Showing papers on "Elementary function published in 1987"

Exact first-order solutions of the nonlinear Schrödinger equation

Abstract: A method is proposed for finding exact solutions of the nonlinear Schroedinger equations. It uses an ansatz in which the real and imaginary parts of the unknown function are connected by a linear relation with coefficients that depend only on the time. The method consists of constructing a system of ordinary differential equations whose solutions determine solutions of the nonlinear Schroedinger equation. The obtained solutions form a three-parameter family that can be expressed in terms of elliptic Jacobi functions and the incomplete elliptic integral of the third kind. In the general case, the obtained solutions are periodic with respect to the spatial variable and doubly periodic with respect to the time. Special cases for which the solutions can be expressed in terms of elliptic Jacobi functions and elementary functions are considered in detail. Possible fields of practical applications of the solutions are mentioned.

On spherically symmetric shear‐free perfect fluid configurations (neutral and charged). I

Abstract: A class of solutions describing a wide variety of nonstatic, spherically symmetric, charged, shear‐free perfect fluid configurations is derived. It is presented in the form of Jacobian elliptic functions characterized by seven free parameters: five constants and two arbitrary functions of time. This class of solutions is the most general charged version of the class derived by Kustaanheimo and Qvist [Comment. Phys. Math. Helsingf. 13, 12 (1948); Exact Solutions of Einstein’s Field Equations (Cambridge U. P., Cambridge, 1980), Chap. 12, Sec. 2]. It is found that many of the charged particular solutions expressible by elementary functions are new. Particular solutions, including neutral and uniform density solutions, are classified in detail. The physical interpretation of these solutions, including the study of their singularity structure, will be presented in a subsequent paper (Part II).

Elementary function expansions for Madelung constants

Abstract: The Madelung constant-essentially the Coulomb energy density of a crystal-is usually calculated via Ewald error function expansions or, for the simpler cubic structures, by the 'cosech' series of modern vintage. By considering generalised functional equations for multidimensional zeta functions, the authors provide explicit expansions for the spatial potential and energy density of three-dimensional periodic structures. These formulae, involving only elementary functions, are suitable for systematic calculation of Madelung constants of arbitrary point-charge crystals. They indicate how zeta function relations may be used for dimensional reduction of certain multiple sums arising in the special cubic cases.

Evaluating elementary functions with Chebyshev polynomials on pipeline nets

Abstract: Fast evaluation of vector-valued elementary functions plays a vital role in many real-time applications. In this paper, we present a pipeline networking approach to designing a Chebyshev polynomial evaluator for the fast evaluation of elementary functions over a string of arguments. In particular, pipeline nets are employed to perform the preprocessing and postprocessing of various elementary functions to boost the overall system performance. Design tradeoffs are analyzed among representational accuracy, processing speed and hardware complexity.

Elementary function arithmetic unit

Abstract: A function arithmetic unit which performs elementary operations at high speeds and which enables required memory capacity to be reduced. The function arithmetic unit includes a constant memory which stores a constant that corresponds to the number of successive iterative operations; a controller which causes addition or subtraction in the successive iterative operations; and arithmetic units which receive, as initial values, an argument of an elementary function value to be found and two constants determined depending upon the elementary functions to be found, and which subject these initial values to n iterative operations to produce elementary function values. A further arithmetic unit subjects the elementary function values to multiplication, addition, subtraction, or division, or a combination thereof, thereby to produce the thus obtained value.

Closed form solution to some mixed boundary value problems for a charged sphere

Abstract: A new method is described which allows an exact solution in a closed form to the following non-axisymmetric mixed boundary-value problem for a charged sphere: arbitrary potential values are given at the surface of a spherical segment while an arbitrary charge distribution is prescribed on the rest of the sphere. The method is founded on a new integral representation of the kernel of the governing integral equation. Several examples are considered. All the results are expressed in elementary functions. Some further applications of the method are discussed. No similar result seems to have been published previously.

Recursion relations for solutions to the Schro¨dinger equation

Abstract: We will consider the general eigenfunction for a second-order differential operator in one variable. For many well-known elementary functions, we can also find a three-term recursion relation in the eigenvalue parameter. For practical computation, this is a very desirable property. Examples include Legendre functions, Bessel functions, etc.In [Math. Z., 29 (1929), pp. 730–736], Bochner showed that the only polynomial solutions to this problem were the well-known ones. This paper will look for solutions that may not be polynomials.It will be shown that, unfortunately, for the simple recursion relation considered here, no really new examples exist.

A New Approximate Solution Valid at Turning Points

Abstract: A new approximate solution for a Helmholtz equation of one dimension is suggested. The solution is valid in cases both of having no turning point and of having only “linear” or “semi-linear” ones. The solution has the form of improved classical WKB solution with compensating functions. The solution is expressed by elementary functions.

An algorithm for the integration of elementary functions

Abstract: Trager (1984) recently gave a new algorithm for the indefinite integration of algebraic functions. His approach was “rational” in the sense that the only algebraic extension computed is the smallest one necessary to express the answer. We outline a generalization of this approach that allows us to integrate mixed elementary functions. Using only rational techniques, we are able to normalize the integrand, and to check a necessary condition for elementary integrability.