Form Of Fourier Series Research Articles

Periodic metastable structures in the discrete straight phi4 model.

For the discrete straight phi(4) model a class of metastable periodic solutions is given in the form of Fourier series. From the symmetry consideration, we eliminate the harmonics with zero amplitudes and thus, reduce the number of degrees of freedom. For the rest of harmonics we establish the hierarchy of their significance. For solutions with a small period we give the exact expression of energy density in terms of amplitudes of harmonics. Conditions of existence and stability of the solutions are discussed. For some limiting cases the approximate solutions of different kinds are given. The analytical results are compared with the results of numerical study. We also discuss a mechanism of transition from a metastable periodic structure into the ground state and apply the results to the lock-in transition in dielectric crystals supporting incommensurate phase.

Mathematical methods with applications

Ordinary differential equations - Classification of first order differential equations First order nonlinear differential equations Singular solutions of differential equations Orthogonal trajectories Higher order linear differential equations The solution of the nonhomogeneous equations The method of variation of parameters The method of differential operator Euler-Cauchy differential equations Applications to practical problems. Fourier series and Fourier transform - Introduction Definition of a periodic function Fourier series and Fourier coefficients Complex form of Fourier series Half-range Fourier sine and cosine series Parseval's theorem Gibbs' phenomenon Development of Fourier integral and transform Relationship of Fourier and Laplace transforms Applications of Fourier transforms Parseval's theorem for energy signals Heaviside unit step function and Dirac delta function Some Fourier transforms involving impulse functions Properties of the Fourier transform The frequency transfer function. Laplace transforms - Introduction Definition of Laplace transform Laplace transform properties Laplace transforms of special functions Some important theorems The unit step function and the Dirac delta function The Heaviside expansion theorems to find inverses The method of residues to find inverses The Laplace transform of a periodic function Convolution. Series solution: method of Frobenius - Introduction Definition of ordinary and singular points Series expansion about an ordinary point Series expansion about a regular singular point. Partial differential equations - Introduction Mathematical formulation of equations Classification of PDE: Method of characteristics The D'Alembert solution of the wave equation The method of separation of variables Laplace and Fourier transform methods Similarity technique Applications to miscellaneous problems Sturm-Liouville problems. Bessel functions and Legendre polynomials - Introduction Series solution of Bessel's equation Modified Bessel functions Ber, Bei, Ker and Kei functions Equations solvable in terms of Bessel functions Recurrence relations of Bessel functions Orthogonality of Bessel functions Legendre polynomials Applications. Applications - Applications of Fourier series Applications of Fourier integrals Applications of Laplace transforms Applications with PDE Transmission lines The heat conduction problem The chemical diffusion problem Vibration of beams The hydrodynamics of waves and tides. Green's function - One-dimensional Green's function Green's function using variation of parameters Developments of Green's function in 2D Development of Green's function in 3D Numerical formulation. Integral transforms - Introduction The Hankel transform The Mellin transform The Z-transform.

The asymmetrical mixed temperature problem for a transversely isotropic elastic layer

The problem of the uniform heating of a two-layer plate is solved. The transversely isotropic elastic layer (soft plate) investigated is in ideal contact with an absolutely rigid layer, deformable only by thermal expansion. The generalized plane temperature problem reduces to determining the stress-strain state of the soft anisotropic layer investigated using the equations of the mixed problem of elasticity theory. At the ends of the boundary layer of the soft plate (a thin contact layer), no conditions are imposed. On the remaining part of the ends of the soft plate, the boundary conditions correspond to a free boundary. The problem has a bounded smooth solution. Unlike the approach described earlier [1], it is proposed to seek an accurate solution in the form of ordinary Fourier series with respect to a single longitudinal coordinate. Solutions in polynomials are also used. It is shown that the existence of these solutions in polynomials enables the convergence of the Fourier series to be improved considerably.

A semi-analytical finite element method for three-dimensional consolidation analysis

An efficient formulation, based on a semi-analytical finite element method, is described for elasto-plastic analyses of consolidation of an axi-symmetric soil body subjected to three-dimensional loading. Expressing the field quantities in the form of discrete Fourier series results in a set of modal equations that can be solved separately. This has the effect of considerably reducing the necessary storage and the cost of solving three-dimensional problems. The numerical method is applied to the problem of a laterally loaded pile in consolidating elasto-plastic soil.

Spherical analog of the complex form of trigonometric fourier series; Approximation

For a function defined on the unit sphere of dimension at least 2, we suggest a spherical analog with algebraic structure similar to the Fourier components of a function defined on the unit circle. The deviations of the analogs of partial Fourier sums from the function are estimated with respect to various norms.

Topographic effect of the misfit dislocation dissociation in threefold symmetry epitaxial systems

Abstract Epitaxial systems with semicoherent heterointerfaces contain sometimes complex biperiodic networks of misfit dislocations (MDs). Analytical solutions for the elastic fields of threefold symmetry structures can in fact be derived in the form of Fourier series, in which explicit constants takes account of the detailed geometry of the MD pattern. In principle, any complex planar pattern can be treated. The material is considered as elastically heterogeneous, each medium having its own isotropic elastic constants. As an application of the theory, the In As GaAs(111)A system is treated in detail as observed by Yamaguchi et al. (1997. Phys. Rev. B, 55, 1337), for which the InAs epitaxial layer has a thickness of 1.3nm. The dislocation network consists of edge MDs arranged in a regular hexagonal-based pattern, each alternate node being dissociated into a faulted triangle. Three different networks can be produced according to the dissociation extent: a honeycomb network (no dissociation) formed by MDs wi...

Orthotropic pressure vessels subject to local loads

Differential equations for unsymmetrical loading of simply supported orthotropic circular cylindrical shell are obtained in terms of displacement components. Then displacement components are expressed in the form of double trigonometric Fourier series that satisfies the boundary conditions at the end. Uniformly distributed load on rectangular areas is considered, which is again expressed in terms of double trigonometric Fourier series. Expressions are obtained for displacement components, stress-resultants and stress-couples. Numerical results are obtained and plotted for a particular case.

A resonance case of the existence of solutions of a quasilinear second-order differential system, which are represented by Fourier series with slowly varying parameters

For a quasilinear second-order differential system, whose coefficients have the form of Fourier series with slowly varying coefficients and frequency, we prove, under certain conditions, the existence of a particular solution having a similar structure. This result is obtained in the case where the characteristic equation possesses purely imaginary roots, which satisfy a certain resonance relation.

Free vibration behaviour of helically wound cylindrical shells

In this paper a modified exact analytical method is employed to study the vibration characteristics of high modulus cylindrical shells laminated of a composite material. Here the axial dependence of modal forms is chosen in the form of simple Fourier series. This choice is found to be greatly helpful in the study of the influence of boundary conditions on the various vibration characteristics of such structures. Transverse shear and rotary inertia terms are included in the analysis for a more reliable prediction of response characteristics in general for the composite shells. The effectiveness and the accuracy of the present analytical method are displayed by comparing the analytical frequencies obtained here with some previously published experimental as well as theoretical results. The two sets of results were found to be in good agreement.

The classical limit of quantum mechanics and the Fejér sum of the Fourier series expansion of a classical quantity

In quantum mechanics, the expectation value of a quantity on a quantum state, provided that the state itself gives in the classical limit a motion of a particle in a definite path, in the classical limit goes over to a Fourier series form of the classical quantity. In contrast to this widely accepted point of view, a rigorous calculation shows that the expectation value on such a state in the classical limit exactly gives the Fejér arithmetic mean of the partial sums of the Fourier series.

Inelastic stability of conical tanks

The aim of this investigation is to study the effect of different imperfection shapes on the inelastic stability of liquid-filled conical tanks and to determine the critical imperfection shape that would lead to the minimum inelastic limit load. The study is carried out numerically using a self-developed shell element used to simulate a number of conical tanks having an imperfection shape in the form of Fourier series of equal coefficients. The Fourier analysis of the buckling modes indicates that the existence of axisymmetric imperfection will lead to the critical inelastic limit load for conical tanks.

Natural frequency response of vertical cantilever composite shells containing fluid

An analytical investigation of the natural frequency response of multi-layered orthotropic circular cylindrical shells partially submerged in or filled with non-viscous incompressible fluid has been developed here. The typical examples could be of liquid storage tanks or vertical pipe conveying fluid or submerged in fluid. Since these shells are considered to be laminated of composite material, the effect of transverse shear deformation and rotary inertia terms have been taken into account in the analysis. Axial dependence of modal forms is assumed in the form of a simple Fourier series instead of the commonly used exponential axial dependence. A modified variational method is used here to obtain a polynomial frequency equation. This equation can be solved for an arbitrary choice of end conditions. As an illustration the important case of a vertical cantilever shell with bottom end clamped and top end free is considered and analyzed in detail. The effect of variation in the fluid height level on the frequency response is studied. The influence of axial and meridional stiffnesses of the shell on the natural frequencies is also discussed.

On solutions of a second-order quasilinear differential system representable by Fourier series with slowly varying parameters

For a second-order quasilinear differential system whose coefficients have the form of Fourier series with slowly varying coefficients and frequency, we prove that, under certain conditions, there exists a particular solution with a similar structure in the case of purely imaginary roots of the characteristic equation for the matrix of coefficients of the linear part.

On necessary conditions for the weak regularity of a linear extension of a dynamical system on a torus

We give a necessary condition for the weak regularity of a linear extension of a dynamical system on a torus in the form of Fourier series with respect to a part of the variables.

Unperturbed chandler motion and perturbation theory of the rotation motion of deformable celestial bodies

Abstract New unperturbed motions are suggested for the study of the rotational motion of deformable celestial bodies. This motion describes the rotation of an isolated celestial body deformed by its own rotation. By some natural simplifications and by using special forms of canonical variables (similar to Andoyer's variables) the problem is reduced to the classical Euler-Poinsot problem for a rigid body, but with different moments of inertia. The suggested unpertubed motion describes Chandler's pole motion and we shall call it Chandler or Euler-Chandler motion. The development of the unperturbed theory is described in this paper. The solution of the Chandler problem (Andoyer's variables, components of angular velocity of the body's axes, and their direction cosines) is presented in elliptical and θ- functions, and in the form of Fourier series in the angle-action variables. Similar Fourier series were obtained for products and squares of the diraction cosines. The coefficients of these series are expresse...

Analytical Method for Solving the One-Dimensional Wave Equation With Moving Boundary - Abstract

It is shown that a large class of dynamic systems with non-stationary dimensions can be solved analytically. The corresponding mathematical models of these systems are initial-boundary-value-problems for the wave equation with moving boundary conditions. The method given in a previous paper is generalized and used for solving such problems. It consists of transforming the wave equation for the domain with moving boundary into a form-invariant wave equation for a domain with fixed boundary. The transformation is accomplished through a formal change of variable, and an analytical function F(W) of a complex variable W, permitting a conformal mapping between the two domains. For the transformation to exist, F(W) has to satisfy the condition F*(W) = F(W*). We show that the solution of such systems is obtained in the form of a functional Fourier series,, and the modes are dynamical, each having an instantaneous frequency. The problem of initial conditions is also treated. Furthermore, it is shown that all possible transformation functions permitting the covariant transformation of the wave equation have to satisfy two fundamental relations. As applications, some examples of F(W) and transformation functions are given.

Theoretical Study of Pressure Fluctuations Downstream of a Diffuser Pump Impeller—Part 1: Fundamental Analysis on Rotor-Stator Interaction

A theoretical method was developed to calculate the unsteady flow caused by the interaction between impeller and diffuser vanes in a diffuser pump by using the singularity method. The unsteady flow in the diffuser vane is assumed to be induced by three kinds of unsteady vortices: bound vortices distributed on the impeller blades and diffuser vanes, and free vortices shed from the trailing edge of diffuser vanes. In order to make clear the contribution of each harmonic component of unsteady vortices to unsteady pressure, all the unsteady vortices are expressed in the form of Fourier series. The calculated unsteady pressures downstream of impeller agree well with the corresponding measured ones. Moreover, it was shown that impulsive pressure plays a predominant role for unsteady pressures.

An analytical study of buckling of composite tubes with various boundary conditions

A theoretical investigation of the buckling problem of composite tubes with different boundary conditions is presented. Different types of loading such as compressive, torsional, internal, external or any combination of them can be applied. To increase the accuracy of results, the modal forms are assumed to have axial dependency in the form of simple Fourier series. The derivatives of the Fourier series are legitimised using Stoke's transformation. Flügge's shell equations, modified for anisotropic laminated materials, are used. The method developed in this work for finding the buckling loads yields an exact equation which is simpler than the exact method adopted by other research workers. To check the validity of the present analysis, the results are compared with some available theoretical and experimental results.

Experimental Identification of Input Torque Property of Link Mechanism.

It is difficult to calculate input torque of a complicated link mechanism theoretically. It is also difficult to calculate the input torque of a link mechanism with friction. In this study, we propose a method which can be used to identify the input torque property of link mechanism, such as the equivalent moment of inertia, experimentally. For this purpose, the input torque property of a system is expressed in the form of Fourier series with respect to the crank angle. The measured torque is also expanded in Fourier series. By comparing the coefficients of the series, we can determine the input torque property. Friction torque is separated by comparing the input torque during clockwise or counterclockwise rotation. As an example, we apply the proposed method to a simple sliderclank mechanism, and the validity of the proposed method is confirmed.

On the expansions of intermediate orbit for general planetary theory

Intermediate orbit for general planetary theory is constructed in the form of multivariate Fourier series with numerical coefficients. The structure and efficiency of the derived series are illustrated by giving various statistical properties of the coefficients. The ability of the recently proposed elliptic function approach to compress the Fourier series representing the intermediate orbit is investigated. Our results confirm that when mutual perturbations of a pair of planets are considered the elliptic function approach is quite efficient and allows one to compress the series substantially. However, when perturbations of three or more planets are under study the elliptic function approach does not give any advantages.