Generalized Maxwell equation

On this basis it was possible to obtain equations which agreed with known relativistic equations of the gravitational held and generalized Maxwell equations of electromagnetism. Writing (4.16) in the form... 

In summary, if G t), which is contained in Eqs. (4.30), (4.34)-(4.37), (4.49)-(4.51), (4.63) and (4.73), is known, all the linear viscoelastic quantities can be calculated. In other words, all the various viscoelastic properties of the polymer are related to each other through the relaxation modulus G t). This result is of course the consequence of the generalized Maxwell equation or equivalently Boltzmann s superposition principle. The experimental results of linear viscoelastic properties of various polymers support the phenomenological principle. Some viscoelastic properties play more important roles than the others in certain rheological processes related to... 

To study the kinetics of temperature stresses in polymers, to analyse the influence of various factors on the flow of the examined processes, and to model the relaxation behaviour in polymers, a nonlinear constitutive differential equation is used in the paper. This equation was proposed by G.I. Gurevich [1], who called it the nonlinear generalized Maxwell equation out of respect for J. Maxwell s ideas [2] that served as a partial basis for deducing the equation. Total deformation is regarded as the sum of elastic, viscoelastic and temperature deformations ... 

Strictly speaking, differentiation with respeet to a veetor quantity is not allowed. However for the isotropie spherieal samples for whieh equation (A2.1.8) is appropriate, the two veetors have the same direetion and eould have been written as sealars the veetor notation was kept to avoid eonfiision with other thennodynamie quantities sueh as energy, pressure, ete. It should also be noted that the Maxwell equations above are eorreet for either of the ehoiees for eleetromagnetie work diseussed earlier under the other eonvention A is replaeed by a generalized G.)... 

A fiill solution of tlie nonlinear radiation follows from the Maxwell equations. The general case of radiation from a second-order nonlinear material of finite thickness was solved by Bloembergen and Pershan in 1962 [40]. That problem reduces to the present one if we let the interfacial thickness approach zero. Other equivalent solutions involved tlie application of the boundary conditions for a polarization sheet [14] or the... 

The generalized Stefan-Maxwell equations using binary diffusion coefficients are not easily applicable to hquids since the coefficients are so dependent on conditions. That is, in hquids, each Dy can be strongly composition dependent in binary mixtures and, moreover, the binaiy is strongly affected in a multicomponent mixture. Thus, the convenience of writing multicomponent flux equations in terms of binary coefficients is lost. Conversely, they apply to gas mixtures because each is practically independent of composition by itself and in a multicomponent mixture (see Taylor and Krishna for details). 

This equation, based on the generalized Maxwell model (e.g. jL, p. 68), indicates that G (o) can be determined from the difference between the measured modulus and its relaxational part. A prerequisite, however, is that the relaxation spectrum H(t) should be known over the entire relaxation time range from zero to infinity, which is impossible in practice. Nevertheless, the equation can still be used, because this time interval can generally be taken less wide, as will be demonstrated below. 

The theory on the level of the electrode and on the electrochemical cell is sufficiently advanced [4-7]. In this connection, it is necessary to mention the works of J.Newman and R.White s group [8-12], In the majority of publications, the macroscopical approach is used. The authors take into account the transport process and material balance within the system in a proper way. The analysis of the flows in the porous matrix or in the cell takes generally into consideration the diffusion, migration and convection processes. While computing transport processes in the concentrated electrolytes the Stefan-Maxwell equations are used. To calculate electron transfer in a solid phase the Ohm s law in its differential form is used. The electrochemical transformations within the electrodes are described by the Batler-Volmer equation. The internal surface of the electrode, where electrochemical process runs, is frequently presented as a certain function of the porosity or as a certain state of the reagents transformation. To describe this function, various modeling or empirical equations are offered, and they... 

The electromagnetic field is required to satisfy the Maxwell equations at points where e and ju, are continuous. However, as one crosses the boundary between particle and medium, there is, in general, a sudden change in these properties. This change occurs over a transition region with thickness of the order of atomic dimensions. From a macroscopic point of view, therefore, there is a discontinuity at the boundary. At such boundary points we impose the following conditions on the fields ... 

In U(l) gauge theory, the Lagrangian in general [6] contains the mass term (823), but in order to obtain the inhomogeneous Maxwell equations, this is discarded. This procedure is outlined, for example, on pp. 89ff. of Ref. 6. The U(l) Lagrangian in general is, in reduced units... 

The present review is based mainly on our publications [33,35-39,49-53]. In Section II we give a detailed description of the general reduction routine for an arbitrary relativistically invariant systems of partial differential equations. The results of Section II are used in Section III to solve the problem of symmetry reduction of Yang-Mills equations (1) by subgroups of the Poincare group P 1,3) and to construct their exact (non-Abelian) solutions. In Section IV we review the techniques for nonclassical reductions of the STJ 2) Yang-Mills equations, which are based on their conditional symmetry. These techniques enable us to obtain the principally new classes of exact solutions of (1), which are not derivable within the framework of the standard symmetry reduction technique. In Section V we give an overview of the known invariant solutions of the Maxwell equations and construct multiparameter families of new ones. 

Furthermore, the general method presented in this chapter applies directly to solving the full Maxwell equations with currents. It can also be used to construct exact classical solutions of Yang-Mills equations with Higgs fields and their generalizations. Generically, the method developed in this chapter can be efficiently applied to any conformally invariant wave equation, on the solution set of which a covariant representation of the conformal algebra in Eq. (15) is realized. 

The results of exact solutions of nonlinear generalizations of the Maxwell equations are also beyond the scope of the present review. A survey of these results, as well as an extensive list of references, can be found in Fushchych et al. [21]. 

The generalized solution of Maxwell equations in terms of potentials [56,57]... 

The general differential equation which describes the Maxwell model is... 

The application of the same Maxwell model to creep data is described in Figures 2.52 and 2.53. As shown in Figure 2.52, the force on the sample is fixed by suspending a constant weight at the end of the sample. The length of the sample is then measured as a function of time. Since the force is constant, ds/dt = 0 and the general differential equation again simplifies to a solvable form, namely... 

In dilute electrolyte systems, the diffusional interactions can usually be neglected, and the generalized Maxwell-Stefan equations are reduced to the Nemst-Planck equations (B3) ... 

Since the Maxwell equations involve the components of the Jones vector, it is normally easier to derive the Jones matrix, J, for complex, anisotropic media. Once J is obtained, it is generally convenient to transform it to the Mueller matrix representation for the purpose of analyzing the quantities measured in specific optical trains. This is because the components of the Stokes vector are observable, whereas the Jones vector components are not. Since it is the intensity of light that is normally required, only the first element of Sn,... 

Equation (1.35) provides a general solution to the Maxwell equations and is particularly useful for problems in scattering. Rewriting the results for the scattered field, we have... 

According to the nonlocal theory the vector fields E(r), I)(7) and P(r) in Equations (1.119) can be treated as time dependent and they obey the Maxwell equations [1], Within the LRA, most general expressions are valid ... 

More complicated and realistic models which allow the prediction of transport processes in porous media have been suggested, and have been validated in recent years. For example, it was realized that there might be significant contributions to the overall flux by components which are adsorbed at pore walls but possess a certain mobility [30]. To quantify such surface diffusion processes, a Generalized Stefan-Maxwell equation has been proposed [28] ... 

Farrow et al (1923) used the modified Maxwell equation dv/dx = Pn/ . This equation has a certain amount of generality. Using this basic equation for plastic flow in a capillary tube, the following equation was derived,... 

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