper, leads to a powerful h-version mesh refinement algorithm for eigenproblems in system of second-order ODEs, that can be extended to other classes of applications in damage detection of multiple cracks in structures based on the high-precision eigensolutions.

Method for Studying the Asymptotic Properties of Solutions to a System of Second-Order Linear Differential Equations on the Half-Line

A special method is proposed for reducing an $$n $$-dimensional system of second-order ordinary differential equations to a $$2n$$-dimensional system of first-order ones. Using matrix weight functions (which can be chosen arbitrarily when applying the method), we establish sufficient conditions for the boundedness and power-law absolute integrability on the half-line of the components of all solutions to a linear system of second-order differential equations and their first derivatives as well as for their decay (in particular, exponential or power-law) as the independent variable tends to infinity. An illustrative example is provided.

First integrals and exact solutions of some compartmental disease models

Abstract The notions of artificial Hamiltonian (partial Hamiltonian) and partial Hamiltonian operators are used to derive the first integrals for the first order systems of ordinary differential equations (ODEs) in epidemiology, which need not be derived from standard Hamiltonian approaches. We show that every system of first order ODEs can be cast into artificial Hamiltonian system q ˙ = ∂ H ∂ p $\dot{q}=\frac{{\partial H}}{{\partial p}}$ , p ˙ = − ∂ H ∂ q + Γ ( t , q , p ) $\dot{p}=-\frac{{\partial H}}{{\partial q}}+\Gamma(t,\;q,\;p)$ (see [1]). Moreover, the second order equations and the system of second order ODEs can be written in the form of artificial Hamiltonian system. Then, the partial Hamiltonian approach is employed to derive the first integrals for systems under consideration. These first integrals are then utilized to find the exact solutions of models from the epidemiology for a distinct class of population. For physical insights, the solution curves of the closed-form expressions obtained are interpreted in order for readers understand the disease dynamics in a much deeper way. The effects of various pertinent parameters on the prognosis of the disease are observed and discussed briefly.

First integrals and analytical solutions of some dynamical systems

This article investigates the first integrals and closed- form solutions of some nonlinear first-order dynamical systems from diverse areas of applied mathematics. We use the notion of artificial Hamiltonian, and we show that every first-order system of ordinary differential equations (ODEs) can be written in the form of an artificial Hamiltonian system [see Naz and Naeem (ZNA 73(4):323–330, 2018)]. One can also express the second-order ODE or system of second-order ODEs in the form of system of first-order artificial Hamiltonian system. Then the partial Hamiltonian approach is employed to compute the partial Hamiltonian operators and the corresponding first integrals. The first integrals are utilized to construct the closed-form solutions of two-stream model for tuberculosis and dengue fever, Duffing–van der Pol oscillator, nonlinear optical oscillators under parameter restrictions, nonlinear convection model and the two- dimensional galaxy model. We show that how one can apply the existing “partial Hamiltonian approach” for nonstandard Hamiltonian systems. This study provides a new way of solving the dynamical systems of first-order ODEs, second-order ODE and second-order systems of ODEs which are expressed into the artificial Hamiltonian system.

Size-dependent finite strain analysis of cavity expansion in frictional materials

This paper presents unified solutions for elastic–plastic expansion analysis of a cylindrical or spherical cavity in an infinite medium, adopting a flow theory of strain gradient plasticity. Previous cavity expansion analyses incorporating strain gradient effects have mostly focused on explaining the strain localization phenomenon and/or size effects during infinitesimal expansions. This paper is however concerned with the size-dependent behaviour of a cavity during finite quasi-static expansions. To account for the non-local influence of underlying microstructures to the macroscopic behaviour of granular materials, the conventional Mohr–Coulomb yield criterion is modified by including a second-order strain gradient. Thus the quasi-static cavity expansion problem is converted into a second-order ordinary differential equation system. In the continuous cavity expansion analysis, the resulting governing equations are solved numerically with Cauchy boundary conditions by simple iterations. Furthermore, a simplified method without iterations is proposed for calculating the size-dependent limit pressure of a cavity expanding to a given final radius. By neglecting the elastic strain increments in the plastic zone, approximate analytical size-dependent solutions are also derived. It is shown that the strain gradient effect mainly concentrates in a close vicinity of the inner cavity. Evident size-strengthening effects associated with the sand particle size and the cavity radius in the localized deformation zone is captured by the newly developed solutions presented in this paper. The strain gradient effect will vanish when the intrinsic material length is negligible compared to the instantaneous cavity size, and then the conventional elastic perfectly-plastic solutions can be recovered exactly. The present solutions can provide a theoretical method for modeling the size effect that is often observed in small-sized sand-structure interaction problems.

The Artificial Hamiltonian, First Integrals, and Closed-Form Solutions of Dynamical Systems for Epidemics

Abstract The non-standard Hamiltonian system, also referred to as a partial Hamiltonian system in the literature, of the form q ˙ i = ∂ H ∂ p i , p ˙ i = − ∂ H ∂ q i + Γ i ( t , q i , p i ) ${\dot q^i} = \frac{{\partial H}}{{\partial {p_i}}},{\text{ }}{\dot p^i} = - \frac{{\partial H}}{{\partial {q_i}}} + {\Gamma ^i}(t,{\text{ }}{q^i},{\text{ }}{p_i})$ appears widely in economics, physics, mechanics, and other fields. The non-standard (partial) Hamiltonian systems arise from physical Hamiltonian structures as well as from artificial Hamiltonian structures. We introduce the term ‘artificial Hamiltonian’ for the Hamiltonian of a model having no physical structure. We provide here explicitly the notion of an artificial Hamiltonian for dynamical systems of ordinary differential equations (ODEs). Also, we show that every system of second-order ODEs can be expressed as a non-standard (partial) Hamiltonian system of first-order ODEs by introducing an artificial Hamiltonian. This notion of an artificial Hamiltonian gives a new way to solve dynamical systems of first-order ODEs and systems of second-order ODEs that can be expressed as a non-standard (partial) Hamiltonian system by using the known techniques applicable to the non-standard Hamiltonian systems. We employ the proposed notion to solve dynamical systems of first-order ODEs arising in epidemics.

A vortex-induced vibration model for the fatigue analysis of a marine drilling riser

ABSTRACTThe objective is to propose a quick and accurate vortex-induced vibration (VIV) model to predict the fatigue life of marine drilling risers. The nonlinear wake oscillator model was applied to represent the lift force, which is described by a fourth-order partial differential equation set. Next, we adopted the generalised integral transform technique (GITT) to transform it into second-order ordinary differential equation system, which was solved by the Runge-Kutta method. Validation of the numerical model was implemented and a good agreement was achieved. Next, an S-N curve was adopted to conduct fatigue prediction. The results demonstrate that under the conditions set forth in this paper, an increase in the current and outer diameter could reduce fatigue life, except when the outer diameter reaches a certain value, at which point the tendency is reversed. Moreover, when the top tension coefficient reaches a certain value, the fatigue life tendency is also reversed.

Effect of Delamination on Static and Dynamic Behavior of Laminated Composite Plate

The free vibration, bending, and transient responses of laminated composite plates have been investigated with delamination in this present study. The laminated, composite, shear deformable plate has been modeled mathematically using two higher-order shear deformation theories in conjunction with finite element steps. The final form of the governing equations of the bending and the free vibration responses are obtained using the variational method and the classical Hamilton’s principle, respectively. In addition, the transient responses are computed using Newmark’s time integration scheme for the discrete coupled second-order ordinary differential equation system of the dynamic equilibrium. The desired free vibration, bending, and transient responses are computed numerically using a homemade computer code developed in the MATLAB environment and compared with those available published numerical and analytical results. Finally, the importance of the developed higher-order models has been studied by analyzing the responses for different structural design parameters and discussed in detail.

Solving directly special fourth-order ordinary differential equations using Runge–Kutta type method

In this paper, an explicit Runge–Kutta method for solving directly fourth-order ordinary differential equations (ODEs) is constructed and denoted as (RKFD). We present a relevant-colored tree theory and the associated B-series theory for the order conditions. Based on the order conditions a three-stage fourth-order RKFD method and a three-stage fifth-order RKFD method are constructed. Numerical illustrations are presented to show the efficiency of the new RKFD methods by comparing them with other existing Runge–Kutta Nyström (RKN) and Runge–Kutta (RK) methods in the scientific literature after converting the problem into a system of second order ODEs and a system of first order ODEs respectively.

Singular invariant structures for Lie algebras admitted by a system of second-order ODEs

Systems of second-order ordinary differential equations (ODEs) arise in mechanics and have several applications. Differential invariants play a key role in the construction of invariant differential equations as well as the classification of a system of ODEs. Like regular invariants, singular invariant structures also possess an important role in the algebraic analysis of a system of ODEs. In this work, singular invariant equations for a system of two second-order ODEs admitting four-dimensional Lie algebras are investigated. Moreover, by using these singular invariant equations, canonical forms for systems of two second-order ODEs are constructed. Furthermore, it is observed that the same Lie algebra admitted by a system of second-order ODEs has different type of realizations some of which are related to regular invariants and some lead to singular invariant equations. Thus realizations of four-dimensional Lie algebras are associated with a regular invariant manifold as well as to a singular invariant manifold defined by a system of second-order ODEs. In addition, a case of a Lie algebra with realization resulting in a conditional singular invariant structure is presented and those cases of singular invariant equations are discussed which do not form a system of second-order ODEs. An integration procedure for the invariant representation of these canonical forms are presented. Illustrative examples are presented with physical applications to mechanics.

Arbitrary-Order Trigonometric Fourier Collocation Methods for Multi-Frequency Oscillatory Systems

We rigorously study a novel type of trigonometric Fourier collocation methods for solving multi-frequency oscillatory second-order ordinary differential equations (ODEs) $$q^{\prime \prime }(t)+Mq(t)=f(q(t))$$q?(t)+Mq(t)=f(q(t)) with a principal frequency matrix $$M\in \mathbb {R}^{d\times d}$$M?Rd×d. If $$M$$M is symmetric and positive semi-definite and $$f(q) = -\nabla U(q)$$f(q)=-?U(q) for a smooth function $$U(q)$$U(q), then this is a multi-frequency oscillatory Hamiltonian system with the Hamiltonian $$H(q,p)=p^{T}p/2+q^{T}Mq/2+U(q),$$H(q,p)=pTp/2+qTMq/2+U(q), where $$p = q'$$p=q?. The solution of this system is a nonlinear multi-frequency oscillator. The new trigonometric Fourier collocation method takes advantage of the special structure brought by the linear term $$Mq$$Mq, and its construction incorporates the idea of collocation methods, the variation-of-constants formula and the local Fourier expansion of the system. The properties of the new methods are analysed. The analysis in the paper demonstrates an important feature, namely that the trigonometric Fourier collocation methods can be of an arbitrary order and when $$M\rightarrow 0$$M?0, each trigonometric Fourier collocation method creates a particular Runge---Kutta---Nystrom-type Fourier collocation method, which is symplectic under some conditions. This allows us to obtain arbitrary high-order symplectic methods to deal with a special and important class of systems of second-order ODEs in an efficient way. The results of numerical experiments are quite promising and show that the trigonometric Fourier collocation methods are significantly more efficient in comparison with alternative approaches that have previously appeared in the literature.

INVARIANT LINEARIZATION CRITERIA FOR SYSTEMS OF CUBICALLY NONLINEAR SECOND-ORDER ORDINARY DIFFERENTIAL EQUATIONS

Invariant linearization criteria for square systems of second-order quadratically nonlinear ordinary differential equations (ODEs) that can be represented as geodesic equations are extended to square systems of ODEs cubically nonlinear in the first derivatives. It is shown that there are two branches for the linearization problem via point transformations for an arbitrary system of second-order ODEs and its reduction to the simplest system. One is when the system is at most cubic in the first derivatives. One obtains the equivalent of the Lie conditions for such systems. We explicitly solve this branch of the linearization problem by point transformations in the case of a square system of two second-order ODEs. Necessary and sufficient conditions for linearization to the simplest system by means of point transformations are given in terms of coefficient functions of the system of two second-order ODEs cubically nonlinear in the first derivatives. A consequence of our geometric approach of projection is a rederivation of Lie's linearization conditions for a single second-order ODE and sheds light on more recent results for them. In particular we show here how one can construct point transformations for reduction to the simplest linear equation by going to the higher space and just utilizing the coefficients of the original ODE. We also obtain invariant criteria for the reduction of a linear square system to the simplest system. Moreover these results contain the quadratic case as a special case. Examples are given to illustrate our results.

Exact solutions using symmetry methods and conservation laws for the viscous flow through expanding–contracting channels

We present a range of definitions and methods dealing with the reduction of partial differential equations on the basis of the underlying symmetry structure, conservation laws and a combination of these. The method is used to reduce a complex system to an easy-to-handle second-order ordinary differential equation system independent of restrictions on any physical parameters. In particular, we construct exact solutions of a system modelling viscous flow between slowly expanding and contracting walls.

A NEW METHOD FOR THE CONSTRUCTION OF A LAGRANGIAN

In this paper, the necessary condition for the existence of a La-grangian for the ordered direct analytic representation of a system of second-orderordinary differential equations on a connected domain is given. Under that condition,a new method for the construction of the Lagrangian is proposed. The domain ofdefinition of the constructed Lagrangian might be diminished.

Finite element transient analysis of composite and sandwich plates based on a refined theory and implicit time integration schemes

An isoparametric finite element formulation based on a higher order displacement model for dynamic analysis of multilayer unsymmetric composite plates is presented. Newmark and Wilson-θ schemes are used for time integration of the discrete coupled second-order ordinary differential equation system of dynamic equilibrium and results are compared. A special mass diagonalization scheme is used which conserves total mass of the element and also includes the effects of rotary inertia terms. Effects of consistent/diagonal mass matrix, time step, and finite element mesh are investigated. Several numerical examples are presented and results are compared with those available in the literature.