Some Approaches to the Calculation of Conservation Laws for a Telegraph System and Their Comparisons

Buhe, Eerdun; Bluman, George W.; Chen, Alatancang; Hu, Yulan

doi:10.3390/sym10060182

Cited by 4 publications

(3 citation statements)

References 27 publications

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“…For non-variational problems, described by equation ( 6), few methods of finding conservation laws emmerged [135,136]. One of them, called the direct method, consists in finding multipliers Λ i depending on y, u and derivatives of u such that the scalar function Λ i E i is the total divergence of some function C , which then becomes automatically a conserved current [137][138][139].…”

Section: Conservation Lawsmentioning

confidence: 99%

Lie groups and continuum mechanics: where do we stand today?

de Saxcé,

Razafindralandy

2024

Comptes Rendus. Mécanique

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Section: Conservation Lawsmentioning

confidence: 99%

Lie groups and continuum mechanics: where do we stand today?

de Saxcé,

Razafindralandy

2024

Comptes Rendus. Mécanique

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“…Knowledge of CLs can also be useful as a tool to develop new techniques for the solutions of some differential equations [9]. Various approaches [10,11] have been developed to obtain CLs, like direct [12], Noether theorem [13], symmetry [14,15] or multipliers [16].…”

Section: Introductionmentioning

confidence: 99%

Conservation laws for optical Bloch equations for the Λ scheme

Dimitrijević¹,

Arsenović²,

Jelenković³

2020

J. Phys. A: Math. Theor.

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Results are presented for the conservation laws (CLs) for the optical Bloch equations (OBEs) for the Λ scheme under action of two light fields. The method of multipliers (variational derivative method) is used to obtain CLs which are dependent on the density-matrix elements. Results are classified and discussed by the phenomenological parameters and processes characteristic for OBEs, like the relaxation and decoherence, and the detunings of light fields. CLs for the Liouville von Neumann equation, as a special case of OBEs, without relaxation and decoherence terms, are presented and it is shown that traces, characteristic polynomials and their coefficients, eigen-values and determinants of class of matrices represent CLs. We also presented method, which used CLs obtained by the method of multipliers, to construct another set of CLs, which have the explicit time dependence and are valid for the larger set of parameters, with relaxation due to time of flight and spontaneous emission. Presented results could yield better understanding of the processes in the Λ scheme modeled by the OBEs, and as well have practical applications in various solution methods.

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“…Castaings and Navon [5] proposed a numerical method for solving reaction-diffusion problems. These fluid dynamics often relate to the theory of conservation laws [6][7][8][9] with finite volume methods [10][11][12][13] as solving techniques. Overall, we can choose a fluid dynamics model in the literature that best suits the problem we want to solve.…”

Section: Introductionmentioning

confidence: 99%

Weak Local Residuals as Smoothness Indicators in Adaptive Mesh Methods for Shallow Water Flows

Mungkasi

Roberts

2020

Symmetry

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This paper proposes some formulations of weak local residuals of shallow-water-type equations, namely, one-, one-and-a-half-, and two-dimensional shallow water equations. Smooth parts of numerical solutions have small absolute values of weak local residuals. Rougher parts of numerical solutions have larger absolute values of weak local residuals. This behaviour enables the weak local residuals to detect parts of numerical solutions which are smooth and rough (non-smooth). Weak local residuals that we formulate are implemented successfully as refinement or coarsening indicators for adaptive mesh finite volume methods used to solve shallow water equations.

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Some Approaches to the Calculation of Conservation Laws for a Telegraph System and Their Comparisons

Cited by 4 publications

References 27 publications

Lie groups and continuum mechanics: where do we stand today?

Lie groups and continuum mechanics: where do we stand today?

Conservation laws for optical Bloch equations for the Λ scheme

Weak Local Residuals as Smoothness Indicators in Adaptive Mesh Methods for Shallow Water Flows

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