Showing papers on "Elementary function published in 2018"

Performance Analysis of Free-Space Optical Links Over M\'{a}laga ($\mathcal{M}$) Turbulence Channels with Pointing Errors

Abstract: In this work, we present a unified performance analysis of a free-space optical (FSO) link that accounts for pointing errors and both types of detection techniques (i.e. intensity modulation/direct detection (IM/DD) as well as heterodyne detection). More specifically, we present unified exact closed-form expressions for the cumulative distribution function, the probability density function, the moment generating function, and the moments of the end-to-end signal-to-noise ratio (SNR) of a single link FSO transmission system, all in terms of the Meijer's G function except for the moments that is in terms of simple elementary functions. We then capitalize on these unified results to offer unified exact closed-form expressions for various performance metrics of FSO link transmission systems, such as, the outage probability, the scintillation index (SI), the average error rate for binary and $M$-ary modulation schemes, and the ergodic capacity (except for IM/DD technique, where we present closed-form lower bound results), all in terms of Meijer's G functions except for the SI that is in terms of simple elementary functions. Additionally, we derive the asymptotic results for all the expressions derived earlier in terms of Meijer's G function in the high SNR regime in terms of simple elementary functions via an asymptotic expansion of the Meijer's G function. We also derive new asymptotic expressions for the ergodic capacity in the low as well as high SNR regimes in terms of simple elementary functions via utilizing moments. All the presented results are verified via computer-based Monte-Carlo simulations.

Two closed forms for the Apostol–Bernoulli polynomials

Abstract: "In mathematics, a closed form is a mathematical expression that can be evaluated in a finite number of operations. It may contain constants, variables, four arithmetic operations, and elementary functions, but usually no limit." In this note, we shall obtain two closed forms for the Apostol-Bernoulli polynomials.

Fundamental solution of unsteady Stokes equations and force on an oscillating sphere near a wall

Abstract: We derive the Green's function of unsteady Stokes equations near a plane boundary with no-slip boundary conditions. This provides flow due to an oscillating point force acting on fluid bounded by a wall. Our derivation is different from previous theories and resolves the apparent discrepancies of the reported results. Two-dimensional Fourier transform of the solution with respect to horizontal coordinates is given via elementary functions in a more compact form than by the previous theories. The tensorial Green's function in real space is reduced to two Hankel transforms of order zero. We derive a simple form for the real-space solution in the two limiting cases of a distance to the wall much larger and much smaller than the viscous penetration depth. We demonstrate the applicability of this form by obtaining results for the force exerted on a sphere oscillating near the wall. Using the integral equation on surface traction whose kernel is the fundamental solution, we derive the force in the limits of a distant wall and low frequency. The wall correction to the force decays as the inverse third power of the source to the wall separation distance, much faster than the inverse first power of the classical Lorentz solution for the time-independent problem. Our results significantly extend the range of parameters for which the force admits a simple closed-form solution. Small biological swimmers propelled by inherently unsteady swimming gait generate flows driven by derivatives of the point source and we provide an example of a wall-bounded solution of this type. We demonstrate that frequency expansion is an efficient way of studying the Green's functions in confined geometry that gives the complete series solution for channel geometry.

Some Exact Solutions to Non-Fourier Heat Equations with Substantial Derivative

Abstract: One-dimensional equations of telegrapher’s-type (TE) and Guyer–Krumhansl-type (GK-type) with substantial derivative considered and operational solutions to them are given. The role of the exponential differential operators is discussed. The examples of their action on some initial functions are explored. Proper solutions are constructed in the integral form and some examples are studied with solutions in elementary functions. A system of hyperbolic-type inhomogeneous differential equations (DE), describing non-Fourier heat transfer with substantial derivative thin films, is considered. Exact harmonic solutions to these equations are obtained for the Cauchy and the Dirichlet conditions. The application to the ballistic heat transport in thin films is studied; the ballistic properties are accounted for by the Knudsen number. Two-speed heat propagation process is demonstrated—fast evolution of the ballistic quasi-temperature component in low-dimensional systems is elucidated and compared with slow diffusive heat-exchange process. The comparative analysis of the obtained solutions is performed.

Convergent expansions of the Bessel functions in terms of elementary functions

Abstract: This is a post-peer-review, pre-copyedit version of an article published in Advances in Computational Mathematics. The final authenticated version is available online at: https://doi.org/10.1007/s10444-017-9543-y

Fractional derivative of composite functions: exact results and physical applications

Abstract: We examine the fractional derivative of composite functions and present a generalization of the product and chain rules for the Caputo fractional derivative. These results are especially important for physical and biological systems that exhibit multiple spatial and temporal scales, such as porous materials and clusters of neurons, in which transport phenomena are governed by a fractional derivative of slowly varying parameters given in terms of elementary functions. Both the product and chain rules of the Caputo fractional derivative are obtained from the expansion of the fractional derivative in terms of an infinite series of integer order derivatives. The crucial step in the practical implementation of the fractional product rule relies on the exact evaluation of the repeated integral of the generalized hypergeometric function with a power-law argument. By applying the generalized Euler's integral transform, we are able to represent the repeated integral in terms of a single hypergeometric function of a higher order. We demonstrate the obtained results by the exact evaluation of the Caputo fractional derivative of hyperbolic tangent which describes dark soliton propagation in the non-linear media. We conclude that in the most general case both fractional chain and product rules result in an infinite series of the generalized hypergeometric functions.

The gravitational potential and its derivatives of a right rectangular prism with depth-dependent density following an n-th degree polynomial

Abstract: The direct gravity problem and its solution belong to the basis of the gravimetry. The solutions of this problem are well known for wide class of the source bodies with the constant density contrast. The non-uniform density approximation leads to the relatively complicated mathematical formalism. The analytical solutions for this type of sources are rare and currently these bodies are very useful in the gravimetrical modeling. The solution for the vertical component of the gravitational attraction vector for the 3D right rectangular prism is known in the geophysical literature for the density variations described by the 3-rd degree polynomial. We generalized this solution for an n-th degree, not only for the vertical component, but for the horizontal components, the second-order derivatives and the potential as well. The 2D modifications of all given formulae are presented, too. The presented general solutions, which involve a hypergeometric functions, can be used as they are, or as an auxiliary tool to derive desired solution for the given degree of the density polynomial as a sum of the elementary functions. The pros-and-cons of these approaches (the complexity of the programming codes, runtimes) are discussed, too.

Elementary functions : towards automatically generated, efficient, and vectorizable implementations

Abstract: Elementary mathematical functions are pervasive in many high performance computing programs. However, although the mathematical libraries (libms), on which these programs rely, generally provide several flavors of the same function, these are fixed at implementation time. Hence this monolithic characteristic of libms is an obstacle for the performance of programs relying on them, because they are designed to be versatile at the expense of specific optimizations. Moreover, the duplication of shared patterns in the source code makes maintaining such code bases more error prone and difficult. A current challenge is to propose "meta-tools" targeting automated high performance code generation for the evaluation of elementary functions. These tools must allow reuse of generic and efficient algorithms for different flavours of functions or hardware architectures. Then, it becomes possible to generate optimized tailored libms with factorized generative code, which eases its maintenance. First, we propose an novel algorithm that allows to generate lookup tables that remove rounding errors for trigonometric and hyperbolic functions. The, we study the performance of vectorized polynomial evaluation schemes, a first step towards the generation of efficient vectorized elementary functions. Finally, we develop a meta-implementation of a vectorized logarithm, which factors code generation for different formats and architectures. Our contributions are shown competitive compared to free or commercial solutions, which is a strong incentive to push for developing this new paradigm.

Simple but accurate periodic solutions for the nonlinear pendulum equation

Abstract: Despite its elementary structure, the simple pendulum oscillations are described by a nonlinear differential equation whose exact solution for the angular displacement from vertical as a function of time cannot be expressed in terms of an elementary function, so either a numerical treatment or some analytical approximation is ultimately demanded. Such solutions have been thoroughly investigated due to the abundance of distinct pendular systems in nature and, more recently, due to the availability of automatic data acquisition systems in undergraduate laboratories. However, it is well-known that numerical solutions to differential equations usually loose accuracy (due to accumulation of roundoff errors) and polynomial approximations diverge after long time intervals. In this work, I take a few terms of the Fourier series expansion of the elliptic function sn (u;k) as a source of accurate periodic solutions for the pendulum equation. Interestingly, these approximations remain accurate for arbitrarily long time intervals, even for large amplitudes, which shows its adequacy for the analysis of experimental data gathered in classical mechanics classes.

Research on Some New Results Arising from Multiple q-Calculus

Abstract: In this paper, we develop the theory of the multiple q-analogue of the Heine's binomial formula, chain rule and Leibnitz's rule. We also derive many useful definitions and results involving multiple q-antiderivative and multiple q-Jackson's integral. Finally, we list here multiple q-analogue of some elementary functions including trigonometric functions and hyperbolic functions. This may be a good consideration in developing the multiple q-calculus in combinatorics, number theory and other fields of mathematics.

Elementary Functions and Factorizations of Zeons

Abstract: Algebraic properties of zeons are considered, including the existence of elementary factorizations and homogeneous factorizations of invertible zeons. A “zeon division algorithm” is established, showing that every nontrivial invertible zeon can be written as a sum of homogeneously decomposable zeons. Elementary functions (exponential, logarithmic, hyperbolic, and trigonometric) are extended to zeons, and a number of properties and identities are revealed. Finally, fast computation of logarithms is discussed for homogeneously decomposable zeons.

All static and electrically charged solutions with Einstein base manifold in the arbitrary-dimensional Einstein–Maxwell system with a massless scalar field

Abstract: We present a simple and complete classification of static solutions in the Einstein–Maxwell system with a massless scalar field in arbitrary $$n(\ge 3)$$ dimensions. We consider spacetimes which correspond to a warped product $$M^2 \times K^{n-2}$$ , where $$K^{n-2}$$ is a $$(n-2)$$ -dimensional Einstein space. The scalar field is assumed to depend only on the radial coordinate and the electromagnetic field is purely electric. Suitable Ansatze enable us to integrate the field equations in a general form and express the solutions in terms of elementary functions. The classification with a non-constant real scalar field consists of nine solutions for $$n\ge 4$$ and three solutions for $$n=3$$ . A complete geometric analysis of the solutions is presented and the global mass and electric charge are determined for asymptotically flat configurations. There are two remarkable features for the solutions with $$n\ge 4$$ : (i) Unlike the case with a vanishing electromagnetic field or constant scalar field, asymptotically flat solution is not unique, and (ii) The solutions can asymptotically approach the Bertotti–Robinson spacetime depending on the integrations constants. In accordance with the no-hair theorem, none of the solutions are endowed of a Killing horizon.

Exact results for a fractional derivative of elementary functions

Abstract: We present exact analytical results for the Caputo fractional derivative of a wide class of elementary functions, including trigonometric and inverse trigonometric, hyperbolic and inverse hyperbolic, Gaussian, quartic Gaussian, and Lorentzian functions. These results are especially important for multi-scale physical systems, such as porous materials, disordered media, and turbulent fluids, in which transport is described by fractional partial differential equations. The exact results for the Caputo fractional derivative are obtained from a single generalized Euler's integral transform of the generalized hyper-geometric function with a power-law argument. We present a proof of the generalized Euler's integral transform and directly apply it to the exact evaluation of the Caputo fractional derivative of a broad spectrum of functions, provided that these functions can be expressed in terms of a generalized hyper-geometric function with a power-law argument. We determine that the Caputo fractional derivative of elementary functions is given by the generalized hyper-geometric function. Moreover, we show that in the most general case the final result cannot be reduced to elementary functions, in contrast to both the Liouville-Caputo and Fourier fractional derivatives. However, we establish that in the infinite limit of the argument of elementary functions, all three definitions of a fractional derivative - the Caputo, Liouville-Caputo, and Fourier- converge to the same result given by the elementary functions. Finally, we prove the equivalence between Liouville-Caputo and Fourier fractional derivatives.

Analytic approximate eigenvalues by a new technique. Application to sextic anharmonic potentials

Abstract: A new technique to obtain analytic approximant for eigenvalues is presented here by a simultaneous use of power series and asymptotic expansions is presented. The analytic approximation here obtained is like a bridge to both expansions: rational functions, as Pade, are used, combined with elementary functions are used. Improvement to previous methods as multipoint quasirational approximation, MPQA, are also developed. The application of the method is done in detail for the 1-D Schrodinger equation with anharmonic sextic potential of the form V ( x ) = x 2 + λ x 6 and both ground state and the first excited state of the anharmonic oscillator.

Explicit Solution of Elastostatic Boundary Value Problems for the Elastic Circle with Voids

Abstract: We solve the static two-dimensional boundary value problems for an elastic porous circle with voids. Special representations of a general solution of a system of differential equations are constructed via elementary functions which make it possible to reduce the initial system of equations to equations of simple structure and facilitate the solution of the initial problems. Solutions are written explicitly in the form of absolutely and uniformly converging series. The question pertaining to the uniqueness of regular solutions of the considered problems is investigated.

Three-dimensional steady-state general solution for transversely isotropic hygrothermopiezoelectric media and its application in fracture

Abstract: The potential theory method is utilized to derive the steady-state, general solution for three-dimensional (3D) transversely isotropic, hygrothermopiezoelectric media in the present paper. Two displacement functions are introduced to simplify the governing equations. Employing the differential operator theory and superposition principle, all physical quantities can be expressed in terms of two functions, one satisfies a quasi-harmonic equation and the other satisfies a tenth-order partial differential equation. The obtained general solutions are in a very simple form and convenient to use in boundary value problems. As one example, the 3D fundamental solutions are presented for a steady point moisture source combined with a steady point heat source in the interior of an infinite, transversely isotropic, hygrothermopiezoelectric body. As another example, a flat crack embedded in an infinite, hygrothermopiezoelectric medium is investigated subjected to symmetric mechanical, electric, moisture and temperature loads on the crack faces. Specifically, for a penny-shaped crack under uniform combined loads, complete and exact solutions are given in terms of elementary functions, which serve as a benchmark for different kinds of numerical codes and approximate solutions.

Low-Dimensional and Multi-Dimensional Pendulums in Nonconservative Fields. Part 1

Abstract: In this review, we discuss new cases of integrable systems on the tangent bundles of finitedimensional spheres. Such systems appear in the dynamics of multidimensional rigid bodies in nonconservative fields. These problems are described by systems with variable dissipation with zero mean. We found several new cases of integrability of equations of motion in terms of transcendental functions (in the sense of the classification of singularities) that can be expressed as finite combinations of elementary functions.

Certain results related to the N-transform of a certain class of functions and differential operators

Abstract: In this paper, we aim to investigate q-analogues of the natural transform on various elementary functions of special type. We obtain results associated with classes of q-convolution products, Heaviside functions, q-exponential functions, q-hyperbolic functions and q-trigonometric functions as well. Further, we give definitions and derive results involving some q-differential operators.

Discrete inverse Sumudu transform application to Whittaker and Zettl equations

Abstract: In this research article, the Discrete Inverse Sumudu Transform (DIST) multiple shifting properties are used to design a methodology for solving ordinary differential equations. We say ”Discrete” because it acts on the Taylor or Mclaurin series of the function when any. The algorithm applied to solve the Whittaker and Zettl equations and get their exact solutions. A DIST Table for elementary functions is provided.

Low-Dimensional and Multi-Dimensional Pendulums in Nonconservative Fields. Part 2

Abstract: In this review, we discuss new cases of integrable systems on the tangent bundles of finite-dimensional spheres. Such systems appear in the dynamics of multidimensional rigid bodies in non-conservative fields. These problems are described by systems with variable dissipation with zero mean. We found several new cases of integrability of equations of motion in terms of transcendental functions (in the sense of the classification of singularities) that can be expressed as finite combinations of elementary functions.

Hardware Implementation of Basic Arithmetics and Elementary Functions for Unum Computing

Abstract: The universal number format (unum) is a recently proposed floating point format that solves rounding issues with interval arithmetic and provides variable exponent and mantissa widths. Only few approaches for unum hardware implementations have been published so far. In this paper, we propose architectures for comparisons and basic arithmetic operations, as well as for elementary functions using the CORDIC algorithm with unums. The designs are synthesized and evaluated for a 65 nm process and an Artix-7 FPGA.

A simple approximation for the modified Bessel function of zero order I 0 (x)

Abstract: New efficient analytic approximations have been found for the modified zero-order Bessel functions I 0(x). The method, used herein, improves prior techniques, such as multipoint quasi-rational approximations, MPQA. The present work is also an improvement of previous works, in the sense that the form of the approach is more efficient, that is, a smaller relative error is obtained using a lower number of parameters. As in the Pade method rational functions are used with coefficients determined from the powers series, but now also asymptotic expansions are also used simultaneously with that series, and consequently the rational functions have to be combined with elementary functions, in such a way, that the approach is a bridge between both expansions. The approximation now found is a rational function combined with a hyperbolic function. Only three parameters are needed to obtain a relative error smaller than 0.6 percent.

Morphogenesis of the Zeta Function in the Critical Strip by Computational Approach

Abstract: This article proposes a morphogenesis interpretation of the zeta function by computational approach by relying on numerical approximation formulae between the terms and the partial sums of the series, divergent in the critical strip. The goal is to exhibit structuring properties of the partial sums of the raw series by highlighting their morphogenesis, thanks to the elementary functions constituting the terms of the real and imaginary parts of the series, namely the logarithmic, cosine, sine, and power functions. Two essential indices of these sums appear: the index of no return of the vagrancy and the index of smothering of the function before the resumption of amplification of its divergence when the index tends towards infinity. The method consists of calculating, displaying graphically in 2D and 3D, and correlating, according to the index, the angles, the terms and the partial sums, in three nested domains: the critical strip, the critical line, and the set of non-trivial zeros on this line. Characteristics and approximation formulae are thus identified for the three domains. These formulae make it possible to grasp the morphogenetic foundations of the Riemann hypothesis (RH) and sketch the architecture of a more formal proof.

On Schrödinger Equations Equivalent to Constant Coefficient Equations

Abstract: This paper shows that the solution of some classes of Schrodinger equations may be performed in terms of the solution of equations of constant coefficients. In this context, it has been possible to generate new exactly solvable potentials and to show that the Schrodinger equation for some well known potentials may also be solved in terms of elementary functions.

On Exact Multidimensional Solutions of a Nonlinear System of Reaction–Diffusion Equations

Abstract: We study a nonlinear reaction–diffusion system modeled by a system of two parabolic-type equations with power-law nonlinearities. Such systems describe the processes of nonlinear diffusion in reacting two-component media. We construct multiparameter families of exact solutions and distinguish the cases of blow-up solutions and exact solutions periodic in time and anisotropic in spatial variables that can be represented in elementary functions.

Calculation and Analysis of the Pulse Response of Spatially Non-Invariant Projection Systems

Abstract: Characteristics of spatially non-invariant telecentric projection systems, which are widely used in practice, are considered within the framework of wave optics. In the class of the Fresnel functions, the pulse response of the system is precisely calculated for various values of the projection objective and filter apertures. It is found that the response consists of two components, which determine the invariant and non-invariant properties of the system, respectively. Based on the approximation of the Fresnel function by elementary functions proposed previously by the author, an analytical expression for the pulse response is derived for the first time, and the response behavior is studied for various relationships of the objective and filter apertures. The correctness of choosing the parameters of the known quasi-invariant optical systems is analyzed. Recommendations on choosing the filter aperture are given to improve their spatially invariant characteristics. In contrast to available optical and geometrical methods, the proposed approach allows one to obtain reliable information about the character of wave field transformations in the considered systems.

Uniform convergent expansions of the Gauss hypergeometric function in terms of elementary functions

Abstract: We consider the hypergeometric function 2F1(a,b;c;z) for z∈C∖[1,∞). For ℜa≥0, we derive a convergent expansion of 2F1(a,b;c;z) in terms of the function (1−z)−a and of rational functions of z that i...