Peano-Baker Series Research Articles

Willis couplings in continuously varying cross-sectional area duct.

Acoustic wave propagation in a one-dimensional periodic and asymmetric duct is studied theoretically and numerically to derive the effective properties. Closed form expressions for these effective properties, including the asymmetric Willis coupling, are derived through truncation of the Peano-Baker series expansion of the matricant (which links the state vectors at the two sides of the unit-cell) and Padé's approximation of the matrix exponential. The results of the first-order and second-order homogenization (with Willis coupling) procedures are compared with the numerical results. The second-order homogenization procedure provides scattering coefficients that are valid over a much larger frequency range than the usual first-order procedure. The frequency well below which the effective description is valid is compared with the lower bound of the first Bragg bandgap when the profile is approximated by a two-step function of identical indicator function, i.e., two different cross-sectional areas over the same length. This validity limit is then questioned, particularly with a focus on impedance modeling. This article attempts to facilitate the engineering use of Willis materials.

Explicit solutions and asymptotic behaviors of Caputo discrete fractional-order equations with variable coefficients

A new discrete fractional-order Peano-Baker series is established in this paper. Based on this series, the explicit solutions of Caputo linear discrete fractional-order equations (DFOEs) with variable matrix coefficients involving the homogeneous and inhomogeneous cases are obtained. On the basis of the comparison theorem and the explicit solutions of Caputo linear DFOEs with variable coefficients, the asymptotic behaviors of the solutions of the homogeneous and inhomogeneous Caputo linear DFOEs with variable coefficients are presented. Finally, the applicability and effectiveness of the proposed results are illustrated by four examples.

Quantum mechanics of round magnetic electron lenses with Glaser and power law models of [formula omitted

Scalar theory of quantum electron beam optics, at the single-particle level, derived from the Dirac equation using a Foldy–Wouthuysen-like transformation technique is considered. Round magnetic electron lenses with Glaser and power law models for the axial magnetic field B(z) are studied. Paraxial quantum propagator for the Glaser model lens is obtained in terms of the well known fundamental solutions of its paraxial equation of motion. In the case of lenses with the power law model for B(z) the well known fundamental solutions of the paraxial equations, obtained by solving the differential equation, are constructed using the Peano–Baker series also. Quantum mechanics of aberrations is discussed briefly. Role of quantum uncertainties in aberrations, and in the nonlinear part of the equations of motion for a nonparaxial beam, is pointed out. The main purpose of this article is to understand the quantum mechanics of electron beam optics though the influence of quantum effects on the optics of present-day electron beam devices might be negligible.

Bilinear state systems on an unbounded time scale

We demonstrate the existence and uniqueness of solutions to a bilinear state system with locally essentially bounded coefficients on an unbounded time scale. We obtain a Volterra series representation for these solutions which is norm convergent and uniformly convergent on compact subsets of the time scale. We show the associated state transition matrix has a similarly convergent Peano-Baker series representation and identify a necessary and sufficient condition for its invertibility. Finally, we offer numerical applications for dynamic bilinear systems – a frequency modulated signal model and a two-compartment cancer chemotherapy model.

General Analysis of Coupled Nonuniform Transmission Lines Based on State Transition Matrix

This letter presents an equation-based solution for the analysis of coupled nonuniform transmission lines (CNUTLs) in the frequency domain. The proposed methodology, which is based on state-transition matrix, models the Telegrapher's equations in form of linear position varying (LPV) system. Solving these equations with the Peano-Baker series (PBSs) provides the sought for closed-form far-end/near-end voltages and currents in the CNUTLs. Unlike the previous methods, due to the unique advantages of PBSs, such as versatility, performing with low-order terms, and subsequently, the high convergence rate, the proposed method is very fast and reliable. In order to validate the method, two practical CNUTLs of relevance in electromagnetic compatibility applications, namely, lossy/lossless exponentially and linearly tapered microstrips, are analyzed. Simulation results show that the proposed method is very accurate and fast in comparison to traditional methods.

Solution of Time-Variant Fractional Differential Equations With a Generalized Peano–Baker Series

Time-variant fractional systems have many applications. For example, they can be used for system identification of lithium-ion batteries. However, the analytical solution of the time-variant fractional pseudo state space equation is missing so far. To overcome this limitation, this letter introduces a novel matrix approach, namely the generalized Peano-Baker series, which is comparable to the transition matrix in the case of ordinary systems. Using this matrix, the solution of the time-variant fractional pseudo state space equation is derived. The initialization process is taken into account, which has been proven to be a crucial part for fractional operator calculus. Following this initialization, a modified definition of a fractional pseudo state is presented.

Three-dimensional analysis of functionally graded piezoelectric plate with arbitrarily distributed material properties

An orthotropic functionally graded piezoelectric rectangular plate with arbitrarily distributed material properties was studied, which is simply supported and grounded (electrically) on its four lateral edges. The state equations of the functionally graded piezoelectric material were obtained using the state-space approach, and a Peano-Baker series solution was obtained for the coupled electroelastic fields of the functionally graded piezoelectric plate subjected to mechanical and electric loading on its upper and lower surfaces. The influence of different distributions of material properties on the structural response of the plate was studied using the obtained solutions.

Analysis of steady heat conduction for 3D axisymmetric functionally graded circular plate

The thermal conduction behavior of the three-dimensional axisymmetric functionally graded circular plate was studied under thermal loads on its top and bottom surfaces. Material properties were taken to be arbitrary distribution functions of the thickness. A temperature function that satisfies thermal boundary conditions at the edges and the variable separation method were used to reduce equation governing the steady state heat conduction to an ordinary differential equation (ODE) in the thickness coordinate which was solved analytically. Next, resulting variable coefficients ODE due to arbitrary distribution of material properties along thickness coordinate was also solved by the Peano-Baker series. Some numerical examples were given to demonstrate the accuracy, efficiency of the present model, and to investigate the influence of different distributions of material properties on the temperature field. The numerical results confirm that the influence of different material distributions, gradient indices and thickness of plate to temperature field in plate can not be ignored.

The Peano-Baker series

This note reviews the Peano-Baker series and its use to solve the general linear system of ODEs. The account is elementary and self-contained, and is meant as a pedagogic introduction to this approach, which is well known but usually treated as a folklore result or as a purely formal tool. Here, a simple convergence result is given, and two examples illustrate that the series can be used explicitly as well.

Three-dimensional thermoelastic analysis of functionally graded plate

Three-dimensional thermoelastic analysis is presented for an orthotropic functionally graded rectangular plate, which is simply supported and isothermal on its four lateral edges. With the assumption that material properties have arbitrary dependence on the thickness-coordinate, a Peano-Baker series solution is obtained for the thermoelastic fields of the functionally graded plate subjected to mechanical and thermal loads on its upper and lower surfaces by means of state space method. The correctness of the obtained series solution is validated through numerical examples. The influence of different material properties distributions on the structural response of the plate is also studied.

Three-Dimensional Analysis of Simply Supported Functionally Graded Plate with Arbitrary Distributed Elastic Modulus

Three-dimensional bending analysis is presented for a simply supported orthotropic functionally graded rectangular plate in this paper. Assuming that material properties have arbitrary variations along the plate-thickness direction, Peano-Baker series solution is obtained for the elastic fields of the functionally graded plate subjected to mechanical loads on its upper and lower surfaces by means of state space method. The correctness of the obtained series solution is validated through numerical examples. The influence of the structural response of the plate is also studied when material properties have different dependence on the thickness-coordinate. The results show that the solution is valid for the material properties of arbitrary dependence on the thickness-coordinate of the plate.

Plane wave analysis of panel wedges

In the present work, a wedge structure made of absorbing panels has been analyzed by making use of the matrizant analysis with the help of the Boundary-Condition-Transfer (BCT) algorithm. The rectangular panel wedge, as it is called in this manuscript, is simple in geometry. The theoretical model, based on the plane wave acoustical coupling between multiple interacting ducts of variable cross sectional area, is applied to predict the pressure reflection coefficient of the present wedge configuration. Bulk reaction and hence wave propagation in the wedge material has been assumed in the proposed model. An asymptotic solution using the Peano-Baker series of matrix calculus is derived for different variable area ducts and then solved with the aid of the boundary condition transfer algorithm. A brief parametric study is also included as an aid to designers.

Plane wave analysis of acoustic wedges using the boundary-condition-transfer algorithm

The present work delineates the applicability of the boundary-condition-transfer algorithm for analyzing the performance of the acoustic wedge. The theoretical model, based on the plane wave acoustical coupling between two interacting ducts of variable cross-sectional area, outlines the simplicity as well as the usefulness of the current approach. The proposed model assumes bulk reaction and hence a wave propagation in the wedge material. An asymptotic solution using the Peano-Baker series of matrix calculus is derived with the aid of an inherently stable boundary condition transfer (BCT) algorithm. The proposed method is validated extensively against the available literature in order to establish the utility of this elemental approach over its much more complex and time consuming counterparts like FEM and BEM.

Analysis of Multiple-duct Variable Area Perforated Tube Resonators

Conical concentric tube resonators are often used in commercial automotive mufflers. These are characterised by wave-coupling phenomenon across interacting ducts. Using a one-dimensional control volume approach, a mathematical model is presented for a generalised configuration of variable area perforated tube resonators. The analysis is applied to different configurations that account for waves in the incompressible mean flow and in the acoustic coupling between the interacting ducts due to the admittance of the intervening perforates. The problem has been solved by means of the Peano-Baker's series of matrix calculus, and the transmission loss results have been outlined. The predictions have been validated against the three-dimensional finite element analysis.

Transition matrix and generalized matrix exponential via the Peano-Baker series

We give a closed form for the unique solution to the n × n regressive time varying linear dynamic system of the form via use of a newly developed generalized form of the Peano-Baker series. We develop a power series representation for the generalized time scale matrix exponential when the matrix A(t) ≡ A is a constant matrix. We also introduce a finite series representation of the matrix exponential using the Laplace transform for time scales, as well as a theorem which allows us to write the matrix exponential as a series of (n − 1) terms of scalar functions multiplied by powers of the system matrix A.