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Journal articles on the topic 'Maxwell equation'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 April 2023

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1

BARTHELMÉ, RÉGINE, PATRICK CIARLET, and ERIC SONNENDRÜCKER. "GENERALIZED FORMULATIONS OF MAXWELL'S EQUATIONS FOR NUMERICAL VLASOV–MAXWELL SIMULATIONS." Mathematical Models and Methods in Applied Sciences 17, no. 05 (May 2007): 657–80. http://dx.doi.org/10.1142/s0218202507002066.

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When solving numerically approximations of the Vlasov–Maxwell equations, the source terms in Maxwell's equations coming from the numerical solution of the Vlasov equation do not generally satisfy the continuity equation which is required for Maxwell's equations to be well-posed. Hence it is necessary to introduce generalized Maxwell's equations which remain well-posed when there are errors in the sources. Different such formulations have been introduced previously. The aim of this paper is to perform their mathematical analysis and verify the existence and uniqueness of the solution.

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2

Kalauni, Pushpa, and J. C. A. Barata. "Reconstruction of symmetric Dirac–Maxwell equations using nonassociative algebra." International Journal of Geometric Methods in Modern Physics 12, no. 03 (February 27, 2015): 1550029. http://dx.doi.org/10.1142/s0219887815500292.

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In the presence of sources, the usual Maxwell equations are neither symmetric nor invariant with respect to the duality transformation between electric and magnetic fields. Dirac proposed the existence of magnetic monopoles for symmetrizing the Maxwell equations. In the present work, we obtain the fully symmetric Dirac–Maxwell's equations (i.e. with electric and magnetic charges and currents) as a single equation by using 4 × 4 matrix presentation of fields and derivative operators. This matrix representation has been derived with the help of the algebraic properties of quaternions and octonions. Such description gives a compact representation of electric and magnetic counterparts of the field in a single equation.

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3

Rodrigues, Waldyr A. "The relation between Maxwell, Dirac, and the Seiberg-Witten equations." International Journal of Mathematics and Mathematical Sciences 2003, no. 43 (2003): 2707–34. http://dx.doi.org/10.1155/s0161171203210218.

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We discuss unsuspected relations between Maxwell, Dirac, and the Seiberg-Witten equations. First, we present the Maxwell-Dirac equivalence (MDE) of the first kind. Crucial to that proposed equivalence is the possibility of solving for ψ (a representative on a given spinorial frame of a Dirac-Hestenes spinor field) the equation F=ψγ21ψ˜, where F is a given electromagnetic field. Such task is presented and it permits to clarify some objections to the MDE which claim that no MDE may exist because F has six (real) degrees of freedom and ψ has eight (real) degrees of freedom. Also, we review the generalized Maxwell equation describing charges and monopoles. The enterprise is worth, even if there is no evidence until now for magnetic monopoles, because there are at least two faithful field equations that have the form of the generalized Maxwell equations. One is the generalized Hertz potential field equation (which we discuss in detail) associated with Maxwell theory and the other is a (nonlinear) equation (of the generalized Maxwell type) satisfied by the 2-form field part of a Dirac-Hestenes spinor field that solves the Dirac-Hestenes equation for a free electron. This is a new result which can also be called MDE of the second kind. Finally, we use the MDE of the first kind together with a reasonable hypothesis to give a derivation of the famous Seiberg-Witten equations on Minkowski spacetime. A physical interpretation for those equations is proposed.

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CIARLET, PATRICK, and SIMON LABRUNIE. "NUMERICAL ANALYSIS OF THE GENERALIZED MAXWELL EQUATIONS (WITH AN ELLIPTIC CORRECTION) FOR CHARGED PARTICLE SIMULATIONS." Mathematical Models and Methods in Applied Sciences 19, no. 11 (November 2009): 1959–94. http://dx.doi.org/10.1142/s0218202509004017.

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When computing numerical solutions to the Vlasov–Maxwell equations, the source terms in Maxwell's equations usually fail to satisfy the continuity equation. Since this condition is required for the well-posedness of Maxwell's equations, it is necessary to introduce generalized Maxwell's equations which remain well-posed when there are errors in the sources. These approaches, which involve a hyperbolic, a parabolic and an elliptic correction, have been recently analyzed mathematically. The goal of this paper is to carry out the numerical analysis for several variants of Maxwell's equations with an elliptic correction.

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5

Vedenyapin, Victor Valentinovich. "Vlasov-Maxwell-Einstein Equation." Keldysh Institute Preprints, no. 188 (2018): 1–20. http://dx.doi.org/10.20948/prepr-2018-188.

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6

Dzhunushaliev, Vladimir, and Vladimir Folomeev. "Nonperturbative Quantization Approach for QED on the Hopf Bundle." Universe 7, no. 3 (March 11, 2021): 65. http://dx.doi.org/10.3390/universe7030065.

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We consider the Dirac equation and Maxwell’s electrodynamics in R×S3 spacetime, where a three-dimensional sphere is the Hopf bundle S3→S2. In both cases, discrete spectra of classical solutions are obtained. Based on the solutions obtained, the quantization of free, noninteracting Dirac and Maxwell fields is carried out. The method of nonperturbative quantization of interacting Dirac and Maxwell fields is suggested. The corresponding operator equations and the infinite set of the Schwinger–Dyson equations for Green’s functions is written down. We write a simplified set of equations describing some physical situations to illustrate the suggested scheme of nonperturbative quantization. Additionally, we discuss the properties of quantum states and operators of interacting fields.

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7

Vedenyapin, Victor, Nikolay Fimin, and Valery Chechetkin. "The system of Vlasov–Maxwell–Einstein-type equations and its nonrelativistic and weak relativistic limits." International Journal of Modern Physics D 29, no. 01 (January 2020): 2050006. http://dx.doi.org/10.1142/s0218271820500066.

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We considered derivation of Vlasov–Einstein–Maxwell system of equations from the first principles, i.e. using classical Maxwell–Einstein–Hilbert action principle. We know many papers in which the theories indicated as Einstein–Vlasov, Vlasov–Maxwell–Einstein, Einstein–Maxwell–Boltzmann are discussed, and we discuss difficulties of usually used equations. We use another way of derivation and obtain an alternative version based on the generalized Fock–Weinberg form of equation of motion.

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8

Moradzadeh, A., and H. Hassanabadi. "Quasi-Maxwell equation for spin-1 particles." International Journal of Modern Physics E 23, no. 02 (February 2014): 1450007. http://dx.doi.org/10.1142/s0218301314500074.

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In this study, we consider Duffin–Kemmer–Petiau (DKP) equation in three-dimensional, hence we review resemble Maxwell equations where we can derive from DKP equation. An exact solution of the three-dimensional DKP equation is presented in the presence of the pseudo-Coulomb potential-plus-ring-shaped potential. As we derive the energy eigenvalues and corresponding eigenfunctions, we explain about DKP equation under different forms of interactions.

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9

Candemir, Nuray, Murat Tanışlı, Kudret Özdaş, and Süleyman Demir. "Hyperbolic Octonionic Proca-Maxwell Equations." Zeitschrift für Naturforschung A 63, no. 1-2 (February 1, 2008): 15–18. http://dx.doi.org/10.1515/zna-2008-1-203.

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In this study, after introducing the hyperbolic octonionic (counteroctonion) algebra, which is also expressed in the sub-algebra of sedenions, and differential operator, Proca-Maxwell equations and relevant field equations are derived in compact, simpler and elegant forms using hyperbolic octonions. This formalism demonstrates that Proca-Maxwell equations can be expressed in a single equation.

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10

SALTI, MUSTAFA, and ALI HAVARE. "ON THE EQUIVALENCE OF THE MASSLESS DKP EQUATION AND THE MAXWELL EQUATIONS IN THE SHUWER." Modern Physics Letters A 20, no. 06 (February 28, 2005): 451–65. http://dx.doi.org/10.1142/s0217732305015768.

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In this paper, a general relativistic wave equation is written to deal with electromagnetic waves in the background of the Shuwer. We obtain the exact form of this equation in a second-order form. On the other hand, by using spinor form of the Maxwell equations the propagation problem is reduced to the solution of the second-order differential equation of complex combination of the electric and magnetic fields. For these two different approaches, we obtain the spinors in terms of field strength tensor. We show that the Maxwell equations are equivalence to the mDKP equation in the Shuwer.

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11

Kiiranen, K., and V. Rosenhaus. "FROM GROUP TO EQUATION. THE MAXWELL EQUATION." Proceedings of the Academy of Sciences of the Estonian SSR. Physics. Mathematics 38, no. 3 (1989): 294. http://dx.doi.org/10.3176/phys.math.1989.3.08.

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12

Hashimoto, H. "Viscoelastic Squeeze Film Characteristics With Inertia Effects Between Two Parallel Circular Plates Under Sinusoidal Motion." Journal of Tribology 116, no. 1 (January 1, 1994): 161–66. http://dx.doi.org/10.1115/1.2927034.

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In this paper, viscoelastic squeeze film characteristics subjected to fluid inertia effects are investigated theoretically in the case of parallel circular type squeeze films. In the development of modified lubrication equations, the nonlinear Maxwell model combining the Rabinowitsch model and Maxwell model is used as a constitutive equation for the viscoelastic fluids, and the inertia term in the momentum equation is approximated by the mean value averaged over the film thickness. Applying the modified lubrication equation to parallel circular type squeeze films under sinusoidal motion, the variation of the pressure distribution with time is calculated numerically for various types of fluids such as Newtonian, pseudo-plastic, linear Maxwell and nonlinear Maxwell fluids. Some numerical results are presented in graphic form, and the effects of inertia forces on the viscoelastic squeeze film characteristics are discussed.

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13

Heiermann, Jörg, and Monika Auweter‐Kurtz. "Discretization of the magnetic field in MPD thrusters." International Journal of Numerical Methods for Heat & Fluid Flow 14, no. 4 (June 1, 2004): 559–72. http://dx.doi.org/10.1108/09615530410532295.

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For the numerical simulation of magnetoplasmadynamic (MPD) self‐field thruster flow, the solution of one of the two dynamical Maxwell equations – Faraday's law – is required. The Maxwell equations and Ohm's law for plasmas can be summarized in one equation for the stream function so that the two‐dimensional, axisymmetric magnetic field can be calculated. The finite volume (FV) discretization of the equation on unstructured, adaptive meshes is presented in detail and solutions for different thruster currents are shown. The calculated thrust is compared with the experimental data.

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14

Chernitskii, Alexander A. "Born-infeld electrodynamics: Clifford number and spinor representations." International Journal of Mathematics and Mathematical Sciences 31, no. 2 (2002): 77–84. http://dx.doi.org/10.1155/s016117120210620x.

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The Clifford number formalism for Maxwell equations is considered. The Clifford imaginary unit for space-time is introduced as coordinate independent form of fully antisymmetric fourth-rank tensor. The representation of Maxwell equations in massless Dirac equation form is considered; we also consider two approaches to the invariance of Dirac equation with respect to the Lorentz transformations. According to the first approach, the unknown column is invariant and according to the second approach it has the transformation properties known as spinorial ones. The Clifford number representation for nonlinear electrodynamics equations is obtained. From this representation, we obtain the nonlinear like Dirac equation which is the form of nonlinear electrodynamics equations. As a special case we have the appropriate representations for Born-Infeld nonlinear electrodynamics.

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15

Verfürth, Barbara. "Heterogeneous Multiscale Method for the Maxwell equations with high contrast." ESAIM: Mathematical Modelling and Numerical Analysis 53, no. 1 (January 2019): 35–61. http://dx.doi.org/10.1051/m2an/2018064.

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In this paper, we suggest a new Heterogeneous Multiscale Method (HMM) for the (time-harmonic) Maxwell scattering problem with high contrast. The method is constructed for a setting as in Bouchitté, Bourel and Felbacq [C.R. Math. Acad. Sci. Paris347(2009) 571–576], where the high contrast in the parameter leads to unusual effective parameters in the homogenized equation. We present a new homogenization result for this special setting, compare it to existing homogenization approaches and analyze the stability of the two-scale solution with respect to the wavenumber and the data. This includes a new stability result for solutions to time-harmonic Maxwell’s equations with matrix-valued, spatially dependent coefficients. The HMM is defined as direct discretization of the two-scale limit equation. With this approach we are able to show quasi-optimality anda priorierror estimates in energy and dual norms under a resolution condition that inherits its dependence on the wavenumber from the stability constant for the analytical problem. This is the first wavenumber-explicit resolution condition for time-harmonic Maxwell’s equations. Numerical experiments confirm our theoretical convergence results.

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16

Dzhunushaliev, Vladimir, and Vladimir Folomeev. "Nonperturbative QED on the Hopf Bundle." Physical Sciences Forum 2, no. 1 (July 22, 2021): 43. http://dx.doi.org/10.3390/ecu2021-09286.

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We consider the Dirac equation and Maxwell’s electrodynamics in ℝ×S3 spacetime, where a three-dimensional sphere is the Hopf bundle S3→S2. The method of nonperturbative quantization of interacting Dirac and Maxwell fields is suggested. The corresponding operator equations and the infinite set of the Schwinger–Dyson equations for Green’s functions is written down. To illustrate the suggested scheme of nonperturbative quantization, we write a simplified set of equations describing some physical situation. Additionally, we discuss the properties of quantum states and operators of interacting fields.

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17

CHUBYKALO, ANDREW E. "ON THE PHYSICAL ORIGIN OF THE OPPENHEIMER–AHLUWALIA ZERO-ENERGY SOLUTIONS OF MAXWELL EQUATIONS." Modern Physics Letters A 13, no. 26 (August 30, 1998): 2139–46. http://dx.doi.org/10.1142/s0217732398002266.

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By virtue of the Chubykalo–Smirnov–Rueda generalized form of the Maxwell–Lorentz equation, a new form of the energy density of the electromagnetic field is obtained. This result allows us to explain a physical origin of the Oppenheimer–Ahluwalia zero-energy solutions of the Maxwell equations.

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18

Goldman, Mark. "On: “New prospects in shallow depth electrical surveying for archeological and pedological applications” by A. Hesse, A. Jolivet and A. Tabbagh (GEOPHYSICS, 51, 585–594, March 1986)." GEOPHYSICS 54, no. 10 (October 1989): 1355. http://dx.doi.org/10.1190/1.1442599.

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The forward solution given by Hesse et al. is incorrect. The error is a result of the erroneous governing differential equation [their equation (2)], which for the only nonzero component of the magnetic field has the following form: [Formula: see text]Unfortunately, the authors did not show how they arrived at this equation, but the mistake is so frequently encountered that its origin can be reconstructed quite easily. Indeed, by neglecting displacement currents in the fourth Maxwell equation and by applying the vector operator ∇× to both parts of the equation, we obtain [Formula: see text]Making use of the well known vector identity [Formula: see text]and of the first and third Maxwell equations, we obtain [Formula: see text]

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19

Koziol, M. J. "Reevaluation and correction of Maxwell’s Equations: a magnetic field has a source, a moving electric charge." F1000Research 9 (September 4, 2020): 1092. http://dx.doi.org/10.12688/f1000research.26035.1.

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Maxwell’s Equations are considered to summarize the world of electromagnetism in four elegant equations. They summarize how electric and magnetic fields propagate, interact, how they are influenced by other objects and what their sources are. While it is widely accepted that the source of a magnetic field is a moving charge, one of the equations instead states that the magnetic field has no source. However, it is widely accepted that a magnetic field cannot be created without a moving electric charge. As such, here, after carefully reevaluating how Maxwell derived his equation, a limitation was identified. After adjustments, a new equation was derived that instead demonstrates that the source of a magnetic field is a moving charge, confirming experimentally verified and widely accepted observations.

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20

Li, Jian, Tiecheng Xia, and Hanyu Wei. "The N-soliton solutions to the Hirota and Maxwell–Bloch equation via the Riemann–Hilbert approach." International Journal of Modern Physics B 35, no. 11 (April 30, 2021): 2150153. http://dx.doi.org/10.1142/s0217979221501538.

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In this paper, we study the [Formula: see text]-soliton solutions for the Hirota and Maxwell–Bloch equation with physical meaning. From the Lax pair and Volterra integral equations, the Riemann–Hilbert problem of this integrable equation is constructed. By solving the matrix Riemann–Hilbert problem with the condition of no reflecting, the [Formula: see text]-soliton solutions for the Hirota and Maxwell–Bloch equation are obtained explicitly. Finally, we simulate the three-dimensional diagram of [Formula: see text] with 2-soliton solutions and the motion trajectory of [Formula: see text]-axis in the case of different [Formula: see text].

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21

Pak, Hee Chul. "Geometric two-scale convergence on forms and its applications to Maxwell's equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 135, no. 1 (February 2005): 133–47. http://dx.doi.org/10.1017/s0308210500003802.

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We develop the geometric two-scale convergence on forms in order to describe the homogenization of partial differential equations with random variables on non-flat domain. We prove the compactness theorem and some two-scale behaviours for differential forms. For its applications, we investigate the limiting equations of the n-dimensional Maxwell equations with random coefficients, with given initial and boundary conditions, where are symmetric positive-definite matrices for x ∈ M, and M is an n-dimensional compact oriented Riemannian manifold with smooth boundary. The limiting system of n-dimensional Maxwell equations turns out to be degenerate and it is proven to be well-posed. The homogenized coefficients affected by the geometry of the domain are presented, and compared with the homogenized coefficient of the second order elliptic equation. We present the convergence theorem in order to explain the convergence of the solutions of Maxwell system as a parabolic partial differential equation.

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22

Bruce, S. A. "Maxwell-Like Equations for Free Dirac Electrons." Zeitschrift für Naturforschung A 73, no. 4 (March 28, 2018): 331–35. http://dx.doi.org/10.1515/zna-2017-0328.

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AbstractIn this article, we show that the wave equation for a free Dirac electron can be represented in a form that is analogous to Maxwell’s electrodynamics. The electron bispinor wavefunction is explicitly expressed in terms of its real and imaginary components. This leads us to incorporate into it appropriate scalar and pseudo-scalar fields in advance, so that a full symmetry may be accomplished. The Dirac equation then takes on a form similar to that of a set of inhomogeneous Maxwell’s equations involving a particular self-source. We relate plane wave solutions of these equations to waves corresponding to free Dirac electrons, identifying the longitudinal component of the electron motion, together with the corresponding Zitterbewegung (“trembling motion”).

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23

Larsson, Jonas. "An action principle for the Vlasov equation and associated Lie perturbation equations. Part 2. The Vlasov–Maxwell system." Journal of Plasma Physics 49, no. 2 (April 1993): 255–70. http://dx.doi.org/10.1017/s0022377800016974.

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An action principle for the Vlasov–Maxwell system in Eulerian field variables is presented. Thus the (extended) particle distribution function appears as one of the fields to be freely varied in the action. The Hamiltonian structures of the Vlasov–Maxwell equations and of the reduced systems associated with small-ampliltude perturbation calculations are easily obtained. Previous results for the linearized Vlasov–Maxwell system are generalized. We find the Hermitian structure also when the background is time-dependent, and furthermore we may now also include the case of non-Hamiltonian perturbations within the Hamiltonian-Hermitian context. The action principle for the Vlasov–Maxwell system appears to be suitable for the derivation of reduced dynamical equations by expanding the action in various small parameters.

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24

SUCU, YUSUF, and NURI UNAL. "SOLUTION OF MASSLESS SPIN ONE WAVE EQUATION IN ROBERTSON–WALKER SPACE–TIME." International Journal of Modern Physics A 17, no. 08 (March 30, 2002): 1137–47. http://dx.doi.org/10.1142/s0217751x02005852.

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We generalize the quantum spinor wave equation for photon into the curved space–time and discuss the solutions of this equation in Robertson–Walker space–time and compare them with the solution of the Maxwell equations in the same space–time.

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25

Benci, Vieri, and Donato Fortunato. "The nonlinear Klein–Gordon equation coupled with the Maxwell equations." Nonlinear Analysis: Theory, Methods & Applications 47, no. 9 (August 2001): 6065–72. http://dx.doi.org/10.1016/s0362-546x(01)00688-5.

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26

HAVARE, ALI, MURAT KORUNUR, OKTAY AYDOGDU, MUSTAFA SALTI, and TAYLAN YETKIN. "EXACT SOLUTIONS OF THE PHOTON EQUATION IN ANISOTROPIC SPACETIMES." International Journal of Modern Physics D 14, no. 06 (June 2005): 957–71. http://dx.doi.org/10.1142/s0218271805006754.

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In this paper we study the solution of the photon equation (the Massless Duffin–Kemmer–Petiau equation (mDKP)) in anisotropic expanding the Bianchi-I type spacetime using the Fourier analyze method. The harmonic oscillator behavior of the solutions is found. It is shown that Maxwell equations are equivalent to the photon equation.

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27

Fedele, Renato. "From Maxwell's theory of Saturn's rings to the negative mass instability." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 366, no. 1871 (January 25, 2008): 1717–33. http://dx.doi.org/10.1098/rsta.2007.2181.

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The impact of Maxwell's theory of Saturn's rings, formulated in Aberdeen ca 1856, is discussed. One century later, Nielsen, Sessler and Symon formulated a similar theory to describe the coherent instabilities (in particular, the negative mass instability) exhibited by a charged particle beam in a high-energy accelerating machine. Extended to systems of particles where the mutual gravitational attraction is replaced by the electric repulsion, Maxwell's approach was the conceptual basis to formulate the kinetic theory of coherent instability (Vlasov–Maxwell system), which, in particular, predicts the stabilizing role of the Landau damping. However, Maxwell's idea was so fertile that, later on, it was extended to quantum-like models (e.g. thermal wave model), providing the quantum-like description of coherent instability (Schrödinger–Maxwell system) and its identification with the modulational instability (MI). The latter has recently been formulated for any nonlinear wave propagation governed by the nonlinear Schrödinger equation, as in the statistical approach to MI (Wigner–Maxwell system). It seems that the above recent developments may provide a possible feedback to Maxwell's original idea with the extension to quantum gravity and cosmology.

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ANTONOV, A. V., B. L. FEIGIN, and A. A. BELOV. "GEOMETRICAL DESCRIPTION OF THE LOCAL INTEGRALS OF MOTION OF MAXWELL-BLOCH EQUATION." Modern Physics Letters A 10, no. 17 (June 7, 1995): 1209–23. http://dx.doi.org/10.1142/s0217732395001332.

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We represent a classical Maxwell-Bloch equation and relate it to positive part of the AKNS hierarchy in geometrical terms. The Maxwell-Bloch evolution is given by an infinitesimal action of a nilpotent subalgebra n+ of affine Lie algebra [Formula: see text] on a Maxwell–Bloch phase space treated as a homogeneous space of n+. A space of local integrals of motion is described using cohomology methods. We show that Hamiltonian flows associated with the Maxwell–Bloch local integrals of motion (i.e. positive AKNS flows) are identified with an infinitesimal action of an Abelian subalgebra of the nilpotent subalgebra n− on a Maxwell–Bloch phase space. Possibilities of quantization and lattice setting of Maxwell–Bloch equation are discussed.

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Akhmeteli, Andrey. "Some Classical Models of Particles and Quantum Gauge Theories." Quantum Reports 4, no. 4 (November 3, 2022): 486–508. http://dx.doi.org/10.3390/quantum4040035.

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The article contains a review and new results of some mathematical models relevant to the interpretation of quantum mechanics and emulating well-known quantum gauge theories, such as scalar electrodynamics (Klein–Gordon–Maxwell electrodynamics), spinor electrodynamics (Dirac–Maxwell electrodynamics), etc. In these models, evolution is typically described by modified Maxwell equations. In the case of scalar electrodynamics, the scalar complex wave function can be made real by a gauge transformation, the wave function can be algebraically eliminated from the equations of scalar electrodynamics, and the resulting modified Maxwell equations describe the independent evolution of the electromagnetic field. Similar results were obtained for spinor electrodynamics. Three out of four components of the Dirac spinor can be algebraically eliminated from the Dirac equation, and the remaining component can be made real by a gauge transformation. A similar result was obtained for the Dirac equation in the Yang–Mills field. As quantum gauge theories play a central role in modern physics, the approach of this article may be sufficiently general. One-particle wave functions can be modeled as plasma-like collections of a large number of particles and antiparticles. This seems to enable the simulation of quantum phase-space distribution functions, such as the Wigner distribution function, which are not necessarily non-negative.

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XIAO, JIANHUA. "FORMULATING JOSEPHSON EFFECTS AND VORTICES BY REFORMULATED MAXWELL EQUATIONS." International Journal of Modern Physics B 23, no. 20n21 (August 20, 2009): 4384–94. http://dx.doi.org/10.1142/s0217979209063535.

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Maxwell equations are not logical consistent. This problem is caused by the implication that the divergence and the curl of a vector are not related. Based on Chen's S - R decomposition of a rank-two tensor, this logical un-consistency is discarded and, as a consequence, the classical Maxwell equations are reformulated to deduce London equations. From boundary field point, the relations between Josephson current and outside magnetic field are established, which shows that the Josephson current is produced by vortices of boundary magnetic field. From local field point, the first London equation corresponds to the local average rotation of electric field and the second London equation corresponds to the local average rotation of magnetic field. The relation between the Josephson effects and vortices of electromagnetic fields is discussed.

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Gumede, Sfundo C., Keshlan S. Govinder, and Sunil D. Maharaj. "Charged Shear-Free Fluids and Complexity in First Integrals." Entropy 24, no. 5 (May 4, 2022): 645. http://dx.doi.org/10.3390/e24050645.

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The equation yxx=f(x)y2+g(x)y3 is the charged generalization of the Emden-Fowler equation that is crucial in the study of spherically symmetric shear-free spacetimes. This version arises from the Einstein–Maxwell system for a charged shear-free matter distribution. We integrate this equation and find a new first integral. For this solution to exist, two integral equations arise as integrability conditions. The integrability conditions can be transformed to nonlinear differential equations, which give explicit forms for f(x) and g(x) in terms of elementary and special functions. The explicit forms f(x)∼1x51−1x−11/5 and g(x)∼1x61−1x−12/5 arise as repeated roots of a fourth order polynomial. This is a new solution to the Einstein-Maxwell equations. Our result complements earlier work in neutral and charged matter showing that the complexity of a charged self-gravitating fluid is connected to the existence of a first integral.

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Sheng, Xin-Li, Yang Li, Shi Pu, and Qun Wang. "Lorentz Transformation in Maxwell Equations for Slowly Moving Media." Symmetry 14, no. 8 (August 9, 2022): 1641. http://dx.doi.org/10.3390/sym14081641.

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We use the method of field decomposition, a widely used technique in relativistic magnetohydrodynamics, to study the small velocity approximation (SVA) of the Lorentz transformation in Maxwell equations for slowly moving media. The “deformed” Maxwell equations derived using SVA in the lab frame can be put into the conventional form of Maxwell equations in the medium’s co-moving frame. Our results show that the Lorentz transformation in the SVA of up to O(v/c) (v is the speed of the medium and c is the speed of light in a vacuum) is essential to derive these equations: the time and charge density must also change when transforming to a different frame, even in the SVA, not just the position and current density, as in the Galilean transformation. This marks the essential difference between the Lorentz transformation and the Galilean one. We show that the integral forms of Faraday and Ampere equations for slowly moving surfaces are consistent with Maxwell equations. We also present Faraday equation in the covariant integral form, in which the electromotive force can be defined as a Lorentz scalar that is independent of the observer’s frame. No evidence exists to support an extension or modification of Maxwell equations.

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Niu, Chao, Yu Tian, Xiao-Ning Wu, and Yi Ling. "Incompressible Navier–Stokes equation from Einstein–Maxwell and Gauss–Bonnet–Maxwell theories." Physics Letters B 711, no. 5 (May 2012): 411–16. http://dx.doi.org/10.1016/j.physletb.2012.04.029.

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Sheu, Tony W. H., S. Z. Wang, J. H. Li, and Matthew R. Smith. "Simulation of Maxwell's Equations on GPU Using a High-Order Error-Minimized Scheme." Communications in Computational Physics 21, no. 4 (March 8, 2017): 1039–64. http://dx.doi.org/10.4208/cicp.oa-2016-0079.

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AbstractIn this study an explicit Finite Difference Method (FDM) based scheme is developed to solve the Maxwell's equations in time domain for a lossless medium. This manuscript focuses on two unique aspects – the three dimensional time-accurate discretization of the hyperbolic system of Maxwell equations in three-point non-staggered grid stencil and it's application to parallel computing through the use of Graphics Processing Units (GPU). The proposed temporal scheme is symplectic, thus permitting conservation of all Hamiltonians in the Maxwell equation. Moreover, to enable accurate predictions over large time frames, a phase velocity preserving scheme is developed for treatment of the spatial derivative terms. As a result, the chosen time increment and grid spacing can be optimally coupled. An additional theoretical investigation into this pairing is also shown. Finally, the application of the proposed scheme to parallel computing using one Nvidia K20 Tesla GPU card is demonstrated. For the benchmarks performed, the parallel speedup when compared to a single core of an Intel i7-4820K CPU is approximately 190x.

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Ene, Remus-Daniel, Nicolina Pop, Marioara Lapadat, and Luisa Dungan. "Approximate Closed-Form Solutions for the Maxwell-Bloch Equations via the Optimal Homotopy Asymptotic Method." Mathematics 10, no. 21 (November 4, 2022): 4118. http://dx.doi.org/10.3390/math10214118.

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This paper emphasizes some geometrical properties of the Maxwell–Bloch equations. Based on these properties, the closed-form solutions of their equations are established. Thus, the Maxwell–Bloch equations are reduced to a nonlinear differential equation depending on an auxiliary unknown function. The approximate analytical solutions were built using the optimal homotopy asymptotic method (OHAM). These represent the ε-approximate OHAM solutions. A good agreement between the analytical and corresponding numerical results was found. The accuracy of the obtained results is validated through the representative figures. This procedure is suitable to be applied for dynamical systems with certain geometrical properties.

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Benci, Vieri, and Donato Fortunato. "A strongly degenerate elliptic equation arising from the semilinear Maxwell equations." Comptes Rendus Mathematique 339, no. 12 (December 2004): 839–42. http://dx.doi.org/10.1016/j.crma.2004.07.029.

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Besse, Nicolas, and Philippe Bechouche. "Regularity of weak solutions for the relativistic Vlasov–Maxwell system." Journal of Hyperbolic Differential Equations 15, no. 04 (December 2018): 693–719. http://dx.doi.org/10.1142/s0219891618500212.

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We investigate the regularity of weak solutions of the relativistic Vlasov–Maxwell system by using Fourier analysis and the smoothing effect of low velocity particles. This smoothing effect has been used by several authors (see Glassey and Strauss 1986; Klainerman and Staffilani, 2002) for proving existence and uniqueness of [Formula: see text]-regular solutions of the Vlasov–Maxwell system. This smoothing mechanism has also been used to study the regularity of solutions for a kinetic transport equation coupled with a wave equation (see Bouchut, Golse and Pallard 2004). Under the same assumptions as in the paper “Nonresonant smoothing for coupled wave[Formula: see text]+[Formula: see text]transport equations and the Vlasov–Maxwell system”, Rev. Mat. Iberoamericana 20 (2004) 865–892, by Bouchut, Golse and Pallard, we prove a slightly better regularity for the electromagnetic field than the one showed in the latter paper. Namely, we prove that the electromagnetic field belongs to [Formula: see text], with [Formula: see text].

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De Alwis, A. C. Wimal Lalith. "Solution to Stokes-Maxwell-Euler Differential Equation." Applied Mathematics 08, no. 03 (2017): 410–16. http://dx.doi.org/10.4236/am.2017.83033.

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Vedenyapin, Victor Valentinovich, and Ilya Sergeevich Pershin. "Vlasov-Maxwell-Einstein Equation and Einstein Lambda." Keldysh Institute Preprints, no. 39-e (2019): 1–17. http://dx.doi.org/10.20948/prepr-2019-39-e.

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Petrosky, T. "Stochastic Maxwell-Lorentz equation in radiation damping." International Journal of Quantum Chemistry 98, no. 2 (2004): 103–11. http://dx.doi.org/10.1002/qua.10832.

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Feroze, Tooba. "Exact solutions of the Einstein–Maxwell equations with linear equation of state." Canadian Journal of Physics 90, no. 12 (December 2012): 1179–83. http://dx.doi.org/10.1139/p2012-067.

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Two new classes of solutions of the Einstein–Maxwell field equations are obtained by substituting a general linear equation of state into the energy–momentum conservation equation. We have considered static, anisotropic, and spherically symmetric charged perfect fluid distribution of matter with a particular form of gravitational potential. Expressions for the mass–radius ratio, the surface, and the central red shift horizons are given for these solutions.

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Konobeeva, N. N., and M. B. Belonenko. "Extremely short optical pulses in the presence of dilatons." Izvestiya vysshikh uchebnykh zavedenii. Fizika, no. 1 (2022): 165–69. http://dx.doi.org/10.17223/00213411/65/1/165.

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The propagation of an extremely short optical pulse is analyzed based on a numerical solution of the Maxwell`s equation related to the dilaton field in a flat space-time. The dynamics of the pulse turned out to be unstable and the pulse collapses. The influence of the parameter α of the Lagrangian is analyzed for the cases of the Einstein-Maxwell scalar theory, low-energy action of string theory, Kaluza-Klein field equations obtained from dimensional reduction of Einstein’s five-dimensional theory.

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Tuomela, Jukka. "Fourth-order schemes for the wave equation, Maxwell equations, and linearized elastodynamic equations." Numerical Methods for Partial Differential Equations 10, no. 1 (January 1994): 33–63. http://dx.doi.org/10.1002/num.1690100104.

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Ge, Jianchao, Mark E. Everett, and Chester J. Weiss. "Fractional diffusion analysis of the electromagnetic field in fractured media Part I: 2D approach." GEOPHYSICS 77, no. 4 (July 1, 2012): WB213—WB218. http://dx.doi.org/10.1190/geo2012-0072.1.

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We address a 2D finite difference (FD) frequency-domain modeling algorithm based on the theory of fractional diffusion of electromagnetic (EM) fields, which is generated by an infinite line source lying above a fractured geological medium. The presence of fractures in the subsurface, usually containing highly conductive pore fluids, gives rise to spatially hierarchical flow paths of induced EM eddy currents. The diffusion of EM eddy currents in such formations is anomalous, generalizing the classical Gaussian process described by the conventional Maxwell equations. Based on the continuous time random walk (CTRW) theory, the diffusion of EM eddy currents in a rough medium is governed by the fractional Maxwell equations. Here, we model the EM response of a 2D subsurface containing fractured zones, based on the fractional Maxwell equations. The governing equation in the frequency domain is discretized using the FD approach. The resulting equation system is solved by the multifrontal massively parallel solver (MUMPS). We find excellent agreement between the FD and analytic solutions for a rough half-space model. Then, FD solutions are calculated for a 2D fault zone model with variable conductivity and roughness. We illustrate a case in which a rough fault zone would not be resolved by classical diffusion modeling, even if its conductivity contrasts with the background.

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Kremer, Gilberto Medeiros. "Post-Newtonian Jeans Equation for Stationary and Spherically Symmetrical Self-Gravitating Systems." Universe 8, no. 3 (March 13, 2022): 179. http://dx.doi.org/10.3390/universe8030179.

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The post-Newtonian Jeans equation for stationary self-gravitating systems is derived from the post-Newtonian Boltzmann equation in spherical coordinates. The Jeans equation is coupled with the three Poisson equations from the post-Newtonian theory. The Poisson equations are functions of the energy-momentum tensor components which are determined from the post-Newtonian Maxwell–Jüttner distribution function. As an application, the effect of a central massive black hole on the velocity dispersion profile of the host galaxy is investigated and the influence of the post-Newtonian corrections are determined.

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PAPINI, G. "ELECTRODYNAMICS OF THE DIRAC FIELD." Modern Physics Letters A 03, no. 02 (January 1988): 139–45. http://dx.doi.org/10.1142/s0217732388000179.

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Classical electrodynamical models are constructed in which both vector bosons and fermions are generated by massless, topologically singular scalar fields. The Dirac equation supplies Maxwell equations and the constraint necessary to obtain from them charged fermions with stringlike structure.

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Yin, Ze, Yong Jun Jian, Long Chang, Ren Na, and Quan Sheng Liu. "Transient AC Electro-Osmotic Flow of Generalized Maxwell Fluids through Microchannels." Applied Mechanics and Materials 548-549 (April 2014): 216–23. http://dx.doi.org/10.4028/www.scientific.net/amm.548-549.216.

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In this paper, we represent analytical solutions of transient velocity for electroosmotic flow (EOF) of generalized Maxwell fluids through both micro-parallel channel and micro-tube using the method of Laplace transform. We solve the problem including the linearized Poisson-Boltzmann equation, the Cauchy momentum equation and generalized Maxwell constitutive equation. By numerical calculation, the results show that the EOF velocity is greatly depends on oscillating Reynolds number and normalized relaxation time.

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Hojman, Sergio A., and Felipe A. Asenjo. "Unification of massless field equations solutions for any spin." Europhysics Letters 137, no. 2 (January 1, 2022): 24001. http://dx.doi.org/10.1209/0295-5075/ac4621.

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Abstract A unification in terms of exact solutions for massless Klein-Gordon, Dirac, Maxwell, Rarita-Schwinger, Einstein, and bosonic and fermionic fields of any spin is presented. The method is based on writing all of the relevant dynamical fields in terms of products and derivatives of pre-potential functions, which satisfy the d'Alembert equation. The coupled equations satisfied by the pre-potentials are non-linear. Remarkably, there are particular solutions of (gradient) orthogonal pre-potentials that satisfy the usual wave equation which may be used to construct exact non-trivial solutions to Klein-Gordon, Dirac, Maxwell, Rarita-Schwinger, (linearized and full) Einstein and any spin bosonic and fermionic field equations, thus giving rise to a unification of the solutions of all massless field equations for any spin. Some solutions written in terms of orthogonal pre-potentials are presented. Relations of this method to previously developed ones, as well as to other subjects in physics are pointed out.

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TERUEL, GINÉS R. PÉREZ. "GENERALIZED EINSTEIN–MAXWELL FIELD EQUATIONS IN THE PALATINI FORMALISM." International Journal of Modern Physics D 22, no. 04 (March 2013): 1350017. http://dx.doi.org/10.1142/s021827181350017x.

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We derive a new set of field equations within the framework of the Palatini formalism. These equations are a natural generalization of the Einstein–Maxwell equations which arise by adding a function [Formula: see text], with [Formula: see text] to the Palatini Lagrangian f(R, Q). The result we obtain can be viewed as the coupling of gravity with a nonlinear extension of the electromagnetic field. In addition, a new method is introduced to solve the algebraic equation associated to the Ricci tensor.

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YANYUSHKINA, NATALIA N., MIKHAIL B. BELONENKO, NIKOLAY G. LEBEDEV, ALEXANDER V. ZHUKOV, and MAXIM PALIY. "EXTREMELY SHORT OPTICAL PULSES IN CARBON NANOTUBES IN DISPERSIVE NONMAGNETIC DIELECTRIC MEDIA." International Journal of Modern Physics B 25, no. 25 (October 10, 2011): 3401–8. http://dx.doi.org/10.1142/s0217979211101818.

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We consider Maxwell equations for an electromagnetic field propagating in carbon nanotubes (CNTs) placed on a dispersive nonmagnetic dielectric medium. We obtain the effective equation analogous to the classical sine-Gordon equation. Then it has been analyzed numerically. We have revealed the dependence of the pulse on the type of CNT and on the initial pulse amplitude, as well as on the medium dispersion constants.

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