Null-field integral equations and their applications

Chen, J.-T.; Chen, P.-Y.

doi:10.2495/be070091

Cited by 6 publications

(6 citation statements)

References 11 publications

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“…To calculate the dispersion relation I (p) in the thermodynamic limit for arbitrary γ 0, the Fredholm integral equation of the second kind, equation (2.79), has to be solved. Such a problem can be attacked numerically with the method of Neumann series [53]. In the Tonks-Girardeau limit (γ = ∞) the spectrum is given by equation (2.78) and shifted by −μ ch , namely…”

mentioning

confidence: 99%

Quantum dark solitons in ultracold one-dimensional Bose and Fermi gases

Syrwid

2021

J. Phys. B: At. Mol. Opt. Phys.

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Solitons are ubiquitous phenomena that appear, among others, in the description of tsunami waves, fiber-optic communication and ultracold atomic gases. The latter systems turned out to be an excellent playground for investigations of matter-wave solitons in a quantum world. This tutorial provides a general overview of the ultracold contact interacting Bose and Fermi systems in a one-dimensional space that can be described by the renowned Lieb–Liniger and Yang–Gaudin models. Both the quantum many-body systems are exactly solvable by means of the Bethe ansatz technique, granting us a possibility for investigations of quantum nature of solitonic excitations. We discuss in details a specific class of quantum many-body excited eigenstates called yrast states and show that they are strictly related to quantum dark solitons in the both considered Bose and Fermi systems.

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confidence: 99%

Quantum dark solitons in ultracold one-dimensional Bose and Fermi gases

Syrwid

2021

J. Phys. B: At. Mol. Opt. Phys.

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“…The desingularization technique of subtracting and adding back is based on the discretization of the reduced null-field equation [9,10] R…”

Section: The Regularized Meshless Methodsmentioning

confidence: 99%

“…The desingularization technique of subtracting and adding back is based on the discretization of the reduced null‐field equation

\begin{matrix} center \\ center \int_{Γ} A^{(e)} (t_{i}, s) d Γ (s) = 0, & center t_{i} \in D^{e}, \end{matrix}

where A (e) has the opposite normal direction with A and D e is the exterior domain of

\overset{D}{true}

. According to the relation of A (e) and A , there is

{\begin{cases} left & A (t_{i}, s_{j}) = - A^{(e)} (t_{i}, s_{j}), & i \neq j, \\ left & A (t_{i}, s_{j}) = A^{(e)} (t_{i}, s_{j}), & i = j . \end{cases}

…”

Section: The Regularized Meshless Methodsmentioning

confidence: 99%

Analysis of cutoff wavelength of elliptical waveguide by regularized meshless method

Song

Chen

2012

Int J Numerical Modelling

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In this paper, the regularized meshless method (RMM) combined with the determinant rule is taken to analyze the cutoff wavelength of elliptical waveguide with arbitrary eccentricity. First, an improved desingularization technique of subtracting and adding back is introduced for RMM to discretize this problem. Then, the novel local minimum finding technique on the basis of Chebfun is applied to extract the cutoff wavelength from the determinant of the interpolation matrix of RMM. The numerical examples show that the RMM obtains consistent results with the conventional method of fundamental solutions but has advantage of well-conditioning and no fictitious boundary. Our method provides another highly effective and stable candidate to solve the cutoff wavelength of elliptical waveguide. Figure 5. (a)-(e), the f(l) curves approximated by Chebfun; (f), the condition number curves of the method of fundamental solutions (MFS) and regularized meshless method (RMM) with e = 0.9. 424 R. SONG AND X. CHEN

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“…In this section, the technique provided in this paper is used to find numerical solution of three illustrative examples. We illustrate the above algorithm with the assist of numerical example, which encompass 2nd kind Volterra integral equation available in the existing literature [18], [19]. The convergence of every linear Volterra integral equation is calculated through:…”

Section: Numerical Examplesmentioning

confidence: 99%

New Solution Method for Solving Volterra Integral Equation Using Bernstein Polynomials

Zarnan,

Hameed

2024

Preprint

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Linear and nonlinear differential equations extensively used mathematical models for many interesting and important phenomena observed in numerous areas of science and technology. They are inspired by problems in diverse fields such as economics, biology, fluid dynamics, physics, differential geometry, engineering, control theory, materials science, and quantum mechanics. This special issue aims to highlight recent developments in methods and applications of linear and nonlinear differential equations. In addition, there are papers analyzing equations that arise in engineering, classical and fluid mechanics, and finance. In this paper a novel technique implementing Bernstein polynomials is introduced for the numerical solution of a Volterra integral equation. The obtained solutions are novel, and previous literature lacks such derivations. For this aim, we derive a simple and efficient matrix formulation using Bernstein polynomials as trial functions. Numerical examples are considered to verify the effectiveness of the proposed derivations and numerical solutions are compared with the existing methods available in the literature. The Volterra integral equation can be used to describe the capacitance of the parallel plate capacitor (PPC) in the electrostatic field. It is shown that the numerical results are excellent. The efficiency of the proposed numerical technique is exhibited through graphical illustrations and results drafted in tabular form for specific values of the parameters to validate the numerical investigation.

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Null-field integral equations and their applications

Cited by 6 publications

References 11 publications

Quantum dark solitons in ultracold one-dimensional Bose and Fermi gases

Quantum dark solitons in ultracold one-dimensional Bose and Fermi gases

Analysis of cutoff wavelength of elliptical waveguide by regularized meshless method

New Solution Method for Solving Volterra Integral Equation Using Bernstein Polynomials

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