Convergence Of Series Expansion Research Articles

The shape of an axisymmetric meniscus in a static liquid pool: effective implementation of the Euler transformation

Abstract We examine the classical problem of the height of a static liquid interface that forms on the outside of a solid vertical cylinder in an unbounded stagnant pool exposed to air. Gravitational and surface tension forces compete to affect the interface shape as characterized by the Bond number. Here, we provide a convergent power series solution for interface shapes that rise above or fall below the horizontal pool as a function of contact angle and Bond number. We find that the power series solution expressed in terms of the radial distance from the wall is divergent, and thus rewrite the divergent series as a new power series expressed as powers of an Euler transformed variable; this series is modified to match the large distance asymptotic behaviour of the meniscus. The Euler transformation maps non-physical singularities to locations that do not restrict series convergence in the physical domain, while the asymptotic modification increases the rate of convergence of the series overall. We demonstrate that when the divergent series coefficients are used to implement the Euler transformation, finite precision errors are incurred, even for a relatively small number of terms. To avoid such errors, the independent variable in the governing differential equation is changed to that of the Euler transform, and the power series is developed directly without using the divergent series. The resulting power series solution is validated by comparison with a numerical solution of the interface shape and the small Bond number matched asymptotic solution for the height of the interface along the cylinder developed by Lo (1983, J. Fluid Mech., 132, p.65-78). The convergent power series expansion has the ability to exceed the accuracy of the matched asymptotic solution for any Bond number given enough terms, and the recursive nature of the solution makes it straightforward to implement.

A new HEAP-inspired method for embedded power flows and path-related holomorphic properties

Power flow problems can be robustly solved by holomorphic embedding (HE) methods which require the holomorphic property of solution functions near a reference state and also along a path bridging this state and the target state, for determining the effectiveness of series expansion and physical realizability (PRL), respectively, where PRL means the complex conjugacy of solution values at the target state. Existing studies generally assume holomorphic properties beforehand or assert them from a convergence of series expansion or implicit function theorem at separate points, however, which can hardly be applied to confirm the holomorphic property along a path and then PRL (because a path has infinitely many points and cannot be numerically inspected point-by-point). To this end, a new approach PHVACR is enlightened by HEAP and employs an expansion in the rectangular form and also Cauchy–Riemann (CR) conditions to numerically verify holomorphic properties around a path in an arc-by-arc manner. Here, ‘HE’ is short for holomorphic embedding, ‘AP’ in HEAP indicates an arc-length parametrization, and ‘PHVA’ in PHVACR refers to the path-related holomorphy validation with ‘CR’ representing the Cauchy–Riemann conditions. To explain the importance of PHVACR and its performance, counter-intuitive examples are discussed to explain the limitations of existing approaches, and large problems with 10,000+ buses are also simulated to examine the effectiveness and scalability of PHVACR which are compared with other HE methods.

Harmonic analysis meets stationarity: A general framework for series expansions of special Gaussian processes

In this paper, we present a new approach to derive series expansions for some Gaussian processes based on harmonic analysis of their covariance function. In particular, we propose a new simple rate-optimal series expansion for fractional Brownian motion. The convergence of the latter series holds in mean square and uniformly almost surely, with a rate-optimal decay of the remainder of the series. We also develop a general framework of convergent series expansions for certain classes of Gaussian processes with stationarity. Finally, an application to optimal functional quantization is described.

Evaluation of Computational Analysis for Stiffened Slab Using Finite Strip Method

A rectangular slab stiffened by ribs oriented perpendicular to the support of the slab as structurally equivalent to an orthotropic plate has been analyzed using the finite strip method. The analytic formulation was obtained from the governing equation of motion of the plate using the classical plate theory to an arbitrary boundary and initial condition. The stiffness formulation from total internal energy was considered. The force-displacement equation technique by applying stiffness and uniformly distributed loading was used to obtain the modal solution and hence, displacement at an arbitrary point of the plate for the assumed harmonic function. Approximate stresses following the displacement are also obtained. The comparative solution of the equilibrium differential equation is obtained by means of convergent series expansions of levy plates and the application of Timoshenko and Woinowsky-Krieger averaging of bending rigidity, utilizing suitable orthogonal displacement functions. A computer procedure based on the finite strip method is given through the application of numerical examples, which shows that the procedure provides good results, especially with low discretization and computational effort.

Numerical solution of two-dimensional weakly singular Volterra integral equations of the first kind via bivariate rational approximants

We present a numerical method for solving the two-dimensional weakly singular Volterra integral equations of the first kind. The integral equation is first converted into algebraic form using the two-dimensional Laplace transform. We then derived the series expansion for large values, which is inverted term by term to provide the convergent series expansion of the solution for small values. The asymptotic expansion of the solution is extracted from the series expansions of the two-dimensional Laplace transform around the singularity points. The bivariate homogeneous two-point Padé approximants are used to improve convergence. Numerical results are provided to illustrate the accuracy of the method.

A q-Analog of the Class of Completely Convex Functions and Lidstone Series

This paper introduces a q-analog of the class of completely convex functions. We prove specific properties, including that q-completely convex functions have convergent q-Lidstone series expansions. We also provide a sufficient and necessary condition for a real function to have an absolutely convergent q-Lidstone series expansion.

Comparative Study of Guided Electromagnetic Wave Propagation for Two Models of an Open Tape Helix

Rigorous methods to solve the homogeneous boundary value problems and the techniques to resolve illposed mathematical issues arising in the problem formulation of electromagnetic wave propagation of an Open tape helix slow wave structure with anisotropically conducting and completely conducting model is presented. The dispersion equation of both the models require convergent coeefficients of the infinte linear homogeneous equations whose determinant of the coefficent matrix is zero. The coefficient matrix of the anisotropically conducting model is rapidly convergent, while in the case of completely conducting model, the matrix entries which are infinite summation series representations turns out to be divergent, the not so well-posed boundary value problem is regularised with the method of mollification functions. The dispersion characteristics are plotted after truncating the infinite series expansions to adequate number of terms and summing the terms of converging series expansions(after regularization in the case of the completely conducting tape-helix model) in dispersion equations. The tape-current distribution for both models are estimated from the null-space vector of the truncated coefficient matrix corresponding to a particular (β <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> – <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ) root of the dispersion equation. A comparison of the numerical results for both models show that the neglect of the perpendicular tape-current density component entails a substantial modification of the dispersion characteristics of guided waves supported by an open tape helix.

On the Analytical Solution of Fractional SIR Epidemic Model

This article presents the solution of the fractional SIR epidemic model using the Laplace residual power series method. We introduce the fractional SIR model in the sense of Caputo’s derivative; it is presented by three fractional differential equations, in which the third one depends on the first coupled equations. The Laplace residual power series method (LRPSM) is implemented in this research to solve the proposed model, in which we present the solution in a form of convergent series expansion that converges rapidly to the exact one. We analyze the results and compare the obtained approximate solutions to those obtained from other methods. Figures and tables are illustrated to show the efficiency of the LRPSM in handling the proposed SIR model.

Conditional Expanding of Functions by q-Lidstone Series

This paper characterizes those functions given by convergent q-Lidstone series expansion. We give the necessary and sufficient conditions so that the entire function f(z) has such an expansion, in which case convergence is uniform on compact sets.

Convergence of series expansions in rovibrational configuration interaction (RVCI) calculations.

Rotational and rovibrational spectra are a key in astrophysical studies, atmospheric science, pollution monitoring, and other fields of active research. The ab initio calculation of such spectra is fairly sensitive with respect to a multitude of parameters and all of them must be carefully monitored in order to yield reliable results. Besides the most obvious ones, i.e., the quality of the multidimensional potential energy surface and the vibrational wavefunctions, it is the representation of the μ-tensor within the Watson Hamiltonian, which has a significant impact on the desired line lists or simulated spectra. Within this work, we studied the dependence of high-resolution rovibrational spectra with respect to the truncation order of the μ-tensor within the rotational contribution and the Coriolis coupling operator of the Watson operator. Moreover, the dependence of the infrared intensities of the rovibrational transitions on an n-mode expansion of the dipole moment surface has been investigated as well. Benchmark calculations are provided for thioformaldehyde, which has already served as a test molecule in other studies and whose rovibrational spectrum was found to be fairly sensitive. All calculations rely on rovibrational configuration interaction theory and the discussed high-order terms of the μ-tensor are a newly implemented feature, whose theoretical basics are briefly discussed.

The Madelung constant in N dimensions

We introduce two convergent series expansions (direct and recursive) in terms of Bessel functions and the number of representations of an integer as a sum of squares for N -dimensional Madelung constants, M N ( s ) , where s is the exponent of the Madelung series (usually chosen as s = 1 / 2 ). The convergence of the Bessel function expansion is discussed in detail. Values for M N ( s ) for s = 1 2 , 3 2 , 3 and 6 for dimension up to N = 20 are presented. This work extends Zucker’s original analysis on N -dimensional Madelung constants for even dimensions up to N = 8 .

Convergence of perturbation series for unbounded monotone quasiperiodic operators

We consider a class of unbounded quasiperiodic Schrödinger-type operators on ℓ2(Zd) with monotone potentials (akin to the Maryland model) and show that the Rayleigh–Schrödinger perturbation series for these operators converges in the regime of small kinetic energies, uniformly in the spectrum. As a consequence, we obtain a new proof of Anderson localization in a more general than before class of such operators, with explicit convergent series expansions for eigenvalues and eigenvectors. This result can be restricted to an energy window if the potential is only locally monotone and one-to-one. A modification of this approach also allows the potential to be non-strictly monotone and have a flat segment, under additional restrictions on the frequencies.

On Convergent Series Expansions for Solutions of Nonlinear Ordinary Differential Equations

On Convergent Series Expansions for Solutions of Nonlinear Ordinary Differential Equations

Iterative solutions for the gravitational lens equation in the strong deflection limit

Two exact lens equations have been recently shown to be equivalent to each other, being consistent with the gravitational deflection angle of light from a source to an observer, both of which can be within a finite distance from a lens object [Phys. Rev. D 102, 064060 (2020)]. We examine methods for iterative solutions of the gravitational lens equations in the strong deflection limit. It has been so far unclear whether a convergent series expansion can be provided by the gravitational lens approach based on the geometrical optics for obtaining approximate solutions in the strong deflection limit in terms of a small offset angle. By using the ratio of the lens mass to the lens distance, we discuss a slightly different method for iterative solutions and behavior of the convergence. Finite-distance effects begin at the third order in the iterative method. The iterative solutions in the strong deflection limit are estimated for Sgr ${\mathrm{A}}^{*}$ and M87. These results suggest that only the linear-order solution can be relevant with current observations, while the finite-distance effects at the third order may be negligible in the Schwarzschild lens model for these astronomical objects.

Anomaly cancellation condition in an effective nonperturbative electroweak theory

We establish the non-perturbative validity of the gauge anomaly cancellation condition in an effective electroweak theory of massless fermions with finite momentum cut-off and Fermi interaction. The requirement that the current is conserved up to terms smaller than the energy divided by the cut-off scale, which is the natural condition as gauge invariance is only emerging, produces the same constraint on charges as in the Standard Model. The result holds at a non-perturbative level as the functional integrals are expressed by convergent power series expansions and are analytic in a finite domain.

Bonus properties of states of low energy

States of Low Energy (SLEs) are exact Hadamard states defined on arbitrary Friedmann–Lemaître spacetimes. They are constructed from a fiducial state by minimizing the Hamiltonian’s expectation value after averaging with a temporal window function. We show the SLE to be expressible solely in terms of the (state independent) commutator function. They also admit a convergent series expansion in powers of the spatial momentum, both for massive and for massless theories. In the massless case, the leading infrared behavior is found to be Minkowski-like for all scale factors. This provides a new cure for the infrared divergences in Friedmann–Lemaître spacetimes with accelerated expansion. As a consequence, massless SLEs are viable candidates for pre-inflationary vacua, and in a soluble model, they are shown to entail a qualitatively correct primordial power spectrum.

Analytical approximations for the inverse Langevin function via linearization, error approximation, and iteration

This paper details an analytical framework, based on an intermediate function, which facilitates analytical approximations for the inverse Langevin function—a function without an explicit analytical form. The approximations have relative error bounds that are typically much lower than those reported in the literature and which can be made arbitrarily small. Results include convergent series expansions in terms of polynomials and sinusoids which have modest relative error bounds and convergence properties but are convergent over the domain of the inverse Langevin function. An important advance is to use error approximations, and then iterative relationships, which allow simple initial approximations for the inverse Langevin function, with modest relative errors, to generate approximations with arbitrarily low relative errors. One example is that of an initial approximating function, with a relative error bound of 0.00969, which yields relative error bounds of 2.77 × 10−6 and 2.66 × 10−16 after the use of first-order error approximation and then first-order iteration. Functions with much lower error bounds are possible and are detailed. First- and second-order Taylor series can be used to simplify the error- and iteration-based approximations.

Modelling of depth-induced wave breaking in a fully nonlinear free-surface potential flow model

Two methods to treat wave breaking in the framework of the Hamiltonian formulation of free-surface potential flow are presented, tested, and validated. The first is an extension of Kennedy et al. (2000)’s eddy-viscosity approach originally developed for Boussinesq-type wave models. In this approach, an extra term, constructed to conserve the horizontal momentum for waves propagating over a flat bottom, is added in the dynamic free-surface condition. In the second method, a pressure distribution is introduced at the free surface that dissipates wave energy by analogy to a hydraulic jump (Guignard and Grilli, 2001). The modified Hamiltonian systems are implemented using the Hamiltonian Coupled-Mode Theory, in which the velocity potential is represented by a rapidly convergent vertical series expansion. Wave energy dissipation and conservation of horizontal momentum are verified numerically. Comparisons with experimental measurements are presented for the propagation of a breaking dispersive shock wave following a dam break, and then incident regular waves breaking on a mildly sloping beach and over a submerged bar.

Optimal Control of AGC Systems Considering Non-Gaussian Wind Power Uncertainty

Wind power uncertainty poses significant challenges for automatic generation control (AGC) systems. It can enhance control performances to explicitly consider wind power uncertainty distributions within controller design. However, widely accepted wind uncertainties usually follow non-Gaussian distributions, which may lead to complicated stochastic AGC modeling and high computational burdens. To overcome the issue, this paper presents a novel Itô-theory-based model for the stochastic control problem (SCP) of AGC systems, which reduces the computational burden of optimization considering non-Gaussian wind power uncertainty to the same scale as that for deterministic control problems. We present an Itô process model to exactly describe non-Gaussian wind power uncertainty, and then propose an SCP based on the concept of stochastic assessment functions (SAFs). Based on a convergent series expansion of the SAF, the SCP is reformulated as a certain deterministic control problem without sacrificing performance under non-Gaussian wind power uncertainty. The reformulated control problem is proven as a convex optimization, which can be solved efficiently. A case study demonstrates the efficiency and accuracy of the proposed approach compared with several conventional approaches.

On approximation of FBVP by homotopy-based truncation technique

In general, analyzing the potential flow past a wedge is described by the well-known Falkner–Skan boundary value problem (FBVP). However, there exist no findings in the literature concerning the FBVP predicted by homotopy analysis method (HAM) and truncation technique, namely, tHAM. Accordingly, the main focus of this paper is on the validity and effectiveness of tHAM through some direct comparisons. It is concluded that the tHAM is a useful tool for obtaining convergent series expansions as well as reducing the central processing unit (CPU) time.