Auxiliary differential equation

The basis behind separation of variables is the orthogonal expansion technique. The method of separation of variables produces a set of auxiliary differential equations. One of these auxiliary problems is called the eigenvalue problem with its eigenfunction solutions. 

We introduce into (4.1) a small bookkeeping parameter A that will finally be put equal to unity. Thus we get the auxiliary differential equation... 

The auxiliary differential equation (4.2) has two linearly independent solutions fi(z) and f2(z) of the form... 

M. Fujii, M. Tahara, I. Sakagami, W. Freude, and P. Russer, High-order FDTD and auxiliary differential equation formulation of optical pulse propagation in 2-D Kerr and Raman nonlinear dispersive media, IEEEJ. Quantum Electron., vol. 40, no. 2, pp. 175-182, Feb. 2004.doi 10.1109/JQE.2003.821881... 

The simulation of dispersive media is based on the auxiliary differential equation technique [6], while the frequency-dependent constitutive relation D(r, co) = s(ry)E(r, co) governs wave propagation in their interior. Let us now investigate three different cases of such materials characterized by diverse s(co). 

In (6.18), a denotes the electric losses of the medium under study, H = HU H, Hw 7 is the magnetic intensity, and J = JU / J ]1 is a prearranged electric current density source used for the external excitation of the structure. Observe that (6.18c) constitutes the auxiliary differential equation form that provides the mathematical background of the frequency relationship between vectors D and E. Specifically, it is derived via the inverse Fourier transform of the Vn definition considering an eiwt variation. 

As mentioned before, the longitudinal excitation always promotes tunneling, A , =i > A =o- The effect of transversal excitation, however, depends on the behavior of the effective frequency 9 x). For instance, for monotonically growing (decreasing) 9 r) the excitation of the transversal mode suppresses (promotes) the tunneling splitting. We see below that the effective frequency is generally determined as a solution of auxiliary differential equation that describes the nodal structure of the semiclassical wave function. This corresponds to the discussion of Takada and Nakamura [30,31] (see Chapter 4). 

Integration is started with known values of the dependent variables at one value of the independent variable, except when the "shooting method" is needed. Auxiliary algebraic equations can be entered to the program along with the differential equations and the boundary conditions. 

We therefore advise that the reader should consult a recent series of papers published by Galvez et al. [171, 172] encompassing all the mechanisms mentioned in Sect. 7.1, elaborated for both d.c. and pulse polarography. The principles of the Galvez method are clearly outlined in the first part of the series [171]. It is similar to the dimensionless parameter method of Koutecky [161], which enables the series solutions for the auxiliary concentration functions cP and cQ exp (kt) and

combined directly with the partial differential equations of the type of eqn. (203). In some of the treatments, the sphericity of the DME is also accounted for. The results are usually visualized by means of predicted polarograms, some examples of which are reproduced in Fig. 38. Naturally, the numerical description of the surface concentrations at fixed potential are also immediately available, in terms of the postulated power series, and the recurrent relationships obtained for the coefficients of these series. 

To solve the differential equation 2.38 subject to the boundary conditions 2.39 and 2.41 we adopt the standard method of substituting the trial solution C — Aemi and obtain the auxiliary equation ... 

For time-dependent systems again the purely exponential time-dependence of the correlation function allows the derivation of a set of differential equations for the auxiliary operators... 

F(c/c,) denotes the dimensionless form of an arbitrary rate expression./(x) is a nonuniform, normalized catalyst activity distribution inside the pellet. A(x) is an auxiliary function, subject to the following linear differential equation ... 

We note that when the losses are too large, free oscillations cannot be excited. For this reason it is compulsory to use the elastic auxiliary element in order to get information on the viscoelastic functions. A stiff elastic element, with constant k, can be added to reduce the loss. When the loss of the system is sufficiently small, the discrepancies between the results obtained from the former theory and the solution based on the classical second-order differential equation [see Eq. (7.49), for example]... 

In Eq. (7.1) we have neglected the centrifugal force which is proportional to the mass of a segment multiplied by q. It can be solved by changing the variables x and y into f and rj which are the integration constants of the auxiliary linear differential equation belonging to the partial differential equation, f and t] are defined by... 

To calculate dispersed phase particle trajectories it will be necessary to solve a set of coupled ordinary differential equations (Eqs. 4.1 and 4.9). Any standard initial value ODE solvers can be used for this purpose. These methods are not discussed here. Necessary details may be found in texts such as Numerical Recipes (Press et al., 1992). When calculating the trajectories of dispersed phase particles, any other auxiliary equations to account for heat transfer or chemical reactions can also be solved following similar procedures. Care must be taken to ensure that the time steps used for integration are sufficiently small and the trajectory integration is adequately time accurate. It is often necessary to use different time steps to simulate transients in the continuous flow field and trajectories of dispersed phase particles. 

The two roots of the auxiliary equation of the differential equation corresponding to the above expression are... 

Although it is easy to write down the explicit solution of the system (127), here we shall provide only a qualitative discussion of the solution. The main features are then best demonstrated with the help of a figure. Eliminating idler and auxiliary mode variables from Eq. (127) we get a differential equation of the third order for the annihilation operator of the signal mode. Its characteristic polynomial (on substitution as (t)=as (0) exp (iXt))... 

Differential equations are equations that contain the derivatives of the unknown functions. They must be supplemented with auxiliary conditions to completely specify a problem. Auxiliary conditions must be prescribed at one or more points in the domain of the independent variables representing the boundary of the domain interface between different regions, and so on. Those equations with prescribed conditions at one point are called initial-value problems, and those with prescribed conditions on the boundary of the domain are appropriately called boundary-value problems. Initial-value problems generally govern the dynamics of the systems, while boundary-value problems describe the systems in steady state. 

The Schrodinger wave equation and its auxiliary postulates enable us to determine certain functions k of the coordinates of a system and the time. These functions are called the Schrodinger wave functions or probability amplitude functions. The square of the absolute value of a given wave function is interpreted as a probability distribution function for the coordinates of the system in the state represented by this wave function, as will be discussed in Section lOo. The wave equation has been given this name because it is a differential equation of the second order in the coordinates of the system, somewhat similar to the wave equation of classical theory. The similarity is not close, however, and we shall not utilize the analogy in our exposition. 

Given an initial temperature for the node, T, it is possible to find the specific internal energy, u = u(T), and the specific volume, v = v(T), and hence the mass m = V/v. Equation (18.65), taken in conjunction with auxiliary equations (18.63), represents an implicit equation in the nodal pressure, p, which may be solved using the methods already outlined, either iteration or the Method of Referred Derivatives. The upstream and downstream flows, Wyp, and Wj , may then be found, so that it becomes possible to calculate the right-hand side of the temperature differential equation (18.64). Equation (18.64) may then be integrated to find the temperature of the node at the next timestep. The process may then be repeated for the duration of the transient under consideration. 

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