Reformulation of elliptic equations for heat transfer and diffusion in solids with space-time algebra

2021 ◽  
Author(s):  
Victor L. Mironov
Keyword(s):  
Heat Transfer ◽  
Elliptic Equations ◽  
Space Time ◽  
Second Order ◽  
Diffusion In Solids ◽  
Heat Flows ◽  
Vector Potentials ◽  
Commutative Space ◽  
The One ◽  
And Diffusion

In this paper, we demonstrate the application of non-commutative space-time algebra of sedeons to generalize the system of equations describing heat transfer and impurity diffusion in solids at finite velocity. It is shown that by analogy with electrodynamics, these transfer processes can be described using a compact second-order sedeonic equation for generalized scalar and vector potentials. On the one hand, this equation is reduced to the system of first-order differential equations for vortex-less mass and heat flows, and on the other hand, it can be transformed to the second-order elliptical equations for the profiles of temperature and impurity concentration. The comparison of peculiarities in transfer within the frames of parabolic and elliptic equations is discussed.

scholarly journals HOPF-ALGEBRA DESCRIPTION OF NONCOMMUTATIVE-SPACE–TIME SYMMETRIES

2004 ◽  
Vol 19 (30) ◽  
pp. 5187-5219 ◽  
Author(s):  
ALESSANDRA AGOSTINI ◽  
GIOVANNI AMELINO-CAMELIA ◽  
FRANCESCO D'ANDREA
Keyword(s):  
Hopf Algebra ◽  
Equations Of Motion ◽  
Space Time ◽  
Noncommutative Space ◽  
Underlying Space ◽  
Minkowski Space Time ◽  
Commutative Space ◽  
Symmetry Transformations ◽  
The One

In the study of certain noncommutative versions of Minkowski space–time a lot remains to be understood for a satisfactory characterization of their symmetries. Adopting as our case study the κ-Minkowski noncommutative space–time, on which a large literature is already available, we propose a line of analysis of noncommutative-space–time symmetries that relies on the introduction of a Weyl map (connecting a given function in the noncommutative Minkowski with a corresponding function in commutative Minkowski). We provide new elements in favor of the expectation that the commutative-space–time notion of Lie-algebra symmetries must be replaced, in the noncommutative-space–time context, by the one of Hopf-algebra symmetries. While previous studies appeared to establish a rather large ambiguity in the description of the Hopf-algebra symmetries of κ-Minkowski, the approach here adopted reduces the ambiguity to the description of the translation generators, and our results, independently of this ambiguity, are sufficient to clarify that some recent studies which argued for an operational indistinguishability between theories with and without a length-scale relativistic invariant, implicitly assumed that the underlying space–time would be classical. Moreover, while usually one describes theories in κ-Minkowski directly at the level of equations of motion, we explore the nature of Hopf-algebra symmetry transformations on an action.

Dirichlet-to-Neumann Mapping for the Characteristic Elliptic Equations with Symmetric Periodic Coefficients

2014 ◽  
Vol 16 (4) ◽  
pp. 1102-1115
Author(s):  
Jingsu Kang ◽  
Meirong Zhang ◽  
Chunxiong Zheng
Keyword(s):  
Elliptic Equations ◽  
Schrodinger Equation ◽  
Infinite Product ◽  
Second Order ◽  
Periodic Coefficients ◽  
One Dimensional ◽  
Analytical Expression ◽  
Comput Phys ◽  
Dimensional Characteristic ◽  
The One

AbstractBased on the numerical evidences, an analytical expression of the Dirichlet-to-Neumann mapping in the form of infinite product was first conjectured for the one-dimensional characteristic Schrödinger equation with a sinusoidal potential in [Commun. Comput. Phys., 3(3): 641-658, 2008]. It was later extended for the general second-order characteristic elliptic equations with symmetric periodic coefficients in [J. Comp. Phys., 227: 6877-6894, 2008]. In this paper, we present a proof for this Dirichlet-to-Neumann mapping.

scholarly journals Preserving the Shape of Functions by Applying Multidimensional Schoenberg-Type Operators

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1016
Author(s):  
Camelia Liliana Moldovan ◽  
Radu Păltănea
Keyword(s):  
Applied Sciences ◽  
Degree Of Approximation ◽  
Higher Order ◽  
Second Order ◽  
Dimensional Case ◽  
Moduli Of Continuity ◽  
One Dimensional ◽  
Multidimensional Generalization ◽  
The One

The paper presents a multidimensional generalization of the Schoenberg operators of higher order. The new operators are powerful tools that can be used for approximation processes in many fields of applied sciences. The construction of these operators uses a symmetry regarding the domain of definition. The degree of approximation by sequences of such operators is given in terms of the first and the second order moduli of continuity. Extending certain results obtained by Marsden in the one-dimensional case, the property of preservation of monotonicity and convexity is proved.

scholarly journals Post-Newtonian limit: second-order Jefimenko equations

2020 ◽  
Vol 80 (7) ◽  
Author(s):  
David Pérez Carlos ◽  
Augusto Espinoza ◽  
Andrew Chubykalo
Keyword(s):  
Linear Equations ◽  
Space Time ◽  
Second Order ◽  
Field Equations ◽  
Gravitational Fields ◽  
Mass Distributions ◽  
Minkowski Space Time ◽  
Theory Of Gravitation ◽  
Gravity Probe ◽  
Gravitational Equations

Abstract The purpose of this paper is to get second-order gravitational equations, a correction made to Jefimenko’s linear gravitational equations. These linear equations were first proposed by Oliver Heaviside in [1], making an analogy between the laws of electromagnetism and gravitation. To achieve our goal, we will use perturbation methods on Einstein field equations. It should be emphasized that the resulting system of equations can also be derived from Logunov’s non-linear gravitational equations, but with different physical interpretation, for while in the former gravitation is considered as a deformation of space-time as we can see in [2–5], in the latter gravitation is considered as a physical tensor field in the Minkowski space-time (as in [6–8]). In Jefimenko’s theory of gravitation, exposed in [9, 10], there are two kinds of gravitational fields, the ordinary gravitational field, due to the presence of masses, at rest, or in motion and other field called Heaviside field due to and acts only on moving masses. The Heaviside field is known in general relativity as Lense-Thirring effect or gravitomagnetism (The Heaviside field is the gravitational analogous of the magnetic field in the electromagnetic theory, its existence was proved employing the Gravity Probe B launched by NASA (See, for example, [11, 12]). It is a type of gravitational induction), interpreted as a distortion of space-time due to the motion of mass distributions, (see, for example [13, 14]). Here, we will present our second-order Jefimenko equations for gravitation and its solutions.

Radiant Heat Transfer From Isothermal Dispersions With Isotropic Scattering

1967 ◽  
Vol 89 (4) ◽  
pp. 300-308 ◽  
Author(s):  
R. H. Edwards ◽  
R. P. Bobco
Keyword(s):  
Heat Transfer ◽  
Approximate Solutions ◽  
Radiant Heat ◽  
Second Order ◽  
Isotropic Scattering ◽  
Radiant Heat Transfer ◽  
Approximate Methods ◽  
First Order ◽  
Order Solution ◽  
Radiant Transfer

Two approximate methods are presented for making radiant heat-transfer computations from gray, isothermal dispersions which absorb, emit, and scatter isotropically. The integrodifferential equation of radiant transfer is solved using moment techniques to obtain a first-order solution. A second-order solution is found by iteration. The approximate solutions are compared to exact solutions found in the literature of astrophysics for the case of a plane-parallel geometry. The exact and approximate solutions are both expressed in terms of directional and hemispherical emissivities at a boundary. The comparison for a slab, which is neither optically thin nor thick (τ = 1), indicates that the second-order solution is accurate to within 10 percent for both directional and hemispherical properties. These results suggest that relatively simple techniques may be used to make design computations for more complex geometries and boundary conditions.

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