An Efficient Method to Find Solutions for Transcendental Equations with Several Roots

Luck, Rogelio; Zdaniuk, Gregory J.; Cho, Heejin

doi:10.1155/2015/523043

Cited by 19 publications

(10 citation statements)

References 16 publications

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“…In the instant case of the application of CBM approach to implement a numerical solution for SoTE encountered in photovoltaics, two illustrations are hereby highlighted. Equation (2) acted as the power output in the integration of solar, thermal, and electrical exergies of a photovoltaic module, as shown in Equation (3).…”

Section: Application Of Sote In Photovoltaic and Thermophotovoltaic Modelling And Simulationmentioning

confidence: 99%

“…However, encountering non-zero TE of the form f(x) = g(x) in science and engineering poses challenges, particularly when the TE is included in a system of equations to create a system of transcendental equations (SoTE). A TE may have many roots which may require explicit method to find their roots using Cauchy's integral theorem [3]. The computational solution to SoTE may result in a single output in a case where some functions act as functions of the output function.…”

Section: Introductionmentioning

confidence: 99%

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A Computational Approach to Solve a System of Transcendental Equations with Multi-Functions and Multi-Variables

et al. 2021

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A system of transcendental equations (SoTE) is a set of simultaneous equations containing at least a transcendental function. Solutions involving transcendental equations are often problematic, particularly in the form of a system of equations. This challenge has limited the number of equations, with inter-related multi-functions and multi-variables, often included in the mathematical modelling of physical systems during problem formulation. Here, we presented detailed steps for using a code-based modelling approach for solving SoTEs that may be encountered in science and engineering problems. A SoTE comprising six functions, including Sine-Gordon wave functions, was used to illustrate the steps. Parametric studies were performed to visualize how a change in the variables affected the superposition of the waves as the independent variable varies from x1 = 1:0.0005:100 to x1 = 1:5:100. The application of the proposed approach in modelling and simulation of photovoltaic and thermophotovoltaic systems were also highlighted. Overall, solutions to SoTEs present new opportunities for including more functions and variables in numerical models of systems, which will ultimately lead to a more robust representation of physical systems.

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Section: Application Of Sote In Photovoltaic and Thermophotovoltaic Modelling And Simulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Computational Approach to Solve a System of Transcendental Equations with Multi-Functions and Multi-Variables

et al. 2021

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“…The dispersion relation (4.4) is a transcendental equation due to the modified Bessel functions that depend on α through l 1 and l 2 . To solve this equation, the method of Luck, Zdaniuk & Cho (2015) is used. The resulting solutions are then compared with experimental data and previous theoretical studies.…”

Section: Comparison With Experimental Datamentioning

confidence: 99%

Linear stability analysis of a Newtonian ferrofluid cylinder surrounded by a Newtonian fluid

Canu

Renoult

2021

J. Fluid Mech.

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We performed a linear stability analysis of a Newtonian ferrofluid cylinder surrounded by a Newtonian non-magnetic fluid in an azimuthal magnetic field. A wire is used at the centre of the ferrofluid cylinder to create this magnetic field. Isothermal conditions are considered and gravity is ignored. An axisymmetric perturbation is imposed at the interface between the two fluids and a dispersion relation is obtained allowing us to predict whether the flow is stable or unstable with respect to this perturbation. This relation is dependent on the Ohnesorge number of the ferrofluid, the dynamic viscosity ratio, the density ratio, the magnetic Bond number, the relative magnetic permeability and the dimensionless wire radius. Solutions to this dispersion relation are compared with experimental data from Arkhipenko et al. (Fluid Dyn., vol. 15, issue 4, 1981, pp. 477–481) and, more recently, Bourdin et al. (Phys. Rev. Lett., vol. 104, issue 9, 2010, 094502). A better agreement than the inviscid theory and the theory that only takes into account the viscosity of the ferrofluid is shown with the data of Arkhipenko et al. (Fluid Dyn., vol. 15, issue 4, 1981, pp. 477–481) and those of Bourdin et al. (Phys. Rev. Lett., vol. 104, issue 9, 2010, 094502) for small wavenumbers.

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“…given the bounds on 𝑧 0 . Theorem 1 of Ullisch (2020) (see also Jackson 1916Jackson , 1917Luck et al 2015) states that, on a simply-connected open subset 𝑈 of the complex plane, for every simple zero 𝑧 0 ∈ 𝑈 of a non-zero analytic function 𝑓 (𝑧), there exists a curve 𝐶 such that…”

Section: Solutionmentioning

confidence: 99%

A Uniform Spherical Goat (Problem): Explicit Solution for Homologous Collapse's Radial Evolution in Time

Slepian¹,

Philcox²

2021

Preprint

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The homologous collapse from rest of a uniform density sphere under its self gravity is a well-known toy model for the formation dynamics of astronomical objects ranging from stars to galaxies. Equally well-known is that the evolution of the radius with time cannot be explicitly obtained because of the transcendental nature of the differential equation solution. Rather, both radius and time are written parametrically in terms of the development angle 𝜃. We here present an explicit integral solution for radius as a function of time, exploiting methods from complex analysis recently applied to the mathematically-similar "geometric goat problem." Our solution can be efficiently evaluated using a Fast Fourier Transform and allows for arbitrary sampling in time, with a simple implementation that is ∼100× faster than using numerical root-finding to achieve arbitrary sampling. Our explicit solution is advantageous relative to the usual approach of first generating a uniform grid in 𝜃, since this latter results in a non-uniform radial or time sampling, less useful for applications such as generation of sub-grid physics models.

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An Efficient Method to Find Solutions for Transcendental Equations with Several Roots

Cited by 19 publications

References 16 publications

A Computational Approach to Solve a System of Transcendental Equations with Multi-Functions and Multi-Variables

A Computational Approach to Solve a System of Transcendental Equations with Multi-Functions and Multi-Variables

Linear stability analysis of a Newtonian ferrofluid cylinder surrounded by a Newtonian fluid

A Uniform Spherical Goat (Problem): Explicit Solution for Homologous Collapse's Radial Evolution in Time

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