Functions In Series Of Polynomials Research Articles

On constructing of multiple rogue wave solutions to the (3+1)-dimensional Korteweg–de Vries Benjamin-Bona-Mahony equation

Exploring new wave soliton solutions to nonlinear partial differential equations has always been one of the most challenging issues in different branches of science, including physics, applied mathematics and engineering. In this paper, we construct multiple rogue waves of (3+1)-dimensional Korteweg–de Vries Benjamin-Bona-Mahony equation through a symbolic calculation approach. Further, a detailed analysis of the localization features of first-order rogue wave solution is also presented. We discuss the influence of the parameters in the equation on the localization and characteristics of a rogue wave, as well as the control of their amplitude, depth, and width. In order to achieve these desired results, a series of polynomial functions are utilized to construct the generalized multiple rogue waves with a controllable center. Based on the bilinear form of this equation, 3-rogue wave solutions, 6-rogue wave solutions, and 9-rogue wave solutions are generated, respectively. The 3-rogue wave has a ‘triangle-shaped’ structure. The center of the 6-rogue wave forms a circle around a single rogue wave. The 9-rogue wave consists of seven first-order rogue waves and one second-order rogue waves as the center. Taking some appropriate parameters into account, their complex and interesting dynamics are shown in three-dimensional and contour plots. These new results are useful to understand the new features of nonlinear dynamics in real-world applications.

Isogeometric Analysis of Euler-Bernoulli Beam Element

Abstract: The Isogeometric analysis is a computational geometry based on a series of polynomial functions (Non-Uniform Rational B-Spline, NURBS) which are assembled to represent the exact geometry. In the Isogeometric analysis, the curvature geometry of the beam element can be represented exactly. The conventional beam element can be developed by using the Isogeometric approach which is based on Euler-Bernoulli principle which is under the assumption that the dimension of the beam cross section is small compared with the length of the beam. The geometric shape of the beam and the shape functions formulation of the element can be formulated by using the Isogeometric approach. This paper highlights the application of the NURBS for the Euler-Bernoulli beam element in the context of finite element analysis. Examples are given to verify the effectiveness of the Isogeometric approach in static and free vibration problems.

Polynomial approximation pseudo-two-dimensional battery model for online application in embedded battery management system

Physics-based model has been regarded as a promising alternative to equivalent circuit model due to its ability to describe internal electrochemical states of battery. However, the rigorous physics-based model, namely pseudo-two-dimensional (P2D) model, is too complicated for online application in embedded battery management system. In this paper, to simplify the P2D model, a series of polynomial functions are employed to approximate the electrolyte phase concentration profile, solid phase concentration profile, and non-uniform reaction flux profile, respectively. Especially, the accuracy of 2nd-order and 3rd-order polynomial approximations for reaction flux is compared, and the higher-order is validated with more strength. Benefit from the acquisition of above variables, the electrolyte potential is derived directly according to the conservation of charge at electrolyte phase; the accuracy of activation overpotential is also improved by using the non-uniform reaction flux rather than assuming the uniform current density in single particle (SP) model. Finally, the developed model is simulated by different constant current rates, hybrid pulse and driving cycles, and its outputs are compared with P2D model and original SP model. The results demonstrate that the model proposed in this paper could capture the battery characteristics efficiently, and also significantly reduce the computation complexity.

A new method for the calculation of continuous relaxation spectra from dynamic-mechanical data

We present a new method for the determination of highly precise continuous relaxation spectra. The method is based on the use of piecewise cubic Hermite splines, which are fairly easy to tabulate by using their knots. The Hermite splines method allows a continuous description of the spectrum by a series of polynomial functions. The numerical instabilities of the spectrum calculation are minimized by limiting the slope of the spectrum to physically meaningful values. The reproducibility of the spectrum calculation is within an error margin of about ±10% in the physically relevant relaxation time range. This method is able to retrieve the spectrum based on data calculated from a benchmark with high accuracy and precision.

Joseph L. Walsh: Selected papers, edited by Theodore J. Rivlin and Edward B. Saff. Pp. 682. £86. 2000. ISBN 0 387 98782 7 (Springer Verlag).

Joseph L. Walsh: Selected papers, edited by Theodore J. Rivlin and Edward B. Saff. Pp. 682. £86. 2000. ISBN 0 387 98782 7 (Springer Verlag).

Polynomial expansions in the Borel region

This paper deals with certain expansions of analytic functions in series of polynomials. Explicit forms for the polynomials are given in terms of the coefficients of the Taylor's expansion and of prescribed positive constants hk. Under suitable conditions, to be presently discussed, the series converge to the function in the Borel region.

Note on the expansion of analytic functions in series of polynomials and in series of other analytic functions

Note on the expansion of analytic functions in series of polynomials and in series of other analytic functions

Note on the Expansion of Analytic Functions in Series of Polynomials and in Series of Other Analytic Functions

Note on the Expansion of Analytic Functions in Series of Polynomials and in Series of Other Analytic Functions

On the Expansion of Analytic Functions in Series of Polynomials and in Series of Other Analytic Functions

On the Expansion of Analytic Functions in Series of Polynomials and in Series of Other Analytic Functions

On the expansion of analytic functions in series of polynomials and in series of other analytic functions

On several occasionst the present writer has studied the expansion of functions of a complex variable in series of more or less arbitrary functions. The results obtained are connected with Birkhoff's generalization of Taylor's series,t but yield additional results when interpreted, after conformal transformation of the plane, as the expansion of arbitrary functions in series of polynomials. The object of the present note is to prove the following theorem, which is easily established by the methods already used, but seems to go in some directions beyond results proved by other writers? concerning polynomials associated with a region bounded by an arbitrary Jordan curve:

On the Expansion of Analytic Functions in Series of Polynomials

On the Expansion of Analytic Functions in Series of Polynomials

On the expansion of analytic functions in series of polynomials

where the functions Pk(z) are properly chosen. The series (1) converges uniformly in the closed region interior to C. This fundamental result is a direct generalization of Taylor's series, to which Faber's series (1) reduces when C is a circle. On any circle C for which Taylor's selies converges, if the center of C is the point about which the Taylor development is considered, Taylor's series reduces precisely to Fourier's series, both formally and in fact. More generally, Laurent's series similarly reduces to Fourier's series and conversely, if the function considered is defined and integrable on the circle C. The natural generalization of Fourier's series and of Laurent's series to the case of an arbitrary contour C seems not to have been made. It is the object of the present paper to set forth such a generalization, as indicated by the following theorem: