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Journal articles on the topic 'Time-Dependent Maxwell's equations'

Author: Grafiati

Published: 25 January 2025

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1

Huang, Yunqing, Jichun Li, and Qun Lin. "Superconvergence analysis for time-dependent Maxwell's equations in metamaterials." Numerical Methods for Partial Differential Equations 28, no. 6 (September 1, 2011): 1794–816. http://dx.doi.org/10.1002/num.20703.

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2

Feliziani, M., and F. Maradei. "Hybrid finite element solutions of time dependent Maxwell's curl equations." IEEE Transactions on Magnetics 31, no. 3 (May 1995): 1330–35. http://dx.doi.org/10.1109/20.376273.

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3

Ciarlet, Jr, P., and Jun Zou. "Fully discrete finite element approaches for time-dependent Maxwell's equations." Numerische Mathematik 82, no. 2 (April 1, 1999): 193–219. http://dx.doi.org/10.1007/s002110050417.

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4

Łoś, Marcin, Maciej Woźniak, Keshav Pingali, Luis Emilio Garcia Castillo, Julen Alvarez-Arramberri, David Pardo, and Maciej Paszyński. "Fast parallel IGA-ADS solver for time-dependent Maxwell's equations." Computers & Mathematics with Applications 151 (December 2023): 36–49. http://dx.doi.org/10.1016/j.camwa.2023.09.035.

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5

Egger, Herbert, Fritz Kretzschmar, Sascha M. Schnepp, and Thomas Weiland. "A Space-Time Discontinuous Galerkin Trefftz Method for Time Dependent Maxwell's Equations." SIAM Journal on Scientific Computing 37, no. 5 (January 2015): B689—B711. http://dx.doi.org/10.1137/140999323.

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6

Holland, Peter. "Hydrodynamic construction of the electromagnetic field." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 461, no. 2063 (September 19, 2005): 3659–79. http://dx.doi.org/10.1098/rspa.2005.1525.

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We present an alternative Eulerian hydrodynamic model for the electromagnetic field in which the discrete vector indices in Maxwell's equations are replaced by continuous angular freedoms, and develop the corresponding Lagrangian picture in which the fluid particles have rotational and translational freedoms. This enables us to extend to the electromagnetic field the exact method of state construction proposed previously for spin 0 systems, in which the time-dependent wavefunction is computed from a single-valued continuum of deterministic trajectories where two spacetime points are linked by at most a single orbit. The deduction of Maxwell's equations from continuum mechanics is achieved by generalizing the spin 0 theory to a general Riemannian manifold from which the electromagnetic construction is extracted as a special case. In particular, the flat-space Maxwell equations are represented as a curved-space Schrödinger equation for a massive system. The Lorentz covariance of the Eulerian field theory is obtained from the non-covariant Lagrangian-coordinate model as a kind of collective effect. The method makes manifest the electromagnetic analogue of the quantum potential that is tacit in Maxwell's equations. This implies a novel definition of the ‘classical limit’ of Maxwell's equations that differs from geometrical optics. It is shown that Maxwell's equations may be obtained by canonical quantization of the classical model. Using the classical trajectories a novel expression is derived for the propagator of the electromagnetic field in the Eulerian picture. The trajectory and propagator methods of solution are illustrated for the case of a light wave.

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7

Benoit, J., C. Chauvière, and P. Bonnet. "Time-dependent current source identification for numerical simulations of Maxwell's equations." Journal of Computational Physics 289 (May 2015): 116–28. http://dx.doi.org/10.1016/j.jcp.2015.02.033.

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8

Zhang, Ya, Li-Qun Cao, and Yau-Shu Wong. "Multiscale Computations for 3D Time-Dependent Maxwell's Equations in Composite Materials." SIAM Journal on Scientific Computing 32, no. 5 (January 2010): 2560–83. http://dx.doi.org/10.1137/080740337.

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9

Li, Jichun, and Yitung Chen. "Finite element study of time-dependent Maxwell's equations in dispersive media." Numerical Methods for Partial Differential Equations 24, no. 5 (December 14, 2007): 1203–21. http://dx.doi.org/10.1002/num.20314.

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10

Yao, Changhui, and Dongyang Shi. "Nonconforming Mixed Finite Element Method for Time-dependent Maxwell's Equations with ABC." Numerical Mathematics: Theory, Methods and Applications 9, no. 2 (May 2016): 193–214. http://dx.doi.org/10.4208/nmtma.2016.m1427.

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AbstractIn this paper, a nonconforming mixed finite element method (FEM) is presented to approximate time-dependent Maxwell's equations in a three-dimensional bounded domain with absorbing boundary conditions (ABC). By employing traditional variational formula, instead of adding penalty terms, we show that the discrete scheme is robust. Meanwhile, with the help of the element's typical properties and derivative transfer skills, the convergence analysis and error estimates for semidiscrete and backward Euler fully-discrete schemes are given, respectively. Numerical tests show the validity of the proposed method.

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11

Shi, Dongyang, and Changhui Yao. "Nonconforming finite element approximation of time-dependent Maxwell's equations in Debye medium." Numerical Methods for Partial Differential Equations 30, no. 5 (March 17, 2014): 1654–73. http://dx.doi.org/10.1002/num.21843.

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12

Bushnaq, Samia, Asif Ullah Hayat, and Hassan Khan. "Numerical simulation of time-dependent viscous fluid flow with upward and downward fluctuation of spinning disk." Boletim da Sociedade Paranaense de Matemática 42 (May 28, 2024): 1–12. http://dx.doi.org/10.5269/bspm.63089.

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The parametric approach towards time-dependent viscous fluid flow across a gyrating disk with upward and downward fluctuation. The major goal of this research is to assess fluid flow under the influence of magnetic fields and heat propagation processes. Because they provide a thorough description of electromagnetic interactions. Maxwell's equations are at the heart of all contemporary information and communication technologies. The governing equations comprising Navier Stokes equation, energy, concentration, and Maxwell equations have been represented appropriately for this purpose. The governing equations are turned down to the system of non-linear ODEs through a resemblance framework. The obtained system of differential equations has been resolved via numerical procedure Parametric Continuation Method (PCM). For the scale reliability purpose, the outcomes are compared to another numerical Matlab scheme boundary value solver. In the current analysis, the presence of convective boundary conditions correlated to mass and energy is of physical relevance. The numerical findings are provided in tabular and graphical forms. The consequences of suction and wall injection have been also highlighted. The upward motion of the spinning disc is thought to lead to comparable findings as in an injection scenario, whilst the downhill motion is thought to contribute to wall suction-like effects.

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13

Hampshire, Damian P. "A derivation of Maxwell's equations using the Heaviside notation." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376, no. 2134 (October 29, 2018): 20170447. http://dx.doi.org/10.1098/rsta.2017.0447.

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Maxwell's four differential equations describing electromagnetism are among the most famous equations in science. Feynman said that they provide four of the seven fundamental laws of classical physics. In this paper, we derive Maxwell's equations using a well-established approach for deriving time-dependent differential equations from static laws. The derivation uses the standard Heaviside notation. It assumes conservation of charge and that Coulomb's law of electrostatics and Ampere's law of magnetostatics are both correct as a function of time when they are limited to describing a local system. It is analogous to deriving the differential equation of motion for sound, assuming conservation of mass, Newton's second law of motion and that Hooke's static law of elasticity holds for a system in local equilibrium. This work demonstrates that it is the conservation of charge that couples time-varying E -fields and B -fields and that Faraday's Law can be derived without any relativistic assumptions about Lorentz invariance. It also widens the choice of axioms, or starting points, for understanding electromagnetism. This article is part of the theme issue ‘Celebrating 125 years of Oliver Heaviside's ‘Electromagnetic Theory’’.

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14

Fujii, M., and W. J. R. Hoefer. "Application of biorthogonal interpolating wavelets to the Galerkin scheme of time dependent Maxwell's equations." IEEE Microwave and Wireless Components Letters 11, no. 1 (January 2001): 22–24. http://dx.doi.org/10.1109/7260.905956.

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15

Dosopoulos, Stylianos, and Jin-Fa Lee. "Interior Penalty Discontinuous Galerkin Finite Element Method for the Time-Dependent First Order Maxwell's Equations." IEEE Transactions on Antennas and Propagation 58, no. 12 (December 2010): 4085–90. http://dx.doi.org/10.1109/tap.2010.2078445.

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16

Yijun Lu and C. Y. Shen. "A domain decomposition finite-difference method for parallel numerical implementation of time-dependent Maxwell's equations." IEEE Transactions on Antennas and Propagation 45, no. 3 (March 1997): 556–62. http://dx.doi.org/10.1109/8.558671.

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17

Ma, Changfeng. "Finite-element method for time-dependent Maxwell's equations based on an explicit-magnetic-field scheme." Journal of Computational and Applied Mathematics 194, no. 2 (October 2006): 409–24. http://dx.doi.org/10.1016/j.cam.2005.08.008.

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18

Chen, Shuqi, Weiping Zang, Axel Schülzgen, Jinjie Liu, Lin Han, Yong Zeng, Jianguo Tian, Feng Song, Jerome V. Moloney, and Nasser Peyghambarian. "Implicit high-order unconditionally stable complex envelope algorithm for solving the time-dependent Maxwell's equations." Optics Letters 33, no. 23 (November 19, 2008): 2755. http://dx.doi.org/10.1364/ol.33.002755.

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19

El Barkani, Imad, and Mohamed Addam. "Splines finite element solver for one-dimensional time-dependent Maxwell's equations via Fourier Transform Discretization." Boletim da Sociedade Paranaense de Matemática 42 (May 22, 2024): 1–26. http://dx.doi.org/10.5269/bspm.65922.

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In this article, we solve the time-dependent Maxwell coupled equations in their one-dimensional version relatively to space-variable. We effectuate a variable reduction via Fourier transform to make the time variable as a frequency parameter easy and quickly to manage. A Galerkin variational method based on higher-order spline interpolations is used to approximate the solution relatively to the spacial variable. So, the state of existence of the solution, its uniqueness, and its regularity are studied and proved, and the study is also provided by an error estimate and the order of convergence of the proposed method. Also, we use the critical Nyquist frequency to calculate numerically the solution of the Inverse Fourier Transform(IFT); and for all numerical computations, we consider several quadrature methods. Finally, we give some experiments to illustrate the success of such an approach.

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20

Gibson, Nathan L. "A Polynomial Chaos Method for Dispersive Electromagnetics." Communications in Computational Physics 18, no. 5 (November 2015): 1234–63. http://dx.doi.org/10.4208/cicp.230714.100315a.

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AbstractElectromagnetic wave propagation in complex dispersive media is governed by the time dependent Maxwell's equations coupled to equations that describe the evolution of the induced macroscopic polarization. We consider “polydispersive” materials represented by distributions of dielectric parameters in a polarization model. The work focuses on a novel computational framework for such problems involving Polynomial Chaos Expansions as a method to improve the modeling accuracy of the Debye model and allow for easy simulation using the Finite Difference Time Domain (FDTD) method. Stability and dispersion analyzes are performed for the approach utilizing the second order Yee scheme in two spatial dimensions.

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21

Xie, Ziqing, Jiangxing Wang, Bo Wang, and Chuanmiao Chen. "Solving Maxwell's Equation in Meta-Materials by a CG-DG Method." Communications in Computational Physics 19, no. 5 (May 2016): 1242–64. http://dx.doi.org/10.4208/cicp.scpde14.35s.

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AbstractIn this paper, an approach combining the DG method in space with CG method in time (CG-DG method) is developed to solve time-dependent Maxwell's equations when meta-materials are involved. Both the unconditional L2-stability and error estimate of order are obtained when polynomials of degree at most r is used for the temporal discretization and at most k for the spatial discretization. Numerical results in 3D are given to validate the theoretical results.

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22

Galagusz, Ryan, David Shirokoff, and Jean-Christophe Nave. "A Fourier penalty method for solving the time-dependent Maxwell's equations in domains with curved boundaries." Journal of Computational Physics 306 (February 2016): 167–98. http://dx.doi.org/10.1016/j.jcp.2015.11.031.

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23

Cakoni, Fioralba, Shixu Meng, and Jingni Xiao. "A note on transmission eigenvalues in electromagnetic scattering theory." Inverse Problems & Imaging 15, no. 5 (2021): 999. http://dx.doi.org/10.3934/ipi.2021025.

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<p style='text-indent:20px;'>This short note was motivated by our efforts to investigate whether there exists a half plane free of transmission eigenvalues for Maxwell's equations. This question is related to solvability of the time domain interior transmission problem which plays a fundamental role in the justification of linear sampling and factorization methods with time dependent data. Our original goal was to adapt semiclassical analysis techniques developed in [<xref ref-type="bibr" rid="b21">21</xref>,<xref ref-type="bibr" rid="b23">23</xref>] to prove that for some combination of electromagnetic parameters, the transmission eigenvalues lie in a strip around the real axis. Unfortunately we failed. To try to understand why, we looked at the particular example of spherically symmetric media, which provided us with some insight on why we couldn't prove the above result. Hence this paper reports our findings on the location of all transmission eigenvalues and the existence of complex transmission eigenvalues for Maxwell's equations for spherically stratified media. We hope that these results can provide reasonable conjectures for general electromagnetic media.</p>

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24

Yu, Mengjun, and Kun Li. "A data-driven reduced-order modeling approach for parameterized time-domain Maxwell's equations." Networks and Heterogeneous Media 19, no. 3 (2024): 1309–35. http://dx.doi.org/10.3934/nhm.2024056.

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<p>This paper proposed a data-driven non-intrusive model order reduction (NIMOR) approach for parameterized time-domain Maxwell's equations. The NIMOR method consisted of fully decoupled offline and online stages. Initially, the high-fidelity (HF) solutions for some training time and parameter sets were obtained by using a discontinuous Galerkin time-domain (DGTD) method. Subsequently, a two-step or nested proper orthogonal decomposition (POD) technique was used to generate the reduced basis (RB) functions and the corresponding projection coefficients within the RB space. The high-order dynamic mode decomposition (HODMD) method leveraged these corresponding coefficients to predict the projection coefficients at all training parameters over a time region beyond the training domain. Instead of direct regression and interpolating new parameters, the predicted projection coefficients were reorganized into a three-dimensional tensor, which was then decomposed into time- and parameter-dependent components through the canonical polyadic decomposition (CPD) method. Gaussian process regression (GPR) was then used to approximate the relationship between the time/parameter values and the above components. Finally, the reduced-order solutions at new time/parameter values were quickly obtained through a linear combination of the POD modes and the approximated projection coefficients. Numerical experiments were presented to evaluate the performance of the method in the case of plane wave scattering.</p>

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25

Cao, Liqun, Keqi Li, Jianlan Luo, and Yaushu Wong. "A Multiscale Approach and a Hybrid FE-FDTD Algorithm for 3D Time-Dependent Maxwell's Equations in Composite Materials." Multiscale Modeling & Simulation 13, no. 4 (January 2015): 1446–77. http://dx.doi.org/10.1137/140999694.

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26

Araújo, Adérito, Sílvia Barbeiro, and Maryam Khaksar Ghalati. "Stability of a Leap-Frog Discontinuous Galerkin Method for Time-Domain Maxwell's Equations in Anisotropic Materials." Communications in Computational Physics 21, no. 5 (March 27, 2017): 1350–75. http://dx.doi.org/10.4208/cicp.oa-2016-0110.

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AbstractIn this work we discuss the numerical discretization of the time-dependent Maxwell's equations using a fully explicit leap-frog type discontinuous Galerkin method. We present a sufficient condition for the stability and error estimates, for cases of typical boundary conditions, either perfect electric, perfect magnetic or first order Silver-Müller. The bounds of the stability region point out the influence of not only the mesh size but also the dependence on the choice of the numerical flux and the degree of the polynomials used in the construction of the finite element space, making possible to balance accuracy and computational efficiency. In the model we consider heterogeneous anisotropic permittivity tensors which arise naturally in many applications of interest. Numerical results supporting the analysis are provided.

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27

Chun, Sehun. "Method of moving frames to solve time-dependent Maxwell's equations on anisotropic curved surfaces: Applications to invisible cloak and ELF propagation." Journal of Computational Physics 340 (July 2017): 85–104. http://dx.doi.org/10.1016/j.jcp.2017.03.031.

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28

Tosti, Fabio, and Andrea Umiliaco. "FDTD Simulation of the GPR Signal for Preventing the Risk of Accidents due to Pavement Damages." International Journal of Interdisciplinary Telecommunications and Networking 6, no. 1 (January 2014): 1–9. http://dx.doi.org/10.4018/ijitn.2014010101.

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It is well known that road safety issues are closely dependent on both pavement structural damages and surface unevenness, whose occurrence is often related to ineffective pavement asset management. The evaluation of road pavement operability is traditionally carried out through distress identification manuals on the basis of standardized comprehensive indexes, as a result of visual inspections or measurements, wherein the failure causes can be partially detected. In this regard, ground-penetrating radar (GPR) has proven to be over the past decades an effective and efficient technique to enable better management of pavement assets and better diagnosis of the causes of pavement failures. In this study, one of the main causes (i.e. subgrade failures) of surface damage is analyzed through finite-difference time-domain (FDTD) simulation of the GPR signal. The GprMax 2D numerical simulator for GPR is used on three different types of flexible pavement to retrieve the numerical solution of Maxwell's equations in the time domain. Results show the high potential of GPR in detecting the causes of such damage.

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29

Daveau, Christian, Diane Manuel Douady, Abdessatar Khelifi, and Anton Sushchenko. "Numerical solution of an inverse initial boundary-value problem for the full time-dependent Maxwell's equations in the presence of imperfections of small volume." Applicable Analysis 92, no. 5 (May 2013): 975–96. http://dx.doi.org/10.1080/00036811.2011.643782.

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30

Wang, Xiang. "Analysis and Application of Single Level, Multi-Level Monte Carlo and Quasi-Monte Carlo Finite Element Methods for Time-Dependent Maxwell's Equations with Random Inputs." Communications in Computational Physics 29, no. 1 (June 2021): 211–36. http://dx.doi.org/10.4208/cicp.oa-2020-0011.

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31

Kowalski, Marian. "The quantum and electromagnetic process of photon emission by the hydrogen atom." Physics Essays 34, no. 2 (June 7, 2021): 116–49. http://dx.doi.org/10.4006/0836-1398-34.2.116.

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Light emitted from atoms during transitions of electrons from higher to lower discrete states has the form of photons carrying energy and angular momentum. This paper considers the process of emission of a single photon from the hydrogen atom by using quantum theory and Maxwell's equations [W. Gough, Eur. J. Phys. 17, 208, 1996; L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon Press, Oxford, 1965); J. D. Jackson, Classical Electrodynamics (John Wiley & Son, New York, 1975, 1982); P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York, 1953)]. The electric and magnetic fields of a photon arise from the time-dependent quantum probability densities of the orbit and the spin current. This paper is an extension of the semiclassical description of photon emission published by the author earlier in 1999 [M. Kowalski, Phys. Essays 12, 312 (1999)]. In the semiclassical approach, the Coulomb force and a radiation resistance force have been taken into account to get time-dependent emission of the photon. In both the quantum and semiclassical cases, the transition takes place within a time interval equal to one period of the photon's wave. The creation of a one-wavelength-long photon is supported by the results of experiments using ultrafast (ultrashort) laser pulses to generate excited atoms, which emit light pulses shorter than two photon wavelengths [F. Krausz and M. Ivanov, Rev. Mod. Phys. 81, 163 (2009); H. Kapteyn and M. Murnane, Phys. World 12, 31 (1999)].

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32

Stamm, Johann, Juha Vierinen, and Björn Gustavsson. "Observing electric field and neutral wind with EISCAT 3D." Annales Geophysicae 39, no. 6 (November 16, 2021): 961–74. http://dx.doi.org/10.5194/angeo-39-961-2021.

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Abstract. Measurements of height-dependent electric field (E) and neutral wind (u) are important governing parameters of the Earth's upper atmosphere, which can be used to study, for example, how auroral currents close or how energy flows between the ionized and neutral constituents. The new EISCAT 3D (E3D) incoherent scatter radar will be able to measure a three-dimensional ion velocity vector (v) at each measurement point, which will allow less stringent prior assumptions about E and u to be made when estimating them from radar measurements. This study investigates the feasibility of estimating the three-dimensional electric field and neutral wind vectors along a magnetic field-aligned profile from E3D measurements, using the ion momentum equation and Maxwell's equations. The uncertainty of ion drift measurements is estimated for a time and height resolution of 5 s and 2 km. With the most favourable ionospheric conditions, the ion wind at E region peak can be measured with an accuracy of less than 1 m/s. In the worst case, during a geomagnetically quiet night, the uncertainty increases by a factor of around 10. The uncertainty of neutral wind and electric field estimates is found to be strongly dependent on the prior constraints imposed on them. In the lower E region, neutral wind estimates have a lower standard deviation than 10 m/s in the most favourable conditions. In such conditions, also the F region electric field can be estimated with uncertainty of about 1 mV/m. Simulated measurements of v are used to demonstrate the ability to resolve the field-aligned profile of E and u. However, they can only be determined well at the heights where they dominate the ion drift, that is above 125 km for E and below 115 km for u. At the other heights, the results are strongly dependent on the prior assumptions of smoothness.

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33

Shields, Sidney, Jichun Li, and Eric A. Machorro. "Weak Galerkin methods for time-dependent Maxwell’s equations." Computers & Mathematics with Applications 74, no. 9 (November 2017): 2106–24. http://dx.doi.org/10.1016/j.camwa.2017.07.047.

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34

Tsutsumi, Masayoshi, and Hironori Kasai. "The time-dependent Ginzburg–Landau Maxwell equations." Nonlinear Analysis: Theory, Methods & Applications 37, no. 2 (July 1999): 187–216. http://dx.doi.org/10.1016/s0362-546x(98)00043-1.

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35

Assous, F., P. Ciarlet, E. Garcia, and J. Segré. "Time-dependent Maxwell’s equations with charges in singular geometries." Computer Methods in Applied Mechanics and Engineering 196, no. 1-3 (December 2006): 665–81. http://dx.doi.org/10.1016/j.cma.2006.07.007.

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36

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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37

Assous, Franck, and Irina Raichik. "Numerical Solution to the 3D Static Maxwell Equations in Axisymmetric Singular Domains with Arbitrary Data." Computational Methods in Applied Mathematics 20, no. 3 (July 1, 2020): 419–35. http://dx.doi.org/10.1515/cmam-2018-0314.

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AbstractWe propose a numerical method to solve the three-dimensional static Maxwell equations in a singular axisymmetric domain, generated by the rotation of a singular polygon around one of its sides. The mathematical tools and an in-depth study of the problem set in the meridian half-plane are exposed in [F. Assous, P. Ciarlet, Jr., S. Labrunie and J. Segré, Numerical solution to the time-dependent Maxwell equations in axisymmetric singular domains: the singular complement method, J. Comput. Phys. 191 2003, 1, 147–176] and [P. Ciarlet, Jr. and S. Labrunie, Numerical solution of Maxwell’s equations in axisymmetric domains with the Fourier singular complement method, Differ. Equ. Appl. 3 2011, 1, 113–155]. Here, we derive a variational formulation and the corresponding approximation method. Numerical experiments are proposed, and show that the approach is able to capture the singular part of the solution. This article can also be viewed as a generalization of the Singular Complement Method to three-dimensional axisymmetric problems.

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38

Berti, Valeria, and Stefania Gatti. "Parabolic-hyperbolic time-dependent Ginzburg-Landau-Maxwell equations." Quarterly of Applied Mathematics 64, no. 4 (October 16, 2006): 617–39. http://dx.doi.org/10.1090/s0033-569x-06-01044-9.

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39

Bommer, Vera, and Irwin Yousept. "Optimal control of the full time-dependent maxwell equations." ESAIM: Mathematical Modelling and Numerical Analysis 50, no. 1 (January 2016): 237–61. http://dx.doi.org/10.1051/m2an/2015041.

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40

Campos Pinto, Martin, and Eric Sonnendrücker. "Gauss-compatible Galerkin schemes for time-dependent Maxwell equations." Mathematics of Computation 85, no. 302 (February 15, 2016): 2651–85. http://dx.doi.org/10.1090/mcom/3079.

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41

Weinan, E., Jianfeng Lu, and Xu Yang. "Effective Maxwell equations from time-dependent density functional theory." Acta Mathematica Sinica, English Series 27, no. 2 (January 15, 2011): 339–68. http://dx.doi.org/10.1007/s10114-011-0555-0.

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42

Kim, Kwang Ik, and Tong Kang. "A potential-based finite-element method for time-dependent Maxwell’s equations." International Journal of Computer Mathematics 83, no. 1 (January 2006): 107–22. http://dx.doi.org/10.1080/00207160500113074.

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43

Huang, Yunqing, and Jichun Li. "Numerical analysis of a PML model for time-dependent Maxwell’s equations." Journal of Computational and Applied Mathematics 235, no. 13 (May 2011): 3932–42. http://dx.doi.org/10.1016/j.cam.2011.01.039.

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44

Li, Jichun, and Yanping Lin. "A priori and posteriori error analysis for time-dependent Maxwell’s equations." Computer Methods in Applied Mechanics and Engineering 292 (August 2015): 54–68. http://dx.doi.org/10.1016/j.cma.2014.08.009.

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Pahari, Basanta R., and William Oates. "An Entropy Dynamics Approach to Inferring Fractal-Order Complexity in the Electromagnetics of Solids." Entropy 26, no. 12 (December 17, 2024): 1103. https://doi.org/10.3390/e26121103.

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Abstract:

A fractal-order entropy dynamics model is developed to create a modified form of Maxwell’s time-dependent electromagnetic equations. The approach uses an information-theoretic method by combining Shannon’s entropy with fractional moment constraints in time and space. Optimization of the cost function leads to a time-dependent Bayesian posterior density that is used to homogenize the electromagnetic fields. Self-consistency between maximizing entropy, inference of Bayesian posterior densities, and a fractal-order version of Maxwell’s equations are developed. We first give a set of relationships for fractal derivative definitions and their relationship to divergence, curl, and Laplacian operators. The fractal-order entropy dynamic framework is then introduced to infer the Bayesian posterior and its application to modeling homogenized electromagnetic fields in solids. The results provide a methodology to help understand complexity from limited electromagnetic data using maximum entropy by formulating a fractal form of Maxwell’s electromagnetic equations.

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Maikov, A. R., A. G. Sveshnikov, and S. A. Yakunin. "Non-local radiation conditions for the time-dependent Maxwell equations." USSR Computational Mathematics and Mathematical Physics 30, no. 6 (January 1990): 133–41. http://dx.doi.org/10.1016/0041-5553(90)90121-8.

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Nguyen, Hoai-Minh, and Loc X. Tran. "Approximate Cloaking for Time-dependent Maxwell Equations via Transformation Optics." SIAM Journal on Mathematical Analysis 51, no. 5 (January 2019): 4142–71. http://dx.doi.org/10.1137/18m1232395.

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Shang, J. S. "High-Order Compact-Difference Schemes for Time-Dependent Maxwell Equations." Journal of Computational Physics 153, no. 2 (August 1999): 312–33. http://dx.doi.org/10.1006/jcph.1999.6279.

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49

Harshawardhan, W., Q. Su, and R. Grobe. "Numerical solution of the time-dependent Maxwell’s equations for random dielectric media." Physical Review E 62, no. 6 (December 1, 2000): 8705–12. http://dx.doi.org/10.1103/physreve.62.8705.

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50

Monk, Peter. "A Comparison of Three Mixed Methods for the Time-Dependent Maxwell’s Equations." SIAM Journal on Scientific and Statistical Computing 13, no. 5 (September 1992): 1097–122. http://dx.doi.org/10.1137/0913064.

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