190,505 research outputs found

    A Generalized Solution Method to Undamped Constant-Coefficient Second-Order ODEs Using Laplace Transforms and Fourier Series

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    A generalized method for solving an undamped second order, linear ordinary differential equation with constant coefficients is presented where the non-homogeneous term of the differential equation is represented by Fourier series and a solution is found through Laplace transforms. This method makes use of a particular partial fraction expansion form for finding the inverse Laplace transform. If a non-homogeneous function meets certain criteria for a Fourier series representation, then this technique can be used as a more automated means to solve the differential equation as transforms for specific functions need not be determined. The combined use of the Fourier series and Laplace transforms also reinforces the understanding of function representation through a Fourier series and its potential limitations, the mechanics of finding the Laplace transform of a differential equation and inverse transforms, the operation of an undamped system, and through programming insight into the practical application of both tools including information on the influence of the number of terms in the series solution

    Computable Cyclic Functions

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    This dissertation concerns computable analysis where the idea of a representation of a set is of central importance. The key ideas introduced are those commenting on the computable relationship between two newly constructed representations, a representation of integrable cyclic functions, and the continuous cyclic function representation. Also, the computable relationship of an absolutely convergent Fourier series representation is considered. It is observed that the representation of integrable cyclic functions gives rise to a much larger set of computable functions than obtained by the continuous cyclic function representation and that integration remains a computable operation, but that basic evaluation of the function is not computable. Many other representations are acknowledged enhancing the picture of the partial order structure on the space of representations of cyclic functions. The paper can also be seen as a foundation for the study of Fourier analysis in a computable universe and concludes with an investigation into the computability of the Fourier transform

    The unified transform method for linear initial-boundary value problems: a spectral interpretation

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    It is known that the unified transform method may be used to solve any well-posed initial-boundary value problem for a linear constant-coefficient evolution equation on the finite interval or the half-line. In contrast, classical methods such as Fourier series and transform techniques may only be used to solve certain problems. The solution representation obtained by such a classical method is known to be an expansion in the eigenfunctions or generalised eigenfunctions of the self-adjoint ordinary differential operator associated with the spatial part of the initial-boundary value problem. In this work, we emphasise that the unified transform method may be viewed as the natural extension of Fourier transform techniques for non-self-adjoint operators. Moreover, we investigate the spectral meaning of the transform pair used in the new method; we discuss the recent definition of a new class of spectral functionals and show how it permits the diagonalisation of certain non-self-adjoint spatial differential operators.Comment: 3 figure

    Fourier transforms on a semisimple symmetric space

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    Let G=H be a semisimple symmetric space, that is, G is a connected semisimple real Lie group with an involution ?, and H is an open subgroup of the group of xed points for ? in G. The main purpose of this paper is to study an explicit Fourier transform on G=H. In terms of general representation theory the (`abstract') Fourier transform of a compactly supported smooth function f 2 C 1 c (G=H) is given by (see [6]) (1) ^ f(?)() =?(f) = ZG=H f(x)?(x) dx; for (?; H?) a unitary irreducible representation of G and 2 (H ? an H-invariant distribution vector for ?. Here dx is the invariant measure on G=H. Thus ^ f f?)() is a smooth vector for H?, depending linearly on . Our goal is to obtain an explicit version of the restriction of this Fourier transform to representations (?; H?) in the (minimal) unitary principal series (??;; H?;) for G=H, under the assumption that the center of G is nite. In the sequel [10] to this paper it is proved that a function f 2 C 1 c (G=H) is uniquely determined by the restriction of ^ f to this series (a priori it is known that f is determined by ^ f )

    Design of Gradient Index Optical Thin Films

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    Gradient index thin films provide greater flexibility for the design of optical coatings than the more conventional \u27layer\u27 films. In addition, gradient index films have higher damage thresholds and better adhesion properties. This dissertation presents an enhancement to the existing inverse Fourier transform gradient index design method, and develops a new optimal design method for gradient index films using a generalized Fourier series approach. The inverse Fourier transform method is modified to include use of the phase of the index profile as a variable in rugate filter design. Use of an optimal phase function in Fourier-based filter designs reduces the product of index contrast and thickness for desired reflectance spectra. The shape of the reflectance spectrum is recovered with greater fidelity by suppression of Gibbs oscillations and shifting of side-lobes into desired wavelength regions. A new method of gradient index thin film design using generalized Fourier series extends the domain of problems for which gradient index solutions can be found. The method is analogous to existing techniques for layer based coating design, but adds the flexibility of gradient index films. A subset of the coefficients of a generalized Fourier series representation of the gradient index of refraction profile are used as variables in a nonlinear constrained optimization formulation. The optimal values of the design coefficients are determined using a sequential quadratic programming algorithm. This method is particularly well suited for the design of coatings for laser applications, where only a few widely separated wavelength requirements exist. The generalized Fourier series method is extended to determine the minimum film thickness needed, as well as the index of refraction profile for the optimal film
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