Journal articles: 'Nonlinear Maxwell equations'

1

Kotel’nikov, G. A. "Nonlinear Maxwell Equations." Journal of Nonlinear Mathematical Physics 3, no. 3-4 (January 1996): 391–95. http://dx.doi.org/10.2991/jnmp.1996.3.3-4.19.

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2

Krejčí, Pavel. "Periodic solutions to Maxwell equations in nonlinear media." Czechoslovak Mathematical Journal 36, no. 2 (1986): 238–58. http://dx.doi.org/10.21136/cmj.1986.102088.

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3

D'Aprile, Teresa, and Dimitri Mugnai. "Solitary waves for nonlinear Klein–Gordon–Maxwell and Schrödinger–Maxwell equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 134, no. 5 (October 2004): 893–906. http://dx.doi.org/10.1017/s030821050000353x.

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In this paper we study the existence of radially symmetric solitary waves for nonlinear Klein–Gordon equations and nonlinear Schrödinger equations coupled with Maxwell equations. The method relies on a variational approach and the solutions are obtained as mountain-pass critical points for the associated energy functional.

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4

Brizard, Alain J., and Anthony A. Chan. "Nonlinear relativistic gyrokinetic Vlasov-Maxwell equations." Physics of Plasmas 6, no. 12 (December 1999): 4548–58. http://dx.doi.org/10.1063/1.873742.

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5

Babin, Anatoli, and Alexander Figotin. "Nonlinear Maxwell Equations in Inhomogeneous Media." Communications in Mathematical Physics 241, no. 2-3 (September 19, 2003): 519–81. http://dx.doi.org/10.1007/s00220-003-0939-9.

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6

Gupta, Vinay Kumar, and Manuel Torrilhon. "Higher order moment equations for rarefied gas mixtures." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 471, no. 2173 (January 2015): 20140754. http://dx.doi.org/10.1098/rspa.2014.0754.

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The fully nonlinear Grad's N ×26-moment ( N × G 26) equations for a mixture of N monatomic-inert-ideal gases made up of Maxwell molecules are derived. The boundary conditions for these equations are derived by using Maxwell's accommodation model for each component in the mixture. The linear stability analysis is performed to show that the 2×G26 equations for a binary gas mixture of Maxwell molecules are linearly stable. The derived equations are used to study the heat flux problem for binary gas mixtures confined between parallel plates having different temperatures.

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7

DUPLIJ, STEVEN, ELISABETTA DI GREZIA, GIAMPIERO ESPOSITO, and ALBERT KOTVYTSKIY. "NONLINEAR CONSTITUTIVE EQUATIONS FOR GRAVITOELECTROMAGNETISM." International Journal of Geometric Methods in Modern Physics 11, no. 01 (December 16, 2013): 1450004. http://dx.doi.org/10.1142/s0219887814500042.

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This paper studies nonlinear constitutive equations for gravitoelectromagnetism. Eventually, the problem is solved of finding, for a given particular solution of the gravity-Maxwell equations, the exact form of the corresponding nonlinear constitutive equations.

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8

Ciaglia, F. M., F. Di Cosmo, G. Marmo, and L. Schiavone. "Evolutionary equations and constraints: Maxwell equations." Journal of Mathematical Physics 60, no. 11 (November 1, 2019): 113503. http://dx.doi.org/10.1063/1.5109087.

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9

LONG, EAMONN. "EXISTENCE AND STABILITY OF SOLITARY WAVES IN NON-LINEAR KLEIN–GORDON–MAXWELL EQUATIONS." Reviews in Mathematical Physics 18, no. 07 (August 2006): 747–79. http://dx.doi.org/10.1142/s0129055x06002784.

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We prove the existence and stability of non-topological solitons in a class of weakly coupled non-linear Klein–Gordon–Maxwell equations. These equations arise from coupling non-linear Klein–Gordon equations to Maxwell's equations for electromagnetism.

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10

Colin, Thierry, and Boniface Nkonga. "Multiscale numerical method for nonlinear Maxwell equations." Discrete & Continuous Dynamical Systems - B 5, no. 3 (2005): 631–58. http://dx.doi.org/10.3934/dcdsb.2005.5.631.

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11

Bruce, S. A. "Nonlinear Maxwell equations and the Poynting theorem." European Journal of Physics 42, no. 1 (January 1, 2020): 015201. http://dx.doi.org/10.1088/1361-6404/abb296.

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12

Bartsch, Thomas, and Jarosław Mederski. "Nonlinear time-harmonic Maxwell equations in domains." Journal of Fixed Point Theory and Applications 19, no. 1 (February 14, 2017): 959–86. http://dx.doi.org/10.1007/s11784-017-0409-1.

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13

Umurzhakhova, Zh B., M. D. Koshanova, Zh Pashen, and K. R. Yesmakhanova. "QUASICLASSICAL LIMIT OF THE SCHRÖDINGER-MAXWELL- BLOCH EQUATIONS." PHYSICO-MATHEMATICAL SERIES 2, no. 336 (April 15, 2021): 179–84. http://dx.doi.org/10.32014/2021.2518-1726.39.

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The study of integrable equations is one of the most important aspects of modern mathematical and theoretical physics. Currently, there are a large number of nonlinear integrable equations that have a physical application. The concept of nonlinear integrable equations is closely related to solitons. An object being in a nonlinear medium that maintains its shape at moving, as well as when interacting with its own kind, is called a soliton or a solitary wave. In many physical processes, nonlinearity is closely related to the concept of dispersion. Soliton solutions have dispersionless properties. Connection with the fact that the nonlinear component of the equation compensates for the dispersion term. In addition to integrable nonlinear differential equations, there is also an important class of integrable partial differential equations (PDEs), so-called the integrable equations of hydrodynamic type or dispersionless (quasiclassical) equations [1-13]. Nonlinear dispersionless equations arise as a dispersionless (quasiclassical) limit of known integrable equations. In recent years, the study of dispersionless systems has become of great importance, since they arise as a result of the analysis of various problems, such as physics, mathematics, and applied mathematics, from the theory of quantum fields and strings to the theory of conformal mappings on the complex plane. Well-known classical methods of the theory of intrinsic systems are used to study dispersionless equations. In this paper, we present the quasicalassical limit of the system of (1+1)-dimensional Schrödinger-Maxwell- Bloch (NLS-MB) equations. The system of the NLS-MB equations is one of the classic examples of the theory of nonlinear integrable equations. The NLS-MB equations describe the propagation of optical solitons in fibers with resonance and doped with erbium. And we will also show the integrability of the quasiclassical limit of the NLS-MB using the obtained Lax representation.

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14

JIMÉNEZ, S., P. PASCUAL, C. AGUIRRE, and L. VÁZQUEZ. "A PANORAMIC VIEW OF SOME PERTURBED NONLINEAR WAVE EQUATIONS." International Journal of Bifurcation and Chaos 14, no. 01 (January 2004): 1–40. http://dx.doi.org/10.1142/s0218127404009211.

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In this paper we present a panoramic view of numerical simulations associated with nonlinear wave equations which appear in different experimental contexts. Mainly, we deal with scalar wave equations, but also the Maxwell equations in nonlinear media are studied. A basic part of this work is devoted to the construction and verification of numerical schemes on a physical basis. The stochastic perturbations of scalar wave equations are especially analyzed by analytical and numerical approaches. Also, other kinds of perturbations are considered, like nonlocal ones. Finally, a summary of promising experimental results from the numerical simulations of the Maxwell system in a nonlinear media is presented.

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15

Brizard, Alain J. "Variational principle for nonlinear gyrokinetic Vlasov–Maxwell equations." Physics of Plasmas 7, no. 12 (December 2000): 4816–22. http://dx.doi.org/10.1063/1.1322063.

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16

Joly, Jean-Luc, Guy Metivier, and Jeffrey Rauch. "Transparent Nonlinear Geometric Optics and Maxwell–Bloch Equations." Journal of Differential Equations 166, no. 1 (September 2000): 175–250. http://dx.doi.org/10.1006/jdeq.2000.3794.

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17

Huang, Wen-nian, and X. H. Tang. "Semiclassical solutions for the nonlinear Schrödinger–Maxwell equations." Journal of Mathematical Analysis and Applications 415, no. 2 (July 2014): 791–802. http://dx.doi.org/10.1016/j.jmaa.2014.02.015.

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18

Deumens, E. "The Klein-Gordon-Maxwell nonlinear system of equations." Physica D: Nonlinear Phenomena 18, no. 1-3 (January 1986): 371–73. http://dx.doi.org/10.1016/0167-2789(86)90201-0.

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19

Damianou, Pantelis A., and Paschalis G. Paschali. "Symmetries of Maxwell-Bloch Equations." Journal of Nonlinear Mathematical Physics 2, no. 3-4 (January 1995): 269–77. http://dx.doi.org/10.2991/jnmp.1995.2.3-4.6.

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20

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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21

Brizard, A. "Nonlinear gyrokinetic Maxwell-Vlasov equations using magnetic co-ordinates." Journal of Plasma Physics 41, no. 3 (June 1989): 541–59. http://dx.doi.org/10.1017/s0022377800014070.

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A gyrokinetic formalism using magnetic co-ordinates is used to derive self-consistent, nonlinear Maxwell–Vlasov equations that are suitable for particle simulation studies of finite-β tokamak microturbulence and its associated anomalous transport. The use of magnetic co-ordinates is an important feature of this work since it introduces the toroidal geometry naturally into our gyrokinetic formalism. The gyrokinetic formalism itself is based on the use of the action-variational Lie perturbation method of Cary & Littlejohn, and preserves the Hamiltonian structure of the original Maxwell-Vlasov system. Previous nonlinear gyrokinetic sets of equations suitable for particle simulation analysis have considered either electrostatic and shear-Alfvén perturbations in slab geometry or electrostatic perturbations in toroidal geometry. In this present work fully electromagnetic perturbations in toroidal geometry are considered.

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22

Papachristou, C. J. "The Maxwell equations as a Bäcklund transformation." Advanced Electromagnetics 4, no. 1 (July 27, 2015): 52. http://dx.doi.org/10.7716/aem.v4i1.311.

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Bäcklund transformations (BTs) are a useful tool for integrating nonlinear partial differential equations (PDEs). However, the significance of BTs in linear problems should not be ignored. In fact, an important linear system of PDEs in Physics, namely, the Maxwell equations of Electromagnetism, may be viewed as a BT relating the wave equations for the electric and the magnetic field, these equations representing integrability conditions for solution of the Maxwell system. We examine the BT property of this system in detail, both for the vacuum case and for the case of a linear conducting medium.

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23

Kovachev, Lubomir M. "Vortex solutions of the nonlinear optical Maxwell–Dirac equations." Physica D: Nonlinear Phenomena 190, no. 1-2 (March 2004): 78–92. http://dx.doi.org/10.1016/j.physd.2003.08.009.

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24

Wellander, Niklas. "Homogenization of the Maxwell Equations: Case II. Nonlinear Conductivity." Applications of Mathematics 47, no. 3 (June 2002): 255–83. http://dx.doi.org/10.1023/a:1021797505024.

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25

Miano, G., C. Serpico, L. Verolino, and F. Villone. "Numerical solution of the Maxwell equations in nonlinear media." IEEE Transactions on Magnetics 32, no. 3 (May 1996): 950–53. http://dx.doi.org/10.1109/20.497399.

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26

Serre, Denis. "Hyperbolicity of the Nonlinear Models of Maxwell?s Equations." Archive for Rational Mechanics and Analysis 172, no. 3 (May 1, 2004): 309–31. http://dx.doi.org/10.1007/s00205-003-0303-4.

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27

Hu, Liang, Dazhi Zhao, Xun Wang, and MaoKang Luo. "Analytic methods for solving the cylindrical nonlinear Maxwell equations." Optik 170 (October 2018): 287–94. http://dx.doi.org/10.1016/j.ijleo.2018.03.083.

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28

Azzollini, A., and A. Pomponio. "Ground state solutions for the nonlinear Schrödinger–Maxwell equations." Journal of Mathematical Analysis and Applications 345, no. 1 (September 2008): 90–108. http://dx.doi.org/10.1016/j.jmaa.2008.03.057.

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29

Radford, C. J. "The stationary Maxwell–Dirac equations." Journal of Physics A: Mathematical and General 36, no. 20 (May 8, 2003): 5663–81. http://dx.doi.org/10.1088/0305-4470/36/20/321.

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30

Li, Yan, Jian Li, and Ruiqi Wang. "N-soliton solutions for the Maxwell–Bloch equations via the Riemann–Hilbert approach." Modern Physics Letters B 35, no. 21 (June 4, 2021): 2150356. http://dx.doi.org/10.1142/s0217984921503565.

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We mainly study [Formula: see text]-soliton solutions for the Maxwell–Bloch equations via the Riemann–Hilbert (RH) approach in this paper. The relevant RH problem has been constructed by performing spectral analysis of Lax pair. Then the jump matrix of the Maxwell–Bloch equations has been obtained. Finally, we gain the exact solutions of the Maxwell–Bloch equations by solving the special RH problem with reflectionless case.

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31

Ghanaee, Reza, Ahmad Darabi, Arash Kioumarsi, Mohammad Baghayipour, and Mohammad Morshed. "A nonlinear equivalent circuit model for flux density calculation of a permanent magnet linear synchronous motor." Serbian Journal of Electrical Engineering 12, no. 3 (2015): 359–73. http://dx.doi.org/10.2298/sjee1503359g.

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In this paper, a nonlinear magnetic equivalent circuit is presented as an analytical solution method for modeling of a permanent magnet linear synchronous motor (PMLSM). The accuracy of the proposed model is verified via comparing its simulation results with those obtained by two other methods. These two are the Maxwell?s Equations based analytical method and the wellknown finite elements method (FEM). Saturation and any saliency e.g. slotting effects can be considered properly by both nonlinear magnetic equivalent circuit and FEM, where it cannot be taken into account easily by the Maxwell?s Equations based analytical approach. Accordingly, as the simulation results presented in this paper confirm, the proposed nonlinear magnetic equivalent circuit is compatible with FEM regarding the accuracy while it requires very shorter execution time. Therefore, the magnetic equivalent circuit model of the present paper can be considered as a preferable substitute for the time consuming FEM and approximated analytical method built on Maxwell?s Equations in particular when required to be applied for a design optimization problem.

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32

Puta, Mircea. "On the geometric prequantization of Maxwell equations." Letters in Mathematical Physics 13, no. 2 (February 1987): 99–103. http://dx.doi.org/10.1007/bf00955196.

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33

Marchuk, N. G. "Model Dirac-Maxwell equations with pseudounitary symmetry." Theoretical and Mathematical Physics 157, no. 3 (December 2008): 1723–32. http://dx.doi.org/10.1007/s11232-008-0144-2.

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34

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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35

Barna, I. F. "Self-similar shock wave solutions of the nonlinear Maxwell equations." Laser Physics 24, no. 8 (July 15, 2014): 086002. http://dx.doi.org/10.1088/1054-660x/24/8/086002.

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36

Negro, Angelo. "Homogenization of the quasistationary maxwell equations in nonlinear laminated materials (∗)." Applicable Analysis 25, no. 3 (January 1987): 197–227. http://dx.doi.org/10.1080/00036818708839685.

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37

Bartsch, Thomas, and Jarosław Mederski. "Nonlinear time-harmonic Maxwell equations in an anisotropic bounded medium." Journal of Functional Analysis 272, no. 10 (May 2017): 4304–33. http://dx.doi.org/10.1016/j.jfa.2017.02.019.

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38

Donnat, P., and J. Rauch. "Global solvability of the maxwell-bloch equations from nonlinear optics." Archive for Rational Mechanics and Analysis 136, no. 3 (December 1996): 291–303. http://dx.doi.org/10.1007/bf02206557.

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39

Donnat, P., and J. Rauch. "Global Solvability of the Maxwell-Bloch Equations from Nonlinear Optics." Archive for Rational Mechanics and Analysis 136, no. 3 (December 12, 1996): 291–303. http://dx.doi.org/10.1007/s002050050017.

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40

Li, Chuan-Zhong, Jing-Song He, and K. Porseizan. "Nonlinear waves of the Hirota and the Maxwell—Bloch equations in nonlinear optics." Chinese Physics B 22, no. 4 (April 2013): 044208. http://dx.doi.org/10.1088/1674-1056/22/4/044208.

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41

Castin, Yvan, and Klaus Mo/lmer. "Maxwell-Bloch equations: A unified view of nonlinear optics and nonlinear atom optics." Physical Review A 51, no. 5 (May 1, 1995): R3426—R3428. http://dx.doi.org/10.1103/physreva.51.r3426.

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42

Masmoudi, Nader, and Kenji Nakanishi. "Uniqueness of Finite Energy Solutions for Maxwell-Dirac and Maxwell-Klein-Gordon Equations." Communications in Mathematical Physics 243, no. 1 (November 1, 2003): 123–36. http://dx.doi.org/10.1007/s00220-003-0951-0.

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43

Suraiah Palaiah, Suresha, Hussain Basha, Gudala Janardhana Reddy, and Mikhail A. Sheremet. "Magnetized Dissipative Soret Effect on Chemically Reactive Maxwell Fluid over a Stretching Sheet with Joule Heating." Coatings 11, no. 5 (April 29, 2021): 528. http://dx.doi.org/10.3390/coatings11050528.

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The present research paper deals with the study of heat and mass transfer characteristics of steady viscous incompressible two-dimensional Maxwell fluid flow past a stretching sheet under the influence of magnetic field and the Soret effect. A well-known non-Newtonian Maxwell fluid flow model is used to differentiate it from the Newtonian fluids. The present physical problem gives the set of highly nonlinear-coupled partial differential equations that are not amenable to any of the direct techniques. The resultant nonlinear system of partial differential equations is reduced to a set of nonlinear ordinary differential equations by using suitable similarity transformations. Due to the inadequacy of analytical techniques, a bvp4c MATLAB function is used to solve the developed nonlinear system of equations. The simulated results are shown for various values of physical parameters in the flow regime. Additionally, the numerical values of skin-friction coefficient, heat, and mass transfer rates are calculated and tabularized. From the present investigation, it is observed that the normal and axial velocity profiles decreased for the enhancing values of the magnetic parameter. Increasing the Prandtl and Schmidt numbers reduces the temperature and concentration profiles in the flow region, respectively. Increasing the Maxwell fluid parameter decreases the velocity profile and magnifies the temperature field. Additionally, increasing the Soret number increases the concentration profile in the flow regime. Comparison of current similarity solutions with available results indicates the accuracy and guarantee of the present numerical results and the used method.

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44

Hussain, Syed M., Rohit Sharma, Manas R. Mishra, and Sattam S. Alrashidy. "Hydromagnetic Dissipative and Radiative Graphene Maxwell Nanofluid Flow Past a Stretched Sheet-Numerical and Statistical Analysis." Mathematics 8, no. 11 (November 2, 2020): 1929. http://dx.doi.org/10.3390/math8111929.

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The key objective of this analysis is to examine the flow of hydromagnetic dissipative and radiative graphene Maxwell nanofluid over a linearly stretched sheet considering momentum and thermal slip conditions. The appropriate similarity variables are chosen to transform highly nonlinear partial differential equations (PDE) of mathematical model in the form of nonlinear ordinary differential equations (ODE). Further, these transformed equations are numerically solved by making use of Runge-Kutta-Fehlberg algorithm along with the shooting scheme. The significance of pertinent physical parameters on the flow of graphene Maxwell nanofluid velocity and temperature are enumerated via different graphs whereas skin friction coefficients and Nusselt numbers are illustrated in numeric data form and are reported in different tables. In addition, a statistical approach is used for multiple quadratic regression analysis on the numerical figures of wall velocity gradient and local Nusselt number to demonstrate the relationship amongst heat transfer rate and physical parameters. Our results reveal that the magnetic field, unsteadiness, inclination angle of magnetic field and porosity parameters boost the graphene Maxwell nanofluid velocity while Maxwell parameter has a reversal impact on it. Finally, we have compared our numerical results with those of earlier published articles under the restricted conditions to validate our solution. The comparison of results shows an excellent conformity among the results.

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45

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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46

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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47

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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48

Booth, Hilary. "Nonlinear electron solutions and their characteristics at infinity." ANZIAM Journal 44, no. 1 (July 2002): 51–59. http://dx.doi.org/10.1017/s1446181100007902.

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AbstractThe Maxwell-Dirac equations model an electron in an electromagnetic field. The two equations are coupled via the Dirac current which acts as a source in the Maxwell equation, resulting in a nonlinear system of partial differential equations (PDE's). Well-behaved solutions, within reasonable Sobolev spaces, have been shown to exist globally as recently as 1997 [12]. Exact solutions have not been found—except in some simple cases.We have shown analytically in [6, 18] that any spherical solution surrounds a Coulomb field and any cylindrical solution surrounds a central charged wire; and in [3] and [19] that in any stationary case, the surrounding electron field must be equal and opposite to the central (external) field. Here we extend the numerical solutions in [6] to a family of orbits all of which are well-behaved numerical solutions satisfying the analytic results in [6] and [11]. These solutions die off exponentially with increasing distance from the central axis of symmetry. The results in [18] can be extended in the same way. A third case is included, with dependence on z only yielding a related fourth-order ordinary differential equation (ODE) [3].

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49

Revenko, Igor. "On Exact Solutions of the Lorentz-Maxwell Equations." Journal of Nonlinear Mathematical Physics 3, no. 3-4 (January 1996): 417–20. http://dx.doi.org/10.2991/jnmp.1996.3.3-4.24.

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50

Noriega, R. J., and C. G. Schifini. "The equivariant inverse problem and the Maxwell equations." Journal of Mathematical Physics 28, no. 4 (April 1987): 815–17. http://dx.doi.org/10.1063/1.527568.

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Journal articles on the topic 'Nonlinear Maxwell equations'

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