Terms Of Power Series Research Articles

Use of rational approximations for phonon density of states in evaluation of neutron thermal scattering law

We propose and explore a new analytical formulation for evaluation of Thermal Scattering Law (TSL) using continuous phonon spectra for neutron inelastic scattering. It involves approximating Phonon Density of States (PDOS) in terms of rational approximations in energy transfer. Debye Waller coefficient calculated with these forms are in reasonable agreement with LEAPR calculated values. TSL can be expanded as a power series, each term of which can be integrated analytically. The method leads to TSL being a series of exponential cosine integrals which again have closed rational forms in energy transfer. Their shapes are analogous to asymmetric resonance line shapes and show maxima for certain values of energy transfer which could correspond to phonon emission or absorption. The series is shown to be convergent and easy to compute as it does not involve any numerical integration. The approach is applied to Beryllium metal and Graphite which have vastly different PDOS shapes, and is shown to be capable of giving most of the trends shown by TSL and cross section results from NJOY nuclear code processing system, even with smaller number of power series terms. Since TSL can be calculated for both negative and positive energy transfer with equal ease, one can easily check for the validity of the approximation obeying Principle of Detailed Balance (PDB). The results overall have shown promise of our formulation in making a case for rapid calculations of TSL and also for using the rational forms in energy transfer for analysis of experimental/calculated PDOS data.

A Force Function Formula for Solutions of Nonlinear Weakly Singular Volterra Integral Equations

In this paper, we examine the nonlinear Weakly Singular Volterra Integral Equation(WSVIE), $u(x)=f(x)+\int_{0}^{x}\frac{t^{\mu-1}}{x^\mu}[u(t)]^\beta dt$. AL-Jawary and Shehan used Daftardar-Jafari Method(DJM) and solved the above integral equation for the investigation parameter $\mu>1$ using specific force functions with $\mu$ and $\beta$ values and obtained unique solutions. We have discovered a force function $f(x)=x^{k_1}-{\frac{x^{\gamma k_1}}{\gamma k_1+\mu}}$, that allows the introduction of noise terms phenomena discovered by Wazwaz; that cancel out the terms of the power series in the successive solution terms $u_m$, $m=0,1,2,...,n$: we thus obtain a maximum finite power series terms for each solution term called truncation point and denoted by $x^{g(n)}$. Such that the integral solution can be written as $u(x)=u_0+\sum_{m=1}^{n}u_m$, where $n$ is finite. Simplifying the solution terms we get the unique solution $u(x)=x^{k_1}$, irrespective of $n-$value in the truncation point. We discovered a formula relation between the last solution term $u_n$ and the truncation point as $u_n=a_nx^{g(n)}$. Our results confirm the results of the two solution examples of AL-Jawary and Shehan for the investigation parameter $\mu>1$. We extend the parameter range to include $\mu>1$ and $0<\mu\leq 1$ for our solution. In addition, for any chosen rational parameter $k_1$, the solution $u(x)=x^{k_1}$ is extrapolated to be valid for all integer parameter values $\beta\geq 2$, and positive rational parameter value $\mu>0$ and for any finite value of $n\geq2$.

A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine

AbstractIn this study, by virtue of a derivative formula for the ratio of two differentiable functions and with aid of a monotonicity rule, the authors expand a logarithmic expression involving the cosine function into the Maclaurin power series in terms of specific determinants and prove a decreasing property of the ratio of two logarithmic expressions containing the cosine function.

MODELLING THE EVOLUTION OF THE TWO-PLANETARY THREE-BODY SYSTEM OF VARIABLE MASSES

A classical non-stationary three-body problem with two bodies of variable mass moving around the third body on quasi-periodic orbits is considered. In addition to the Newtonian gravitational attraction, the bodies are acted on by the reactive forces arising due to anisotropic variation of the masses. We show that Newtonian’s formalism may be generalized to the case of variable masses and equations of motion are derived in terms of the osculating elements of aperiodic motion on quasiconic sections. As equations of motion are not integrable the perturbative method is applied with the perturbing forces expanded into power series in terms of eccentricities and inclinations which are assumed to be small. Averaging these equations over the mean longitudes of the bodies in the absence of a mean-motion resonances, we obtain the differential equations describing the evolution of orbital parameters over long period of time. We solve the evolution equations numerically and demonstrate that the mass change modify essentially the system evolution.

A Series Expansion of a Logarithmic Expression and a Decreasing Property of the Ratio of Two Logarithmic Expressions Containing Sine

In the paper, by virtue of a derivative formula for the ratio of two differentiable functions and with the help of a monotonicity rule, the authors expand a logarithmic expression involving the sine function into the Maclaurin power series in terms of specific determinants and prove a decreasing property of the ratio of two logarithmic expressions containing the sine function. These results are interesting purely in pure mathematics.

Statistical Convergence of Spliced Sequences in Terms of Power Series on Topological Spaces

In the present paper, $P-$distributional convergence which is defined by power series method has been introduced. We give equivalent expressions for $P-$distributional convergence of spliced sequences. Moreover, convergence of a bounded $\infty$-spliced sequence via power series method is represented in terms of Bochner integral in Banach spaces.

An unsteady flow through porous media leads to a Newtonian fluid presence of CNTS and suction/injection

An action of CNTs flow access an unstable motion of a Newtonian fluid past a stagnation point on a stretched sheet using a penetrable medium is investigated in this paper. Once the fluid commences circulating with the flow of time, it flows. Flow is influenced by heat exchange with the surroundings. The implicitly limited component method is employed to track an unstable system of non-dimensional PDE with an associated boundary layer. Approaches, along with the relevant barrier circumstances, are acquired analytically. An influence of mass transpiration, volume fraction, and thermal radiation is investigated on Newtonian fluid flow using the penetrable medium. Heat transmission in a liquid with electrical conductivity, including heat source /sink across a continually stretched /shrinking surface with a Biot number. The importance of the present exertion is the stream of carbon nanotubes owing to Darcy porous media using linear heat radiation and an analytical approach. The nanoparticles provide important properties in enhancing the heat exchange procedure as well as thoroughly achieved applications in industry. It is discovered that the basic resemblance models accept two-phase for both extending /contracting sheets. In presence of calculation upon momentum with temperature jump conditions is accessible plots aimed at varying predictions of certain physical properties. Analyzing the properties of Darcy porous media of heat transport in the presence of radiation from a flat surface, the problem is solved analytically using incomplete gamma functions. Depending on the kind of boundary heating, the analytical equation for temperature is obtained in terms of power series.

Some identities related to degenerate r-Bell and degenerate Fubini polynomials

Many works have been done in recent years as to explorations for degenerate versions of some special polynomials and numbers, which began with the pioneering work of Carlitz on the degenerate Bernoulli and degenerate Euler polynomials. This paper is focused on the study of some properties, recurrence relations and identities related to the degenerate r-Bell polynomials, the two variable degenerate Fubini polynomials and the degenerate r-Stirling numbers of the second kind. Especially, we express the power series in terms of the degenerate r-Bell polynomials, of the degenerate r-Stirling numbers of the second kind and of the degenerate Fubini polynomials.

Density-functional theory vs density-functional fits: The best of both.

In a recent paper [A. D. Becke, J. Chem. Phys. 156, 214101 (2022)], we compared two Kohn-Sham density functionals based on physical modeling and theory with the best density-functional power series fits in the literature. With only a handful of physically motivated pre-factors, our functionals matched, and even slightly exceeded, the performance of the best power-series functionals on the general main group thermochemistry, kinetics, and noncovalent interactions (GMTKN55) chemical database of Goerigk et al. [Phys. Chem. Chem. Phys. 19, 32184 (2017)]. This begs the question: how much can their performance be improved by adding power-series terms of our own? We address this question in the present work. First, we describe a series expansion variable that we believe contains more local physics than any other variable considered to date. Then we undertake modest, one-dimensional fits to the GMTKN55 data with our theory-based functional corrected by power-series exchange and dynamical correlation terms. We settle on 12 power-series terms (plus six parent terms) and achieve the lowest GMTKN55 "WTMAD2" error yet reported, by a substantial margin, for a hybrid Kohn-Sham density functional. The new functional is called "B22plus."

On the backward and forward error of approximations of analytic functions and applications to the computation of matrix functions

In this paper we give a new formula to write the forward error of Taylor approximations of analytical functions in terms of the backward error of those approximations, considering exact arithmetic in both errors. Using this formula, a method to compute a backward error given by the power series centered in the same expansion point as the Taylor approximation is provided. The application of this method for Padé approximations is also shown. Based on the previous method, a MATLAB implementation for computing the first power series terms of the backward error for Taylor and Padé approximations of a given analytical function is provided, and examples of its use are given. Applications to the computation of matrix functions are given that overcome limitations of other backward error analyses use inverse compositional functions in the literature.

On the calculation of the discrete spectra of one‐dimensional Dirac operators

In this work, we consider the one‐dimensional Dirac operator where is Pauli's matrix, is a ‐matrix representing a regular potential that includes the electrostatic and scalar interactions as well as the anomalous magnetic momentum, is a singular potential consisting of delta distributions ( ) are ‐matrices representing the strengths of Dirac deltas, and is a two‐spinor. We associate to the operator an unbounded in symmetric operator denoted by , where is the support of singular potential . The operator includes only the regular potential together with certain interaction conditions at each point . The paper presents a method for determining the discrete spectrum of the operator for arbitrary potential whose entries are given by ‐functions. The eigenvalues of the operator are the zeros of a dispersion equation , where the characteristic function is determined explicitly in terms of power series involving the spectral parameter . The construction of the characteristic function from a set of monodromy matrices and the interaction conditions is presented in the paper. Moreover, its power series representation leads to an efficient numerical method for calculating the eigenvalues of the Dirac operator from the zeros of certain approximate function which is obtained by truncating the series up to a finite number of terms. Several examples show the applicability and accuracy of the numerical method.

Non-planar elliptic vertex

We consider the problem of obtaining higher order in regularization parameter ε analytical results for master integrals with elliptics. The two commonly employed methods are provided by the use of differential equations and direct integration of parametric representations in terms of iterated integrals. Taking non-planar elliptic vertex as an example we show that in addition to two mentioned methods one can use analytical solution of differential equations in terms of power series. Moreover, in the last case it is possible to obtain the exact in ε results.

An Analytical Model for Molecular Communication Over a Non-Linear Reaction-Diffusion Medium

One of the main challenges in diffusion-based molecular communication is dealing with the non-linearity of reaction-diffusion chemical equations. While numerical methods can be used to solve these equations, a change in the input signals or the parameters of the medium requires one to redo the simulations. This makes it difficult to design modulation schemes and practically impossible to prove the optimality of a given transmission strategy. In this paper, we provide an analytical technique for modeling the non-linearity of chemical reaction equations based on the perturbation method. The perturbation method expresses the solution in terms of an infinite power series. An approximate solution can be found by keeping the leading terms of the power series. The approximate solution is shown to track the true solution if either the simulation time interval or the reaction rate is sufficiently small. Approximate solutions for long time intervals are also discussed. An illustrative example is given. For this example, it is shown that when the reaction rate (or the total time interval) is low, instead of using a continuous release waveform, it is optimal for the transmitters to release molecules at two time instances.

Conjugated Mass Transfer of CO2 Absorption through Concentric Circular Gas–Liquid Membrane Contactors

A new design of gas absorption that winds the permeable membrane onto an inner concentric tube to conduct a concentric circular gas–liquid membrane module has been studied theoretically in the fully developed region. An analytical formulation, referred to as conjugated Graetz problems, is developed to predict the concentration distribution and Sherwood numbers for the absorbent fluid flowing in the shell side and CO2/N2 gas mixture flowing in the tube side under various designs and operating parameters. The analytical solutions to the CO2 absorption efficiency were developed by using a two-dimensional mathematical modeling, and the resultant conjugated partial differential equations were solved analytically using the method of separation variables and eigen-function expansion in terms of power series. The predictions of CO2 absorption rate by using Monoethanolamide (MEA) solution in concentric circular membrane contactors under both concurrent- and countercurrent-flow operations are developed theoretically and confirmed with the experimental results. Consistency in both a good qualitative and quantitative sense is achieved between the theoretical predictions and experimental results. The advantage of the present mathematical treatment provides a concise expression for the chemical absorption of CO2 by MEA solution to calculate the absorption rate, absorption efficiency, and average Sherwood number. The concentration profiles with the mass-transfer Graetz number, inlet CO2 concentration, and both gas feed and absorbent flow rates are also emphasized. Both theoretical predictions and experimental results show that the device performance of the countercurrent-flow operation is better than that of the concurrent-flow device operation. The availability of such simplified expressions of the absorption rate and averaged Sherwood as developed directly from the analytical solutions is the value of the present study.

A finite element approach for the static and vibration analyses of functionally graded material viscoelastic sandwich beams with nonlinear material behavior

A beam finite element model is proposed for the static and free vibration analyses of FGM sandwich beams with viscoelastic nonlinear material behavior. In the analysis, zigzag theory is adopted for the displacement fields. The damping results from the shear properties of the viscoelastic layer and its low stiffness. Timoshenko 1st order and Reddy’s higher order shear models are implemented for static and vibration behaviors. Various viscoelastic frequency-dependent laws are considered. The resulting stiffness matrix is nonlinear and is frequency dependent. Solutions are possible according to a powerful asymptotic method combined with recent method for the power series terms. The efficiency of the present model is proven through simulations using 3D volume elements of Abaqus code, where new module for viscoelastic FGM materials is now available. The effects of power law index and boundary conditions on static, vibration and the damping properties of the viscoelastic sandwich FGM beam are investigated successfully. It is shown that the beam behavior is very sensitive on the loss factor. In vibration, the damping properties are nonlinearly power law index dependent. The boundary conditions have an incidence on the vibration modes. The cantilever case is particular and interesting that needs an optimization process.

Hopf Algebra Structure of Generalized Quasi-Symmetric Functions in Partially Commutative Variables

We introduce a coloured generalization $\mathrm{NSym}_A$ of the Hopf algebra of non-commutative symmetric functions described as a subalgebra of the of rooted ordered coloured trees Hopf algebra. Its natural basis can be identified with the set of sentences over alphabet $A$ (the set of colours). We present also its graded dual algebra $\mathrm{QSym}_A$ of coloured quasi-symmetric functions together with its realization in terms of power series in partially commutative variables. We provide formulas expressing multiplication, comultiplication and the antipode for these Hopf algebras in various bases — the corresponding generalizations of the complete homogeneous, elementary, ribbon Schur and power sum bases of $\mathrm{NSym}$, and the monomial and fundamental bases of $\mathrm{QSym}$. We study also certain distinguished series of trees in the setting of restricted duals to Hopf algebras.

An asymptotic model based on matching far and near field expansions for thin gratings problems

In this paper, we devise an asymptotic model for calculating electromagnetic diffraction and absorption in planar multilayered structures with a shallow surface-relief grating. Far from the grating, we assume that the solution can be written as a power series in terms of the grating thickness δ, the coefficients of this expansion being smooth up to the grating. However, the expansion approximates the solution only sufficiently far from the grating (far field approximation). Near the grating, we assume that there exists another expansion in powers of δ (near field approximation). Moreover, there is an overlapping zone where both expansion are valid. The proposed model is based on matching the two expansions on this overlapping domain. Then, by truncating terms of order δ2 or higher, we obtain explicitly the equations satisfied by the lowest order terms in the power series. Under appropriate assumptions, we prove second order convergence of the error with respect to δ. Finally, an alternative form, more convenient for implementation, is derived and discretized with finite elements to perform some numerical tests.

Study of Navier-Stokes equation by using Iterative Laplace Transform Method (ILTM) involving Caputo- Fabrizio fractional operator

This article arrangement with N-S equation containing the Caputo-Fabrizio differential operator of fractional order. The Iterative Laplace Transform Method (ILTM) has been applied to found numerical solution of time-fractional N-S equation in a tube with unsteady fluid flow in the Caputo-Fabrizio sense. The ILTM is an elegant coupling of transform of the Laplace and new Iterative method (NIM). This scheme provides numerical solution in the terms of power series with easily computable terms. It is observed that the solutions of N-S equations obtained by the ILTM rapidly convergent to exact solutions.

Linear and Weakly Nonlinear Analyses of Magneto-Convection in a Sparsely Packed Porous Medium under Gravity Modulation

This paper deals with the linear stability analysis and weakly non-linear analysis of Magneto-convection in a sparsely packed porous medium with constant vertical Magnetic field and gravity modulation. A linear stability analysis reported here and shows that the gravity modulation has significant effect on the stability limits of the system. The gravity modulation is know to have effect and is treated by a perturbation expansion in powers of the amplitude of modulation. The shift in the critical Rayleigh number is evaluated and depends on the prandtl number and frequency of modulation, using the Venezian method. It is also shown that the onset of convection can be advance or delay by the regulation of various parameters. Weakly nonlinear analysis is performed based on the method of power series, where the disturbance is expressed in terms of power series. A nonlinear Ginzburg-Landau equation to investigate the three different types of gravity modulation on heat transfer is derived as part of this work. Heat transfer have been shown to depend on Nusselt number, further the effect of different types of parameter on heat transport have been studied graphically. Nusselt number graph is also shown for different parameter and explain in detail. The effect of magnetic prandtl number and Chandrasekhar number are stabilize the system. The control of convection is a major issue in systems with fluids as a working media. This is all the more difficult if the fluid system is housed in a porous medium. The paper presents three mechanisms of controlling onset of convection and thereby the heat transfer in such fluid systems. In order the modulation effect is effective in its role, we have considered the system to be a fluid-saturated porous media.

On the applicability of the perturbation method for the random viscous Burgers’ equation

A classical problem in uncertainty quantification is the analysis of the steady-state solution to the viscous Burgers’ equation subject to a small random perturbation at the boundary. The perturbation method, introduced in the sixties by Crandall to study nonlinear vibrations, expands the stochastic solution as a mean square power series in terms of the centered perturbation. We show that the perturbation method does work to quantify the uncertainty for the location of the transition layer of the steady-state solution.