Order System Of Differential Equations Research Articles

Comparison of Least Squares FEM, Mixed Galerkin FEM and an Implicit FDM Applied to Acoustic Scattering

AbstractThis paper compares the performances of a least squares finite element method (LSFEM), a mixed Galerkin finite element method (MGFEM), and an implicit finite difference method (IFDM), when they are applied to an acoustic scattering problem modelled by the one‐dimensional Helmholtz equation. First, the boundary value problem is written as a first order system of differential equations, and variational formulations for the LSFEM and MGFEM are derived. Then, by using appropriate basis functions, the stiffness matrices and load vectors which define the corresponding linear algebraic systems are obtained. In all cases, the stiffness matrix is banded and for the LSFEM it is also Hermitian. Numerical tests show that all these methods have quadratic convergence to the exact solution. However, their efficiency assessed in terms of the number of nodes and computing time required to reach quadratic convergence varies. It is observed that LSFEM requires many more nodes and employs much more time than MGFEM and IFDM to reach the quadratic convergence for high frequencies. It is also found that MGFEM has less numerical dispersion and, as a consequence, performs better than IFDM and LSFEM for high frequencies. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Dynamic analyses of plane frames by integral equations for bars and Timoshenko beams

Since, especially for higher frequencies, Timoshenko's theory gives more reliable results than Euler–Bernoulli's theory, systems of beams, like frames, under arbitrary dynamic excitations should be analyzed on the basis of this refined theory. In this paper, after deriving the basic fundamental solutions of a lateral unit point force and of a unit single moment, the deflection and the rotation of the beam cross-section are given as integral equation forms of the governing second order differential equation system in the Laplace and the frequency domain. Since in frameworks axial displacements also occur, the well-known integral equations for bars under tension or compression are added to complete the modelling of plane frame structures. From these, the unknown boundary values may be exactly determined via point collocation at the beam ends and solution of the resulting algebraic system. With these boundary values, the exact deflection and rotation can also be found at arbitrary interior points. Finally, the convolution quadrature method is applied to the Laplace domain integral equation formulation in order to analyze the time-dependent wave propagation process. Two examples are presented to demonstrate the method's exactness compared with conventional finite element results: a single beam where the shift of the resonance frequencies when using the Timoshenko theory instead of Euler–Bernoulli's theory is shown, and a two-storey framework where, in addition to the frequence response, the causality of the propagating wave is studied.

MODELING THE SLIDING/FALLING LADDER PARADOX

ABSTRACT Recently we were presented with an interesting twist to the sliding ladder problem viewed in the related rates section of most calculus textbooks. Our problem concerning a sliding ladder that eventually hits the ground. At first, those attempting this problem fell into the calculus trap using only related rates. Previous work for this problem included higher order systems of differential equations in the solution. This paper discusses a new direction with modeling issues, several modeling based solutions, and numerical methods to solve these models using a spreadsheet.

Decay of the polarization field in a Maxwell Bloch system

The Maxwell-Bloch equations describing the propagation of electromagnetic waves in a gas of quantum mechanical systems with two energy levels is investigated. The system under consideration consists of a generally nonlinear second order system of differential equations for the dielectrical polarization and the density coupled with Maxwell's equations for the electromagnetic field. The goal is to show decay of the polarization field for $t\rightarrow\infty$.

A Leray-Schauder Alternative for Mönch Maps on Closed Subsets of Fréchet Spaces

In this paper, a continuation principle is obtained for maps defined on a closed, convex subset which may have empty interior in a Fréchet space, and satisfying a condition of Mönch type. An application to first order systems of differential equations is presented to illustrate our theory.

An eight-dimensional approach to G2 manifolds

We develop a systematic approach to G 2 holonomy manifolds with an SU(2)× SU(2) isometry using maximal eight-dimensional gauged supergravity to describe D6-branes wrapped on deformed three-spheres. A quite general metric ansatz that generalizes the celebrated Bryant–Salamon metric involves nine functions. We show that only six of them are the independent ones and derive the general first order system of differential equations that they obey. As a byproduct of our analysis, we generalize the notion of the twist that relates the spin and gauge connections in a way that involves non-trivially the scalar fields.

Essential self-adjointness of symmetric linear relations associated to first order systems

The purpose of this note is to present several criteria for essential self-adjointness. The method is based on ideas due to Shubin. This note is divided into two parts. The first part deals with symmetric first order systems on the line in the most general setting. Such a symmetric first order system of differential equations gives rise naturally to a symmetric linear relation in a Hilbert space. In this case even regularity is nontrivial. We will announce a regularity result and discuss criteria for essential self-adjointness of such systems. A byproduct of the regularity result is a short proof of a result due to Kogan and Rofe-Beketov (V. Kogan and F. Rofe-Beketov: “On square-integrable solutions of symmetric systems of differential equations of arbitrary order”, Proc. Roy. Soc. Edinburgh Sect. A 74 (1974/75), 5-40): the so-called formal deficiency indices of a symmetric first order system are locally constant on ℂ∖ℝ. The regularity and its corollary are based on joint work with Mark Malamud. Details will be published elsewhere. In the second part we consider a complete riemannian manifold, M, and a first order differential operator, D:C 0 ∞ (E)→C 0 ∞ (F), acting between sections of the hermitian vector bundles E,F. Moreover, let V:C ∞ (E)→L loc ∞ (E) be a self-adjoint zero order differential operator. We give a sufficient condition for the Schrödinger operator H=D t D+V to be essentially self-adjoint. This generalizes recent work of I. Oleinik (I. Oleinik: “On the essential self-adjointness of the Schrödinger operator on a complete Riemannian manifold”, Math. Notes 54 (1993), 934-939), (I. Oleinik: “On the connection of the classical and quantum mechanical completeness of a potential at infinity on complete riemannian manifolds”, Math. Notes 55 (1994), 380-386), (I. Oleinik: “On the essential self-adjointness of the general second order elliptic operators”, Proc. Amer. Math. Soc. 127 (1999), 889-900), M. Shubin (M. Shubin: “Classical and quantum completeness for the Schrödinger operators on non-compact manifolds”, In: B. Booß-Bavnbek and K. P. Wojciechowski (eds.), “Geometric aspects of partial differential equations” (Roskilde, 2000), vol. 242 of Contemp. Math. Amer. Math. Soc., Providence, RI (1999), pp. 257-269), (M. Shubin: “Essential self-adjointness for magnetic Schrödinger operators on non-compact manifolds”, In: Seminaire: Équations aux Dérivées Partielles, 2000-1999). École Polytech., Palaiseau (1999), pp. Exp. No. XV, 24), and M. Braverman (M. Braverman: “On self-adjointness of a Schrödinger operator on differential forms”, Proc. Amer. Math. Soc. 126 (2000), 617-623). We essentially use the method of Shubin. Our presentation shows that there is a close link between Shubin’s self-adjointness condition for the Schrödinger operator and Chernoff’s self-adjointness condition for powers of first order operators. We also discuss non-elliptic operators. However, in this case we need an additional assumption. We conjecture that the additional assumption turns out to be obsolete in general. The criteria we are going to present in the first and second part of this note are very closely related. In fact, after we had done the second part, we saw that the theory can be extended to symmetric linear relations associated to symmetric first order systems.

Elastic waves in nonhomogeneous media under 2D conditions: I. Fundamental solutions

The purpose of this work is to present three methods of analysis for elastic waves propagating in two dimensional, elastic nonhomogeneous media. The first step, common to all methods, is a transformation of the governing equations of motion so that derivatives with respect to the material parameters no longer appear in the differential operator. This procedure, however, restricts analysis to a very specific class of nonhomogeneous media, namely those for which Poisson's ratio is equal to 0.25 and the elastic parameters are quadratic functions of position. Subsequently, fundamental solutions are evaluated by: (i) conformal mapping in conjunction with wave decomposition, which in principle allows for both vertical and lateral heterogeneities; (ii) wave decomposition into pseudo-dilatational and pseudo-rotational components, which results in an Euler-type equation for the transformed solution if medium heterogeneity is a function of one coordinate only; and (iii) Fourier transformation followed by a first order differential equation system solution, where the final step involving inverse transformation from the wavenumber domain is accomplished numerically. Finally, in the companion paper numerical examples serve to illustrate the above methodologies and to delineate their range of applicability.

THE NONLINEAR SUPERPOSITION PRINCIPLE AND THE WEI-NORMAN METHOD

Group theoretical methods are used to study some properties of the Riccati equation, which is the only differential equation admitting a nonlinear superposition principle. The Wei–Norman method is applied to obtain the associated differential equation in the group SL(2, ℝ). The superposition principle for first order differential equation systems and Lie–Scheffers theorem are also analyzed from this group theoretical perspective. Finally, the theory is applied in the solution of second order differential equations like time independent Schrödinger equation.

Slowly varying function method applied to quartz crystal oscillator transient calculation

By using an approach based on the full nonlinear Barkhausen criterion, it is possible to describe oscillator behavior under the form of a nonlinear characteristic polynomial whose coefficients are functions of the circuit components and of the oscillation amplitude. Solving the polynomial in the frequency domain leads to the steady state oscillation amplitude and frequency. In the time domain, the characteristic polynomial represents a nonlinear differential equation whose solution gives the oscillator signal transient. It is shown how symbolic manipulation capabilities of commercially available softwares can be used to automatically generate the coding of the oscillator characteristic polynomial from the SPICE description netlist. The numerical processing of such an equation in the time domain leads to unacceptable computer time because of the high quality factor of the oscillator circuits involved. Nevertheless, by using the slowly varying amplitude and phase method, it is possible to transform the initial nonlinear differential equation into a nonlinear first order differential equation system in the amplitude and phase variables. The solution of this system directly gives the designer the most relevant features of the oscillation; that is, the amplitude, phase, or frequency transients which can be accurately obtained within a short computer time by using classical numerical algorithms.

High frequency Wien-type chaotic oscillator

A Wien-type oscillator based on a current feedback opamp (CFOA) employed as a non-inverting voltage-controlled voltage source is modified for chaos by adding an inductor and a general purpose signal diode. By using a CFOA as the active element, high operating frequencies are possible. The chaotic oscillator is described by a fourth order system of differential equations. Experimental results, PSpice simulations and numerical simulation of the proposed model are included.

Explicit Runge-Kutta methods for initial value problems with oscillating solutions

New pairs of embedded Runge-Kutta methods specially adapted to the numerical solution of first order systems of differential equations which are assumed to possess oscillating solutions are obtained. These pairs have been derived taking into account not only the usual properties of accuracy, stability and reliability of the local error estimator to adjust the stepsize of the underlying formulas but also the dispersion and dissipation orders of the advancing formula as defined by Van der Houwen and Sommeijer (1989). Three nine-stage embedded pairs of Runge-Kutta methods with algebraic orders 7 and 5 and higher orders of dispersion and/or dissipation are selected among the members of a family of pairs depending on several free parameters. Some numerical results are presented to show the efficiency of the new methods.

Boundedness of the solution of a degenerate second order differential equation system

Boundedness of the solution of a degenerate second order differential equation system

Boundedness of the solution of a degenerate second order differential equation system

We consider a linear system of second order differential equations $$A x(t) = B x (t),$$ (1) whereA andB are singularn×n matrices,x is ann-dimensional vector with coordinates twice differentiable int. Using the Drazin inverse matrix theory we find a solution of the Cauchy problem that is bounded on the whole axis.

On conjugate points for second order systems

We study the existence of first conjugate points relative to second order systems of differential equations. We also prove a result on the relation between positivity of certain solutions of the adjoint systems and conjugate points.

Solving higher order Fuchs type differential systems avoiding the increase of the problem dimension

In this paper, we develop a Frobenius matrix method for solving higher order systems of differential equations of the Fuchs type. Generalized power series solution of the problem are constructed without increasing the problem dimension. Solving appropriate algebraic matrix equations a closed form expression for the matrix coefficient of the series are found. By means of the concept of a k‐fundamental set of solutions of the homogeneous problem an explicit solution of initial value problems are given.

Part I Parallel architectures and algorithms

This paper provides an introduction to algorithms for fundamental linear algebra problems on various parallel computer architectures, with the emphasis on VLSI systolic array machines. To illustrate the basic concepts and key issues, we consider the problem of the parallel solution of a nonsingular linear systems derived from the finite difference/element discretisation of anN th order system of differential equations.

On the Carlitz Expansion in Products of Bessel Functions

by a rearrangement of infinite series, where v and fL are complex numbers and a and b are nonzero complex numbers. The coefficients en are explicitly given, but it was not shown that the series converges. In the case where a = b the convergence is known, see e. g. [7], p. 145 or [6], p.45. The result of the present paper is based on the theory of eigenfunction expansions for first order systems of differential equations in the complex domain as developed by H. Langer, R. Mennicken, M. Moller and A. Sattler in [3], [4], [5]. Let Clog be the Riemann surface of the logarithm and, for K E C, let P,,( Clog) be the space of those holomorphic functions J on Clog satisfying the Floquet condition J(ze21ri ) = e21ri" J(z) for all z E Clog. In [5] expansions of the form

A NUMERICAL HOMOTOPY METHOD AND INVESTIGATIONS OF A SPRING-MASS SYSTEM

A numerical technique is developed to determine the behavior of periodic solutions to highly nonlinear non-autonomous systems of ordinary differential equations. The method is based on shooting in conjunction with a probability one homotopy method and an implementation of the topological index. It is shown that solutions may be characterized a priori in terms of an index and this is developed into a powerful numerical and investigative tool. This method is used to investigate the periodic solutions of a nonlinear fourth order system of differential equations. These equations describe the motion of a forced mechanical oscillator and are extremely difficult to evaluate numerically. Solutions are presented which could not be found using local methods. These include flip, saddle node and symmetry breaking pitchfork bifurcations.

A COMPREHENSIVE MATHEMATICAL MODEL FOR A MULTIZONE TUBULAR HIGH PRESSURE LDPE REACTOR

Abstract In this study a comprehensive mathematical model of high pressure tubular ethylene polymerization reactors is presented. A fairly general reaction mechanism is employed to describe the complex kinetics of ethylene polymerization. To determine the variation of molecular properties along the reactor length the method of moments is applied to the infinite set of species balance equations to transform it into a low order system of differential equations in terms of the leading moments of the number chain length distribution. Detailed algebraic equations are given describing the variation of kinetic rate constants, thermodynamic and transport properties of the reaction mixture with temperature, pressure and composition. A new correlation is derived to describe the change of reaction viscosity with reactor operating conditions. The model permits a realistic calculation of temperature and pressure profiles, monomer and initiator concentrations, molecular properties of LDPE (i.e. Mn, Mm, LCB and SCB) as ...