Series representation of generalized temperature functions.
Maclaurin series expansion representation of generalized heat equation
HEAT EQUATION↗
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Application of vector-valued rational approximations to the matrix eigenvalue problem and connections with Krylov subspace methods
Let F(z) be a vectored-valued function F: C approaches C sup N, which is analytic at z=0 and meromorphic in a neighborhood of z=0, and let its Maclaurin series be given. We use vector-valued rational approximation procedures for F(z) that are based on its Maclaurin series in conjunction with power iterations to develop bona fide generalizations of the power method for an arbitrary N X N matrix that may be diagonalizable or not. These generalizations can be used to obtain simultaneously several of the largest distinct eigenvalues and the corresponding invariant subspaces, and present a detailed convergence theory for them. In addition, it is shown that the generalized power methods of this work are equivalent to some Krylov subspace methods, among them the methods of Arnoldi and Lanczos. Thus, the theory provides a set of completely new results and constructions for these Krylov subspace methods. This theory suggests at the same time a new mode of usage for these Krylov subspace methods that were observed to possess computational advantages over their common mode of usage.
Sidi, Avram↗
Rational approximations from power series of vector-valued meromorphic functions
Let F(z) be a vector-valued function, F: C yields C(sup N), which is analytic at z = 0 and meromorphic in a neighborhood of z = 0, and let its Maclaurin series be given. In this work we developed vector-valued rational approximation procedures for F(z) by applying vector extrapolation methods to the sequence of partial sums of its Maclaurin series. We analyzed some of the algebraic and analytic properties of the rational approximations thus obtained, and showed that they were akin to Pade approximations. In particular, we proved a Koenig type theorem concerning their poles and a de Montessus type theorem concerning their uniform convergence. We showed how optical approximations to multiple poles and to Laurent expansions about these poles can be constructed. Extensions of the procedures above and the accompanying theoretical results to functions defined in arbitrary linear spaces was also considered. One of the most interesting and immediate applications of the results of this work is to the matrix eigenvalue problem. In a forthcoming paper we exploited the developments of the present work to devise bona fide generalizations of the classical power method that are especially suitable for very large and sparse matrices. These generalizations can be used to approximate simultaneously several of the largest distinct eigenvalues and corresponding eigenvectors and invariant subspaces of arbitrary matrices which may or may not be diagonalizable, and are very closely related with known Krylov subspace methods.
Sidi, Avram↗
Predictions of nucleation theory applied to Ehrenfest thermodynamic transitions
A modified nucleation theory is used to determine a critical nucleus size and a critical activation-energy barrier for second-order Ehrenfest thermodynamic transitions as functions of the degree of undercooling, the interfacial energy, the heat-capacity difference, the specific volume of the transformed phase, and the equilibrium transition temperature. The customary approximations of nucleation theory are avoided by expanding the Gibbs free energy in a Maclaurin series and applying analytical thermodynamic expressions to evaluate the expansion coefficients. Nonlinear correction terms for first-order-transition calculations are derived, and numerical results are presented graphically for water and polystyrene as examples of first-order and quasi-second-order transitions, respectively.
Barker, R. E., Jr.↗
A study of the applicability of nucleation theory to quasi-thermodynamic transformations of second and higher Ehrenfest-order
Transient and steady-state phenomena in temperature, stress, and electric, field intensity in ferroelectric polymers were investigated. The application and extension of the theory in the primary stage to the polarization domain nucleation and growth in ferroelectric polymers were developed. The kinetics of this growth were investigated. Expressions describing nucleation under the influence of an electric field were found through the expansion of the Gibbs' free energy in a Maclaurin series. The series was expanded in the electric field strength rather than the degree of undercooling. The resulting expressions were manipulated and applied to the case of nucleation of polarized domains in ferroelectric polymers. The kinetics of the nucleation and growth of polarized domains are also investigated. This was accomplished through the modification of the Johnson-Mehl-Avrami treatment of crystallization kinetics to be applicable to the growth of polarization domains in ferroelectric materials.
Barker, R. E., Jr.↗
On the fundamental frequency of rectangular simply-supported symmetric laminated plates
A new closed-form approximate solution for the fundamental frequency of symmetric rectangular laminates which are simply supported on all four edges is derived. The solution, obtained from eigensensitivity analysis, is expressed as a truncated Maclaurin series in the coupling stiffnesses D16 and D26. Results show that the predicted fundamental frequency is remarkably accurate. A comparison of the fundamental frequencies of four-ply symmetric angle-ply laminates calculated from the new formula with those determined from a Rayleigh-Ritz procedure yielded a maximum differential of 0.6 percent over a wide range of principal stiffness ratios, plate aspect ratios, and design angles.
Reiss, Robert↗
Further Development of a Function Generator Technique
Further features and properties of functions generated by generalization of the process originally uncovered to describe the evolution of pressure in a rigid volume due to outgassing or desorption are explored. Properties presented include development of a general Maclaurin series associated with these functions, a limitation in using integration by parts to produce asymptotic expansions, and a general description of an implicit Adams-Moulton method for numerical quadrature of transformed functions. These developments are then applied to develop and explore functions that solve the Sievert integral and the modified Bessel function of the first kind, order zero. Features of the error function and the incomplete lower gamma function are also considered.
Michael S. Woronowicz↗
The gamma function with varying difference interval.
Gamma function concept for varying difference interval and complex argument
GAMMA FUNCTION↗
A series expansion of the acoustic power radiated from planar sources
A series expansion in ascending powers of the wavenumber k is derived for the acoustic power delivered by baffled or unbaffled planar sources. This series provides a relatively simple means of derving expressions for the power radiated by a baffled source with a known velocity distribution and can be used for unbaffled plates when the velocity field outside the plate is also known. The terms in the series are calculated from the moments of this velocity distribution in the plane containing the source. If these moments are written as derivaties in wavenumber space, it is shown that a MacLaurin expansion of the Fourier transformed velocity provides an easy technique for computing the first few terms of the acoustic power. Examples are provided for baffled, rectangular plates with various boundary conditions. The arbirarily shaped plate with free boundaries is particularly interesting. It is proven that the volume flow across it surface must be zero and as a result corner and edge mode radiation cannot exist for this kind of source.
Willams, E. G.↗