Null Initial Conditions Research Articles

Exact solutions of fractional oscillation systems with pure delay

In this paper we study the exact solutions of a class of fractional delay differential equations. We consider the fractional derivative of the order between 1 and 2 in the sense of Caputo. In the first part, we introduce two novel matrix functions, namely, the generalized cosine-type and sine-type delay Mittag-Leffler matrix functions. Then we obtain the explicit solutions for the linear homogeneous equations subjecting to corresponding initial conditions, by means of undetermined coefficients. In the second part, we first obtain a particular solution by means of the Laplace transform for the inhomogeneous equations with null initial conditions. Then we give an analytical representation of the general solution of the inhomogeneous equations through the sum of its particular solution and the general solution of the corresponding homogeneous equation.

Reduced-Order Aggregate Model for Large-Scale Converters With Inhomogeneous Initial Conditions in DC Microgrids

In practical microgrids, the inhomogeneous initial values are widely appeared due to soft-starting operation. If traditional model order reduction approaches are applied, the input-output maps error between the original system and reduced-order system is large. To address this problem, this paper proposes a reduced-order aggregate model based on balanced truncation approach to provide the preprocessing approach for the real-time simulation of large-scale converters with inhomogeneous initial conditions in DC microgrid. Firstly, the standard linear time-invariant model with inhomogeneous initial conditions is established through non-leader multiagents concept. To end this, it is convenient for scholars to build complex system modeling with switched topology. Furthermore, the full system is divided into two components, i.e., the unforced component with nontrivial initial conditions and forced component with null initial conditions. Moreover, this paper presents an aggregated approach that involves independent reducing component responses and combining reducing component responses. Based on this, the input-output maps error is reduced. Then, the approximated error estimate of the reduced-order aggregate model regarding large-scale converters in DC microgrid is first provided, which provides prior knowledge and theoretical basis for DC microgrid designers. Finally, the simulation results illustrate the accuracy of the proposed approach.

Propagation of seismic waves in a continuum modeled as a granular material

The interaction between near-surface geology and seismic radiation determines the features of wave propagation in terms of path, amplification or damping of the seismic signal, which in certain cases might cause the so called site effect. Here, we consider the propagation of seismic waves in a geological site modeled as a granular material. The model equations have been integrated with null initial conditions by means of a finite element method for the spatial components and a finite difference scheme with respect to time evolution. The numerical results so obtained refer to the case of a homogeneous not layered domain with different physical conditions.

Model reduction for systems with inhomogeneous initial conditions

We consider the model reduction problem for linear time-invariant dynamical systems having nonzero (but otherwise indeterminate) initial conditions. Building upon the observation that the full system response is decomposable as a superposition of the response map for an unforced system having nontrivial initial conditions and the response map for a forced system having null initial conditions, we develop a new approach that involves reducing these component responses independently and then combining the reduced responses into an aggregate reduced system response. This approach allows greater flexibility and offers better approximation properties than other comparable methods.

Approximation of a stochastic wave equation in dimension three, with application to a support theorem in Hölder norm: The non-stationary case

This paper is a continuation of (Bernoulli 20 (2014) 2169–2216) where we prove a characterization of the support in Hölder norm of the law of the solution to a stochastic wave equation with three-dimensional space variable and null initial conditions. Here, we allow for non-null initial conditions and, therefore, the solution does not possess a stationary property in space. As in (Bernoulli 20 (2014) 2169–2216), the support theorem is a consequence of an approximation result, in the convergence of probability, of a sequence of evolution equations driven by a family of regularizations of the driving noise. However, the method of the proof differs from (Bernoulli 20 (2014) 2169–2216) since arguments based on the stationarity property of the solution cannot be used.

Spline Based Pseudo‐Inversion of Sampled Data Non‐Minimum Phase Systems for an Almost Exact Output Tracking

AbstractThis paper considers the problem of achieving a very accurate tracking of a pre‐specified desired output trajectory , for linear, multiple input multiple output, non‐minimum phase and/or non hyperbolic, sampled data, and closed loop control systems. The proposed approach is situated in the general framework of model stable inversion and introduces significant novelties with the purpose of reducing some theoretical and numerical limitations inherent in the methods usually proposed. In particular, the new method does not require either a preactuation or null initial conditions of the system. The desired and the corresponding sought input are partitioned in a transient component ( and ut(k), respectively) and steady‐state ( and us(k), respectively). The desired transient component is freely assigned without requiring it to be null over an initial time interval. This drastically reduces the total settling time. The structure of ut(k) is a priori assumed to be given by a sampled smoothing spline function. The spline coefficients are determined as the least‐squares solution of the over‐determined system of linear equations obtained imposing that the sampled spline function assumed as reference input yield the desired output over a properly defined transient interval. The steady‐state input us(k) is directly analytically computed exploiting the steady‐state output response expressions for inputs belonging to the same set of .

On an analytical representation for the solution of the neutron point kinetics equation free of stiffness

In this work, we report on an analytical representation for the solution of the neutron point kinetics equation, free of stiffness and assuming that reactivity is a continuous or sectionally continuous function of time. To this end, we cast the point kinetics equation in a first order linear differential equation. Next, we split the corresponding matrix into a diagonal matrix plus a matrix that contains the remaining terms. Expanding the neutron density and the delayed neutron precursor concentrations in a truncated series allows one to construct a recursive system in form of a first order matrix differential equation with source. The initialization of the recursion procedure is of diagonal form and has no source but satisfies the initial conditions. The remaining equations are subject to null initial conditions and include the time-dependent diagonal elements together with the off-diagonal elements as a source term. The solution is obtained in an analytical representation which may be evaluated for any time value, because it is free of stiffness. We present numerical simulations and comparisons against results from the literature for a constant, a step, a ramp, a quadratic, and a sine shaped reactivity function.

Frequency-domain analysis of exponentially damped linear systems

In this paper, frequency-domain analysis of linear exponentially damped systems subjected to arbitrary initial conditions is studied. The standard Discrete Fourier Transform (DFT) procedure combined with Fast Fourier Transform (FFT) algorithms is applied to compute dynamic responses, after transforming the original motion equations subjected to non-zero initial conditions into new ones with completely null initial conditions by using the pseudo-force approach. A multi-degree-of-freedom (MDOF) system and a continuum system with exponential damping are considered as the illustrative problems. The accuracy of the proposed method is evaluated by comparing the results with those obtained from Simulink based on the state-space method.

FFT‐based spectral element analysis for the linear continuum dynamic systems subjected to arbitrary initial conditions by using the pseudo‐force method

AbstractIn this paper, a fast Fourier transform (FFT)‐based spectral element method (SEM) is developed for the linear continuum dynamic systems subjected to arbitrary, non‐null initial conditions. In the FFT‐based SEM, the original equations of motion subjected to arbitrary initial conditions are transformed into a new set of equations of motion subjected to completely null initial conditions by using the pseudo‐force method so that the conventional spectral element analysis can be applied to obtain desired dynamic responses. A simply supported beam and a cantilevered beam are considered as the illustrative problems to evaluate the FFT‐based SEM. The dynamic responses obtained by using the FFT‐based SEM are shown to be in good agreement with the analytical solutions obtained by using the mode superposition method. Copyright © 2007 John Wiley & Sons, Ltd.

ON THE ASYMPTOTICS OF THE DENSITY IN PERTURBED SPDE'S WITH SPATIALLY CORRELATED NOISE

We consider a general type of perturbed stochastic partial differential equations: [Formula: see text] with null initial conditions, [Formula: see text] a second-order partial differential operator and [Formula: see text] a Gaussian noise, white in time and correlated in space. It has been proved that there exists a smooth density [Formula: see text], t > 0, x ∈ ℝd, for the law of the solution of above-mentioned equation. Here, we find the Taylor expansion of this density [Formula: see text] on the diagonal.

Initial conditions contribution in a BEM formulation based on the convolution quadrature method

This work presents a two-dimensional boundary element method (BEM) formulation for the analysis of scalar wave propagation problems. The formulation is based on the so-called convolution quadrature method (CQM) by means of which the convolution integral, presented in time-domain BEM formulations, is numerically substituted by a quadrature formula, whose weights are computed using the Laplace transform of the fundamental solution and a linear multistep method. This BEM formulation was initially developed for scalar wave propagation problems with null initial conditions. In order to overcome this limitation, this work presents a general procedure that enables one to take into account non-homogeneous initial conditions, after replacing the initial conditions by equivalent pseudo-forces. The numerical results included in this work show the accuracy of the proposed BEM formulation and its applicability to such kind of analysis. Copyright © 2006 John Wiley & Sons, Ltd.

Behaviour of the density in perturbed SPDE's with spatially correlated noise

Consider the following general type of perturbed stochastic partial differential equations: Lu ε t,x=εα u ε t,x F ̇ (t,x)+β u ε t,x , (t,x)∈ R +× R d, ε>0, with null initial conditions, L a second-order partial differential operator and F a Gaussian noise, white in time and correlated in space. In a previous work we proved the existence of smooth density p t,x ε(y), t>0, x∈ R d, for the law of the solution of above-mentioned equation. In this paper we study the logarithmic estimates for this density, that means to establish the behaviour of 2 ε 2log p t, x ε ( y), as ε↓0. This kind of estimates is also called Varadhan–Léandre estimates.

On stochastic partial differential equations with spatially correlated noise: smoothness of the law

We deal with the following general kind of stochastic partial differential equations: Lu(t,x)=α(u(t,x)) F ̇ (t,x)+β(u(t,x)), t⩾0, x∈ R d with null initial conditions, L a second-order partial differential operator and F a Gaussian noise, white in time and correlated in space. Firstly, we prove that the solution u( t, x) possesses a smooth density p t, x for every t>0, x∈ R d . We use the tools of Malliavin Calculus. Secondly, we apply this general result to two particular cases: the d-dimensional spatial heat equation, d⩾1, and the wave equation, d∈{1,2}.

The Spectral Element Method in Structural Dynamics

Preface. Part One Introduction to the Spectral Element Method and Spectral Analysis of Signals. 1 Introduction. 1.1 Theoretical Background. 1.2 Historical Background. 2 Spectral Analysis of Signals. 2.1 Fourier Series. 2.2 Discrete Fourier Transform and the FFT. 2.3 Aliasing. 2.4 Leakage. 2.5 Picket-Fence Effect. 2.6 Zero Padding. 2.7 Gibbs Phenomenon. 2.8 General Procedure of DFT Processing. 2.9 DFTs of Typical Functions. Part Two Theory of Spectral Element Method. 3 Methods of Spectral Element Formulation. 3.1 Force-Displacement Relation Method. 3.2 Variational Method. 3.3 State-Vector Equation Method. 3.4 Reduction from the Finite Models. 4 Spectral Element Analysis Method. 4.1 Formulation of Spectral Element Equation. 4.2 Assembly and the Imposition of Boundary Conditions. 4.3 Eigenvalue Problem and Eigensolutions. 4.4 Dynamic Responses with Null Initial Conditions. 4.5 Dynamic Responses with Arbitrary Initial Conditions. 4.6 Dynamic Responses of Nonlinear Systems. Part Three Applications of Spectral Element Method. 5 Dynamics of Beams and Plates. 5.1 Beams. 5.2 Levy-Type Plates. 6 Flow-Induced Vibrations of Pipelines. 6.1 Theory of Pipe Dynamics. 6.2 Pipelines Conveying Internal Steady Fluid. 6.3 Pipelines Conveying Internal Unsteady Fluid. Appendix 6.A: Finite Element Matrices: Steady Fluid. Appendix 6.B: Finite Element Matrices: Unsteady Fluid. 7 Dynamics of Axially Moving Structures. 7.1 Axially Moving String. 7.2 Axially Moving Bernoulli-Euler Beam. 7.3 Axially Moving Timoshenko Beam. 7.4 Axially Moving Thin Plates. Appendix 7.A: Finite Element Matrices for Axially Moving String. Appendix 7.B: Finite Element Matrices for Axially Moving Bernoulli-Euler Beam. Appendix 7.C: Finite Element Matrices for Axially Moving Timoshenko Beam. Appendix 7.D: Finite Element Matrices for Axially Moving Plate. 8 Dynamics of Rotor Systems. 8.1 Governing Equations. 8.2 Spectral Element Modeling. 8.3 Finite Element Model. 8.4 Numerical Examples. Appendix 8.A: Finite Element Matrices for the Transverse Bending Vibration. 9 Dynamics of Multi-Layered Structures. 9.1 Elastic-Elastic Two-Layer Beams. 9.2 Elastic-Viscoelastic-elastic-Three-Layer (PCLD) Beams. Appendix 9.A: Finite Element Matrices for the Elastic-Elastic Two-Layer Beam. Appendix 9.B: Finite Element Matrices for the Elastic-VEM-Elastic Three-Layer Beam. 10 Dynamics of Smart Structures. 10.1 Elastic-Piezoelectric Two-Layer Beams. 10.2 Elastic-Viscoelastic-Piezoelctric Three-Layer (ACLD) Beams. 11 Dynamics of Composite Laminated Structures. 11.1 Theory of Composite Mechanics. 11.2 Equations of Motion for Composite Laminated Beams. 11.3 Dynamics of Axial-Bending-Shear Coupled Composite Beams. 11.4 Dynamics of Bending-Torsion-Shear Coupled Composite Beams. Appendix 11.A: Finite Element Matrices for Axial-Bending-Shear Coupled Composite Beams. Appendix 11.B: Finite Element Matrices for Bending-Torsion-Shear Coupled Composite Beams. 12 Dynamics of Periodic Lattice Structures. 12.1 Continuum Modeling Method. 12.2 Spectral Transfer Matrix Method. 13 Biomechanics: Blood Flow Analysis. 13.1 Governing Equations. 13.2 Spectral Element Modeling: I. Finite Element. 13.3 Spectral Element Modeling: II. Semi-Infinite Element. 13.4 Assembly of Spectral Elements. 13.5 Finite Element Model. 13.6 Numerical Examples. Appendix 13.A: Finite Element Model for the 1-D Blood Flow. 14 Identification of Structural Boundaries and Joints. 14.1 Identification of Non-Ideal Boundary Conditions. 14.2 Identification of Joints. 15 Identification of Structural Damage. 15.1 Spectral Element Modeling of a Damaged Structure. 15.2 Theory of Damage Identification. 15.3 Domain-Reduction Method. 16 Other Applications. 16.1 SEM-FEM Hybrid Method. 16.2 Identification of Impact Forces. 16.3 Other Applications. References. Index.

Spatial decay estimates of saint-venant type in generalized thermoelasticity

Decay estimates of Saint-Venant type are established within the context of dynamic generalized thermoelasticity. A thermoelastic body is deformed under null body loads and initial conditions and subject to non-zero boundary data only on a plane end. The behaviour is studied for the total energy stored in that part of the body beyond a distance z from the basis. An estimate is established that describes the spatial decay of end effects. When the harmonically vibrations of a right cylinder are considered, it is shown that an exponential decay of activity holds away from the excited end provided that the exciting frequency is less than a certain critical frequency.

Method of reflections applied to the construction of the asymptote of Green's function for parabolic equations with variable coefficients

Two studies [1, 2] have been devoted to topics related to the construction of the asymptote of the fundamental solution of Eq. (1). In [3, 4] a multiplicative asymptote was constructed for the mixed initial-boundary problem of the first and second kind under null initial conditions on the second-order parabolic equations with the higher derivatives multiplied by a small parameter. In the present study a method is proposed for the construction of Green's function based on a representation of the delta function in the form [2] (2 h)_1/2 o+,2 (3) ~(z)

A polynomial approach to deriving a state-space model of a periodic process described by difference equations

When the problem is considered of obtaining a periodic description in state-space form of a linear process which can be modelled by linear difference equations with periodic coefficients, it is natural to ask whether it is possible to preliminarily derive a polynomial equivalent form of such equations, which in the periodic case plays a role similar to the Rosenbrock's polynomial matrix description of a linear time-invariant process. In this paper a polynomial time-invariant description of a linear periodic process is introduced. It is shown that such a polynomial description gives a simple characterization of the dimension of the space of the solutions corresponding to the null input function, i.e., of the order of the periodic model under consideration. In addition, it allows us to introduce a transfer matrix for the computation of the output responses corresponding to null initial conditions, and to deduce conditions for the periodic model to be causal. These results, as well as the possibility of defining strict system equivalence between two periodic models through their time-invariant polynomial descriptions, in a similar sense as in the time-invariant case, show the relevance of such a polynomial time-invariant description for the problem under consideration.

THE NON‐LINEAR INVERSION OF SEISMIC WAVEFORMS CAN BE PERFORMED EITHER BY TIME EXTRAPOLATION OR BY DEPTH EXTRAPOLATION1

ABSTRACTThe classical aim of non‐linear inversion of seismograms is to obtain the earth model which, for null initial conditions and given sources, best predicts the observed seismograms. This problem is currently solved by an iterative method: each iteration involves the resolution of the wave equation with the actual sources in the current medium, the resolution of the wave equation, backwards in time, with the current residuals as sources; and the correlation, at each point of space, of the two wavefields thus obtained.Our view of inversion is more general: we want to obtain a whole set of earth model, initial conditions, source functions, and predicted seismograms, which are the closest to some a priori values, and which are related through the wave equation. It allows us to justify the previous method, but it also allows us to set the same inverse problem in a different way: what is now searched for is the best fit between calculated and a priori initial conditions, for given sources and observed surface displacements. This leads to a completely different iterative method, in which each iteration involves the downward extrapolation of given surface displacements and tractions, down to a given depth (the‘bottom’), the upward extrapolation of null displacements and tractions at the bottom, using as sources the initial time conditions of the previous field, and a correlation, at each point of the space, of the two wavefields thus obtained. Besides the theoretical interest of the result, it opens the way to alternative numerical methods of resolution of the inverse problem. If the non‐linear inversion using forward‐backward time propagations now works, this non‐linear inversion using downward‐upward extrapolations will give the same results but more economically, because of some tricks which may be used in depth extrapolation (calculation frequency by frequency, inversion of the top layers before the bottom layers, etc.).

Nonstationary motion of a viscous fluid in a long tube

A number of problems are solved for the nonstationary motion of a viscous compressible fluid in a tube with elastic walls. It is assumed that the tube is semi-infinite, its axis horizontal, and that at one of its ends the flow rate of the fluid can change. The solution of each of the problems is reduced to the finding a generalized solution to a nonlinear system of partial differential equations for two functions — the mean values of the velocity and pressure in the tube section — with certain constant or null initial conditions and with a boundary condition specifying the time dependence of some function of the velocity and the pressure at the end of the tube. It is noted that the same problems can be solved by successive approximation.