Nonlinear second-order

Nonlinear second order optical properties such as second harmonic generation and the linear electrooptic effect arise from the first non-linear term in the constitutive relation for the polarization P(t) of a medium in an applied electric field E(t) = E cos ot. 

The first and third order terms in odd powers of the applied electric field are present for all materials. In the second order term, a polarization is induced proportional to the square of the applied electric field, and the. nonlinear second order optical susceptibility must, therefore, vanish in crystals that possess a center of symmetry. In addition to the noncentrosymmetric structure, efficient second harmonic generation requires crystals to possess propagation directions where the crystal birefringence cancels the natural dispersion leading to phase matching. 

M. Gevers and G. Bastin. Analysis and Optimization of Systems, chapter A Stable Adaptive Observer for a Class of Nonlinear Second-order Systems, pages 143-155. Springer-Verlag, 1986. 

The classical Yang-Mills equations of SU(2) gauge theory in the Minkowski spacetime R1,3 form the system of nonlinear second-order partial differential equations of the form... 

Systems (87) and (91) contain 12 nonlinear second-order ordinary differential equations with variable coefficients. That is why there is little hope for constructing their general solutions. Nevertheless, it is possible to obtain particular solutions of system (87), whose coefficients are given by formulas 2-4 from (91). 

In Figure 5.9 we observe that the theoretical solution (+) and the numerical solution (-) of the DE (5.24) coincide for the first-order reaction. But for the nonlinear second-order reaction, there is no known theoretical solution, hence we cannot compare. 

The mechanism of the third-order three-body induction interactions is somewhat more complicated. It can be shown that one can distinguish three principal categories. The first mechanism is simply the interaction of permanent moments on the monomer C with the moments induced on B by the nonlinear (second-order) effect of the electrostatic potential of the monomer A plus contributions obtained... 

Teng, C.C. and Garito A.F., Dispersion of the nonlinear second-order optical susceptibility of organic systems. Phys. Rev. B (1983) 28 6766-6773. 

Amacher et al. (1988), Selim and Amacher (1988), and Harter (1989) developed nonlinear, second-order, and first-order multireaction models, respectively, to describe element retention by soils. Reactions are distinguished on the basis of mass-action kinetics and these models were found to describe the kinetics of element retention in soils when single-reaction rate functions failed (Amacher et at, 1986). 

Equation (14-53 is a nonlinear second order ODE that is solred on the FEMLAB CD-ROM. 

The axial dispersion is neglected in this model and the system of nonlinear second order partial differential equations can be integrated using numerical techniques e.g. orthogonal collocation. Of course axial dispersion can also be included into the model. However, in general, in most industrial catalytic reactors axial dispersion is negligible. 

Note that the term accounting for effective transport in the axial direction has been neglected in this model, for the reasons given already in Sec. 11.6. This system of nonlinear second order partial differential equations was integrated by Froment using a Crank-Nicolson procedure [76,77], to simulate a multitubular fixed bed reactor for a reaction involving yield problems. 

This is a nonlinear, second-order, second-degree diflerential equation, which can be readily solved by the substitution ... 

The velocity distribution fijr a situation where == 0.5 m/s and R = 0.1 m is shown in Figure 18.6. From Figure 18.6, it is evident that the velocity equation is a nonlinear second-order polynomial and the slope of thu type of model is not constant (it changes with r). For the gwm example, for any 0.01 m change in r, the dependent variable u changes by different amounts depending on where in the pipe you evaluate the change. 

We begin this section by illustrating the types of nonlinear problems that can be resolved using standard methods. Some important nonlinear second order... 

The multiple regression analysis was performed for constructing the RSM-based burr size mathematical models. In the present investigation, the second order burr height and burr thickness mathematical models have been developed in terms of four process parameters, namely, cutting speed (v) feed rate (/), drill diameter (d) and point angle (. The nonlinear second order response surface equation is given by ... 

The Poisson-Boltzmann equation (23.5) is a nonlinear second-order differential equation from which you can compute ip if you know the charge density on P and the bulk salt concentration, rioo- This equation can be solved numerically by a computer. However, a linear approximation, which is easy to solve without a computer, applies when the electrostatic potential is small. For small potentials, zeip/kT 1, you can use the approximation sinh(x) s [(I + x) - (I - x)] 12 = X (which is the first term of the Taylor series expansion for the two exponentials in sinh(x) (see Appendix C, Equation (C.l)). Then Equation (23.5) becomes... 

Equation (1), with the associated boundary conditions, is a nonlinear second-order boundary-value ODE. This was solved by the method of collocation with piecewise cubic Hermite polynomial basis functions for spatial discretization, while simple successive substitution was adequate for the solution of the resulting nonlinear algebraic equations. The method has been extensively described before [9], and no problems were found in this application. 

COPET technique, 391 optical nonlinearity second order, 390 third order, 394... 

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