Series Expansion Of Functions Research Articles

The sequential power flow in complex plane for solving the VSC-MTDC hybrid AC/DC transmission grids

This paper presents new methodologies and modeling aimed at studying interwoven AC and DC subsystems of a power system in the complex plane. It is well known that the development of any former power flow application in real domain addressed to embedding FACTS devices is preceded by an arduous algebra task. To overcome this difficulty, a novel power flow solution method is proposed in this work. Thanks to the Wirtinger calculus, a Newton–Raphson method based on Taylor series expansions of nonlinear functions of complex variables and their complex conjugates is developed. Thus, the voltage-source-converter multi-terminal direct current is formulated in complex plane without any loss of accurateness and is termed as VSC-MTDC. Its performance is assessed through IEEE-test systems interconnected across a DC network prone to several scenarios, e.g., topology, loading magnitude and interchanging of active power.

Two identities and closed-form formulas for the Bernoulli numbers in terms of central factorial numbers of the second kind

Abstract In this article, the authors present two identities involving products of the Bernoulli numbers, provide four alternative proofs for these two identities, derive two closed-form formulas for the Bernoulli numbers in terms of central factorial numbers of the second kind, and supply simple proofs of series expansions of (hyperbolic) cosecant and cotangent functions.

Propagation characteristics of an elastic bar coupled with a discrete snap-through element

Wave propagation characteristics of an elastic bar coupled at one end with a single degree of freedom, bi-stable, essentially nonlinear snap-through element are considered. The free end of the bar is subjected to sinusoidal excitations. A novel approach based on multiple time scales and harmonic balance method has been proposed to analytically investigate the reflected wave from the nonlinear interface and the dynamic response of the snap-through element. A unified approach to the non-dimensional representation of the governing equations of motion, boundary conditions and system parameters, which is consistent across all the externally applied excitation frequencies and excitation amplitudes, has been developed. Through Taylor series expansion of the non-autonomous forcing functions arising in the governing differential equations and natural boundary condition about an initial stable configuration of the system and the proposed asymptotic method, approximate closed-form analytical solutions have been derived for sufficiently small amplitudes of the excitation pulse. Numerical results obtained through a finite difference algorithm validate the asymptotic model for the same small amplitudes of the excitation pulse. A stability analysis has been subsequently performed for the discrete snap-through element by using the extended Floquet theory for sufficiently large amplitudes of the excitation pulse by approximating the displacement at the nonlinear interface as a sinusoidal function of time, and the Mathieu plot of the excitation frequency vs the excitation amplitude showing the stable and unstable regions for the motion of the snap-through element has been generated. The expressions derived here give the most comprehensive and consistent description of the wave propagation characteristics and the motion of the snap-through element, which can be directly used in finite difference analysis over a wide range of parameter values of the excitation pulse.

New solutions for interfacial capillary waves of permanent form

Progressive capillary waves on the interface between two homogeneous fluids confined in a channel with rigid walls parallel to the undisturbed interface are investigated. This problem is formulated as a system of integrodifferential equations that can be solved numerically via a boundary integral equation method coupled with series expansions of the unknown functions. With this highly accurate scheme and numerical continuation, we explore the global bifurcation of periodic travelling waves. It is found that there are two types of limiting profile, self-intersecting and boundary-touching, which appear either along a primary branch bifurcating from infinitesimal periodic waves or on an isolated branch existing above a certain finite amplitude. For particular sets of parameters, these two types of bifurcation curves can intersect, which can be viewed as a secondary bifurcation phenomenon occurring on the primary branch. Based on asymptotic and numerical analyses of the almost limiting waves, it is found that the boundary-touching solutions feature a circular geometry, i.e. the interface is pieced together by circular arcs of the same radius. A theoretical investigation yields the necessary conditions for the existence of these extreme waves, whereby we can predict the limiting configurations for most parameter sets. The comparisons between theoretical predictions and numerical results show good agreement.

Differentiation identities for hypergeometric functions

It is well-known that differentiation of hypergeometric function multiplied by a certain power function yields another hypergeometric function with a different set of parameters. Such differentiation identities for hypergeometric functions have been used widely in various fields of applied mathematics and natural sciences. In this expository note, we provide a simple proof of the differentiation identities, which is based only on the definition of the coefficients for the power series expansion of the hypergeometric functions.

On linear spline algorithms of computerized tomography in the space of n-orbits

Abstract We fix a continuous linear operator A : H → M {A:H\rightarrow M} acting between the Hilbert spaces H and M that admits a singular value decomposition (SVD). We consider the following ill-posed problem: for an element f ∈ M {f\in M} , find u ∈ H {u\in H} such that A ⁢ u = f {Au=f} and a generalized solution in the sense of Moore–Penrose u is sought that satisfies the equation A * ⁢ A ⁢ u = A * ⁢ f {A^{*}Au=A^{*}f} . Moreover, we fix an integer n ∈ ℕ 0 = { 0 , 1 , 2 , … } {n\in\mathbb{N}_{0}=\{0,1,2,\dots\}} and transfer this equation to a special Hilbert space D ⁢ ( ( A * ⁢ A ) - n ) {D((A^{*}A)^{-n})} of n-orbits. For an approximate solution of this equation in the case of a nonadaptive information on the right-hand side f, a linear spline algorithm is constructed. The specificity of the considered norm is that the approximate solution is the truncated singular value decomposition (TSVD) and does not depend on n. In the case n = 0 {n=0} , the space D ⁢ ( ( A * ⁢ A ) - n ) {D((A^{*}A)^{-n})} coincides with H and we obtain the results for the latter space. In the limiting case of the Fréchet–Hilbert space of all orbits D ⁢ ( ( A * ⁢ A ) - ∞ ) {D((A^{*}A)^{-\infty})} , the equation A * ⁢ A ⁢ u = A * ⁢ f {A^{*}Au=A^{*}f} becomes well-posed and was considered in [D. Ugulava and D. Zarnadze, On a linear generalized central spline algorithm of computerized tomography, Proc. A. Razmadze Math. Inst. 168 2015, 129–148]. It is also noted that the space D ⁢ ( ( A * ⁢ A ) - ∞ ) {D((A^{*}A)^{-\infty})} is the projective limit of the sequence of Hilbert spaces { D ⁢ ( ( A * ⁢ A ) - n ) } {\{D((A^{*}A)^{-n})\}} . The application of the obtained results for the computerized tomography problem, i.e., for the inversion of the Radon transform ℜ {\mathfrak{R}} with the SVD of Louis [A. K. Louis, Orthogonal function series expansions and the null space of the Radon transform, SIAM J. Math. Anal. 15 1984, 3, 621–633] in the space D ( ( ℜ * ℜ ) - n ) ) {D((\mathfrak{R}^{*}\mathfrak{R})^{-n}))} is given.

Expansion of some well-known functions and classical polynomials in series of the generalized Laguerre polynomials.1

The generalized Laguerre polynomials form a complete set (orthogonal and normalized) in the space ** with respect to a certain weighting function because they are the Eigen functions of a second-order differential operator. Here we shall show how to expand some well-known classical polynomials such as the Legendre and the Hermite polynomials in series of generalized Laguerre polynomials. Since the generalized Laguerre orthogonal polynomials are set of polynomials that are mutually orthogonal to each other with respect to a measure of weighting function that is just the integrand of the gamma function, thus it grants us the guarantee of the ability to expand the first kind of Bessel functions and the gamma function in terms of the generalized Laguerre polynomials. The series expansion in terms of the generalized Laguerre polynomials can be achieved by following various approaches. For instance, the series expansion of the first kind Bessel functions is gained by the generalized hypergeometric function approach, whereas the series expansion of gamma function was obtained directly by the usual way that is by calling the orthogonality and orthonormality properties of the generalized Laguerre polynomials. Another powerful technique to gain the series expansion of the classical polynomials is the generating function approach as it has been followed here to obtain a series expansion of the Legendre and the Hermite polynomials in terms of the generalized Laguerre polynomials. The series expansion in series of the generalized Laguerre polynomials has a variety of applications in mathematics, physics, and engineering

On q-scale functions of spectrally negative Lévy processes

AbstractWe obtain series expansions of the q-scale functions of arbitrary spectrally negative Lévy processes, including processes with infinite jump activity, and use these to derive various new examples of explicit q-scale functions. Moreover, we study smoothness properties of the q-scale functions of spectrally negative Lévy processes with infinite jump activity. This complements previous results of Chan et al. (Prob. Theory Relat. Fields150, 2011) for spectrally negative Lévy processes with Gaussian component or bounded variation.

Winter (or [formula omitted]-shell) model at small and intermediate volumes

We consider Winter (or δ-shell) model at finite volume, describing a small resonating cavity weakly coupled to a large one, for small and intermediate volumes (lengths). By defining N as the ratio of the length of the large cavity over the small one, we study the symmetric case N=1, in which the two cavities actually have the same length, as well as the cases N=2,3,4.By increasing N in the above range, the transition from a simple quantum oscillating system to a system having a resonance spectrum is investigated. We find that each resonant state is represented, at finite volume, by a cluster of states, each one resonating in a specific coupling region, centered around a state resonating at very small couplings.We derive high-energy expansions for the particle momenta in the above N cases, which (approximately) resum their perturbative series to all orders in the coupling among the cavities. These new expansions converge rather quickly with the order, provide, surprisingly, a uniform approximation in the coupling and also work, again surprisingly, at low energies.We construct a first resummation scheme having a clear physical picture, which is based on a function-series expansion, as well as a second scheme based on a recursion equation. The two schemes coincide at leading order, while they differ from next-to-leading order on. In particular, the recursive scheme realizes an approximate resummation of the function-series expansion generated within the first scheme.

Алгоритм синтеза пленочного аттенюатора с равномерным рассеянием мощности по его длине

The problem of synthesizing a film attenuator made by applying an absorbing material to a conductor is considered. Since the absorbing material has a parameter of maximum dissipated power per unit area, then when synthesizing an attenuator to attenuate high powers, the problem of exceeding the maximum dissipated power per unit area may arise. A sectional attenuator structure has been proposed, which is a set of sections having different attenuation to reduce the power dissipated per unit area. An algorithm for the synthesis of a sectional attenuator is proposed. The attenuator sections are rectangles of various widths. The attenuation distribution is performed using the Taylor series expansion of mathematical functions. The properties of the functions used to distribute the attenuation related to the problem under consideration were considered. The graphs of the dependence of power on the attenuator section are presented. Power distribution has been confirmed by simulation results. Theoretical calculations were confirmed by simulation results in the CST Microwave Studio software.

A quadratic Wiener path integral approximation for stochastic response determination of multi-degree-of-freedom nonlinear systems

A Wiener path integral (WPI) technique is developed for determining the stochastic response of multi-degree-of-freedom (MDOF) nonlinear systems. Specifically, the nonlinear system response joint transition probability density function (PDF) is expressed as a WPI over the space of paths satisfying the initial and final conditions in time. Next, a functional series expansion is considered for the WPI and a quadratic approximation is employed. Further, relying on a variational principle yields a functional optimization problem to be solved for the most probable path, which is used for determining approximately the joint response transition PDF. It is shown that compared to the standard (semiclassical) WPI solution approach, which accounts only for the most probable path, the quadratic approximation developed herein exhibits enhanced accuracy. This is due to the fact that fluctuations around the most probable path are also accounted for by considering a localized state-dependent factor in the calculation of the WPI. Furthermore, the PDF normalization step of the most probable path approach is bypassed, and thus, probabilities of rare events (e.g., failures) can be determined in a direct manner without the need for obtaining the complete joint response PDF first. The herein developed technique can be construed as an extension of earlier efforts in the literature to account for MDOF systems. Several numerical examples are considered for demonstrating the accuracy of the technique. These pertain to various dynamical systems exhibiting diverse nonlinear behaviors. Comparisons with pertinent Monte Carlo simulation data are included as well.

The Natural Approaches of Shafer-Fink Inequality for Inverse Sine Function

In this paper, we obtain some new natural approaches of Shafer-Fink inequality for arc sine function and the square of arc sine function by using the power series expansions of certain functions, which generalize and strengthen those in the existing literature.

A scaled wave finite element method for computing scalar wave radiation and scattering in exterior domains

The calculation of high-frequency wave radiation in exterior domains by finite element methods can lead to large computations. In this paper, it is shown that the solution in the exterior domain can be decomposed as a series expansion of functions with an analytical part made from the product of harmonic waves and polynomials in a scaled variable and a numerical part made of a finite element approximation vector. The solution of the radiation or scattering problem can be found by solving a sparse linear system which is set from the dynamic stiffness matrices of several scaled layers around the radiating body. These dynamic stiffness matrices are classical finite element matrices obtained from any finite element software. Moreover, accurate results can be obtained from a small number of terms in the series expansion. Several examples are given to estimate the efficiency of the proposed method.

Study of Differential Cross-Section and S-Matrix Using Volkov Function and Taylor Series Expansion for Elastic Scattering

This paper deals with S-matrix, born first approximation, amplitude, and differential cross-section (DCS), using Volkov function and Taylor series expansion in laser field, scattering. Equation (30) copes-with DCS and Equation (36) deals with S-matrix, with different parameters, moreover, both equations contain real and imaginary parts. The DCS increases with increasing angle and polarizabilities, constant with dipole distance for both emission and absorption of single-photon. The DCS for both emission and absorption is responded to low incidence energy (30 eV - 60 eV) and photon energy (15 eV) while at high energy only emission and absorption are responded for DCS. The DCS between absorption and emission of a photon with angle variation, dipole distance, and atomic polarizabilities was found 1.098 a.u.2 and at high incidence, energies were found 0.1 a.u.2.

Exponentials of general multivector in 3D Clifford algebras

Closed form expressions to calculate the exponential of a general multivector (MV) in Clifford geometric algebras (GAs) Clp;q are presented for n = p + q = 3. The obtained exponential formulas were applied to find exact GA trigonometric and hyperbolic functions of MV argument. We have verified that the presented exact formulas are in accord with series expansion of MV hyperbolic and trigonometric functions. The exponentials may be applied to solve GA differential equations, in signal and image processing, automatic control and robotics.

Математическое моделирование нестационарности биотехнологического получения молочной кислоты: устойчивость

The article presents the calculated ratios of indicators determining the stationary states of the lactic acid production process. Three technologies that are most often mentioned in scientific publications are identified: the technology of using strains of microorganisms to produce biomass is a technology that is extremely rarely used; the fairly common technology of using strains of microorganisms to produce lactic acid with the consumption of the main substrate (most often glucose); the promising technology of obtaining lactic acid using, in addition to the main substrate, a component that reproduces the main substrate in the synthesis process. For each technology, the equations of material balance for stationary and non-stationary conditions, a generalized differential equation for non-stationary conditions, and a characteristic equation are given. The formulas for estimating the coefficients of differential equations and the coefficients of the characteristic equation are also given. The equations for non-stationary conditions according to the last two technologies are based on the use of the Taylor series expansion of functions with the preservation of only the first terms of the expansion, i. e. deviations from stationarity in small. The characteristic equation is formed using the eigenvalues . The methodology for all three technologies is given, which allows us to assess the stability of the considered stationary state – the Hurwitz method. For all three technologies, numerical results are obtained for estimating the coefficients of the characteristic equations Pi. Tabular values of the coefficients are given, according to which stability estimates for the dilution rate of 0.1 h–1, 0.2 h–1, 0.3 h–1 are obtained using determinants according to the Hurwitz matrix. The results of numerical estimates for the stability of stationary states for all three technologies are presented. The estimates were based on the indicators of constants published in scientific studies.

Buckling of partially-compressed laminated composite plates

The web buckling is one failure mode of web crippling for thin-walled fiber-reinforced plastic (FRP) composite beams subjected to transverse concentrated or distributed compression load from both the top and bottom bearing plates, and the web between the top and bottom flanges can be modeled as a plate with the rotationally-restrained (RR) and partially-compressed loaded edges and the simply-supported (SS) unloaded edges. Explicit solution and universal analytical charts/formulas for buckling analysis of such RRSS composite plates partially compressed along the transverse direction are obtained. The unique Heaviside function and trigonometric series expansion are used to express the partial edge load with effective load length in the governing equilibrium equation. The explicit solution is established for the orthotropic plates subjected to concentrated point compression load in the middle; while the generic buckling analytical curves are developed for infinitely long laminated composite plates transversely compressed partially at the middle of edges as well as near the corners. The accuracy of the present method is validated with available simple analytical solution, numerical finite element method and experimental data. The effect of rotational restraint stiffness, plate aspect ratio, the location and length of partial edge compression load on buckling behavior of laminated composite plates is investigated in the parametric study. A practical guideline for laminated composite plates with the elastic rotational restraints along the loaded edges and partially compressed along the transverse direction is proposed, and it can be further applied in discrete plate analysis to predict the web buckling strength (one failure mode in web crippling) of FRP composite structural shapes under transverse loading at the locations of load bearing or supports.

Boundary integral formulation of the standard eigenvalue problem for the 2-D Helmholtz equation

In this paper, a boundary integral formulation is presented for obtaining the standard eigenvalue problem for the two-dimensional (2-D) Helmholtz equation. The formulation is derived by using the series expansions of zero-order Bessel functions for the fundamental solution to the Helmholtz equation. The proposed approach leads to a series of new fundamental functions which are independent of the wave number k of the Helmholtz equation. The coefficient matrix of the resulting homogeneous system of boundary element equations is of the form of a polynomial matrix in k which allows a much faster search for the eigenvalues by scanning k over an interval of interest or the standard eigenvalue problem to be formulated for directly solving for the eigenvalues without resort to iterative methods. The proposed technique was used to solve some known problems with available analytical solutions: 2-D domains with circular and rectangular geometries under Dirichlet and/or Neumann boundary conditions. The outcomes demonstrate that the proposed approach is computationally efficient and highly accurate.

Elastostatic problems of crack lying along the interface of an elliptical inclusion embedded in elastic media

The problems of an elliptical arc crack lying along the interface of an anisotropic inclusion are considered in this article. By means of the complex series expansion of the stress functions, the related coefficients in the stress functions are obtained using the continuity condition for the stresses at the imperfect interface and using equilibrium condition based on relationships of stresses and displacements at the imperfect interface between matrix and inclusion. The displacements and stresses at the interface in the matrix and in the inclusion can be calculated. The imperfection at the interface between matrix and inclusion is expressed by the Fourier series, which can be used to represent the crack lying along the interface and the perfect interface. The method presented in this article can be used to analyze the problems of crack, inclusion, and hole in the anisotropic materials due to separate or combined effect of eigenstrains and far-field tension.

Synthesis of the digital regulator of the main contour of the three-circuit system of the linear electric drive of the working body of the mechanism of onboard aviation equipment

Goal The purpose of the article is to further develop analytical methods for calculating and synthesizing power electronics systems with deep pulse width modulation (PWM). A three-circuit linear electric drive system for positioning the working body of the mechanism of onboard aircraft equipment, in which the linear electric motor is controlled from a pulse width converter (PWC), is considered. The power converter is included in the current loop. It has a noticeable effect on the level of current ripple, travel speed and positioning accuracy of the operating mechanism of a linear electric drive. Methodology. To analyze the processes in the current loop, a discrete transfer function of a pulse-width converter for PWM in the final zone and «in the large» is obtained on the basis of the statistical linearization of the modulation characteristics of the multi-loop PWM model. The modulation characteristic of each circuit of the model is obtained as a result of the Fourier series expansion in Walsh functions of the output voltage of the PWM during the PWM process. Statistical linearization of modulation characteristics is performed based on Hermite polynomials. Results. During the analysis, discrete transfer functions of closed current loops, velocity and open loop position were obtained, for which a digital controller was synthesized in the form of a recursive filter. Originality. The parameters of the regulator links are found, which make it possible to complete the transient process in four PWC switching periods with an overshoot of no more than 6 %. The analysis of the speed-optimized positioning process of a linear electric drive based on the LED AT 605TU motor is carried out. Practical significance. The purpose of the analysis was to establish the relationship between the switching period of the PWM and the value of the uncompensated constant, at which the pulsations of the positioning process are minimal while ensuring the minimum overshoot and maximum speed. It was found that the specified requirements are satisfied by the ratio between the switching period, PWC and uncompensated constant in the range of one or two.