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Power Series Solutions and Special Functions
Published in George F. Simmons, Differential Equations with Applications and Historical Notes, 2016
Most of the specific functions encountered in elementary analysis belong to a class known as the elementary functions. In order to describe this class, we begin by recalling that an algebraic function is a polynomial, a rational function, or more generally any function y = f(x) that satisfies an equation of the form Pn(x)yn+Pn−1(x)yn−1+⋯+P1(x)y+P0(x)=0,
Meta-Analysis 2
Published in Michael Humphreys, Fergus Nicol, Susan Roaf, Adaptive Thermal Comfort: Foundations and Analysis, 2015
Michael Humphreys, Fergus Nicol, Susan Roaf
So, insofar as anything was expected, it would be a broadly horizontal band within which the comfort temperatures would lie, but having a ‘kink’ around the most pleasant outdoor conditions. Polynomials were used as a convenient tool to explore the data, but it would not be appropriate to use a polynomial finally to express the expected shape of the data. Polynomials tend towards plus or minus infinity at the extremes – not at all what one would expect of a comfort temperature. So a suitable algebraic function was chosen that would have the right general shape (like the shape of zone B in Figure 9.1).
Variables, functions and mappings
Published in Alan Jeffrey, Mathematics, 2004
An algebraic function arises when attempting to from the inverse of a rational function. The function y=+√x for x ≥ 0 provides a typical example here. More complicated examples are the functions: y=x2/3y=x2+2√x−1y=x√[x/(2−x)].
Optical solitons solution of resonance nonlinear Schrödinger type equation with Atangana's-conformable derivative using sub-equation method
Published in Waves in Random and Complex Media, 2021
H. Yépez-Martínez, J. F. Gómez-Aguilar
Nonlinear Schrödinger equations (NLSEs) appear in various areas of engineering sciences, physical and biological sciences. In particular, the NLSEs appear in fluid dynamics, nonlinear optics, plasma and nuclear physics [12–19]. The space-time resonant nonlinear Schrödinger equation (R-NLSE) portrays the dynamics of Madelung's fluids in several nonlinear systems [20–26]. Recently, we have seen many progresses in the field of nonlinear optics [27,28]. The dimensionless form of the resonant nonlinear Schrödinger equation (R-CNLSE) [29] that will be studied in this paper is given by [20–26] where the dependent variable is the normalized electric-field envelope, and are the Atangana's-conformable derivatives [30], while the independent variables z and t represent the longitudinal coordinate along the optical fibers and γ is the coefficient of resonant nonlinearity. The function is a real valued algebraic function.
Assessment of Engineering Turbulence Models in Buoyant Diabatic Turbulent Flow
Published in Nuclear Technology, 2023
One approach to modeling Reynolds stresses is to transport all six Reynolds stress components with production and dissipation terms designed to capture local Reynolds stress anisotropy. Development of this approach stretches back 50 years.[10] However, Reynolds stress modeling is highly sensitive to model error and model calibration, and it has not been shown to generalize well. Instead, industrial CFD users usually choose two-equation turbulence models, such as the k-ω model or the k-ε model. These approaches model the Reynolds stress terms as an algebraic function of local flow quantities. The full nonlinear eddy viscosity model of Baglietto and Ninokata[15] is Eq. (4):
On the uniform algebraic observability of multi-switching linear systems
Published in International Journal of Control, 2021
Laura Menini, Corrado Possieri, Antonio Tornambè
By Lemma 4.2, for each , there exists a non-constant polynomial such that , which shows that is an algebraic function over . If , then is a linear function over , whence it is defined for almost all values of .