Ordinary differential equations expansions

Equation (3.5) is a set of ordinary differential equations of second order for each radial wavefunction Xri(-R) the different expansion functions Xn (R> Ef>n) are coupled to all other functions by the real and symmetric potential matrix V. 

APPLIED COMPLEX VARIABLES, John W. Dettman. Step-by-step coverage of fundamentals of analytic function theory—plus lucid exposition of 5 important applications Potential Theory Ordinary Differential Equations Fourier Transforms Laplace Transforms Asymptotic Expansions. 66 figures. Exercises at chapter ends. 512pp. 5)4 x 8)4. 64670-X Pa. 10.95... 

ASYMPTOTIC EXPANSIONS FOR ORDINARY DIFFERENTIAL EQUATIONS, Wolfgang Wasow. Outstanding text covers asymptotic power series, Jordan s canonical form, turning point problems, singular perturbations, much more. Problems. 3B4pp. 5K x 8X. 65456-7 Pa. 8.95... 

Near the point where the two streams first meet the chemical reaction rate is small and a self-similar frozen-flow solution for Yp applies. This frozen solution has been used as the first term in a series expansion [62] or as the first approximation in an iterative approach [64]. An integral method also has been developed [62], in which ordinary differential equations are solved for the streamwise evolution of parameters that characterize profile shapes. The problem also is well suited for application of activation-energy asymptotics, as may be seen by analogy with [65]. The boundary-layer approximation fails in the downstream region of flame spreading unless the burning velocity is small compared with u it may also fail near the point where the temperature bulge develops because of the rapid onset of heat release there,... 

Main region 6, which begins after some transition 5 of the flow, lasts infinitely and is distinctive by the fact that the internal portions of all the distributions stop changing while the expansion of the external boundary layer d2(x) is going on. This means that the drag force came to an equilibrium with the shear stress. Imposing the conditions d/dz 0 and V 0 on the first system of equations (3.33), the following reduction to one ordinary differential equation takes place ... 

The final stage in the adiabatic reduction is the solution of Eq. (4.24). Given the adiabatic potential of Eq. (4.26) this cannot be done analytically, but the resulting ordinary differential equation may be solved numerically using the finite difference method. As an example, we show in Fig. 20 a comparison between the even-parity adiabatic eigenvalues and the exact ones, obtained by solving the full coupled channels expansion, using the artificial channel method.69... 

Wasow, W., Asymptotics Expansions for Ordinary Differential Equations, Mineola, NY Dover Publications, 1987. 

To find the unknown functions, one writes out a system of linked boundary value problems for ordinary differential equations. The numerical solution of these problems for the first six terms of the expansion (1.8.14) is tabulated in detail in [427], The corresponding analytical expressions for the velocity components in the boundary layer can be calculated by formulas (1.1.6). 

Substituting the expansion (3.4.12) into (3.4.1) and then separating the variables, we obtain the following ordinary differential equation for the functions Hm ... 

Substitution of this expansion into (4.16) yields the hierarchy of ordinary differential equations ... 

Some work on distributed reactors and their optimum control has been undertaken by means of an expansion in orthogonal modes. If the expansion is terminated and a finite number of terms used, the model reduces essentially to a system of ordinary differential equations with an increased number of elements in the state and costate vectors. However, we shall give a more general account. At the same time, this general account can meaningfully be reduced to the steady state for certain problems of interest. Another motive for this section is to demonstrate a connection between Pontryagin s work and some well-established results of reactor theory. 

Both approaches have been applied to low-speed spinning of a Newtonian fluid, with equivalent results. The conclusion is that the spinline is stable to finite disturbances below a draw ratio of 20.21, while at higher draw ratios the system is unstable and approaches a limit cycle characterized by sustained oscillations of the force and takeup area. Figure 11.4 shows the sustained oscillations computed by the expansion method with truncation at one term ior Dr = 23.34, using Galerkin s method to obtain the ordinary differential equations for the real and imaginary parts of the area perturbation. This method is effective only for draw ratios close to the critical value because of the need to keep the number of terms in the expansion small, whereas direct numerical solution can produce wave forms hke those in Figure 11.1. 

A numerical tool, sensitivity analysis, which can be used to study the effects of parameter perturbations on systems of dynamical equations is briefly described. A straightforward application of the methods of sensitivity analysis to ordinary differential equation models for oscillating reactions is found to yield results which are difficult to physically interpret. In this work it is shown that the standard sensitivity analysis of equations with periodic solutions yields an expansion that contains secular terms. A Lindstedt-Poincare approach is taken instead, and it is found that physically meaningful sensitivity information can be extracted from the straightforward sensitivity analysis results, in some cases. In the other cases, it is found that structural stability/instability can be assessed with this modification of sensitivity analysis. Illustration is given for the Lotka-Volterra oscillator. 

Analytical solutions (e.g., obtained by eigenfunction expansion, Fourier transform, similarity transform, perturbation methods, and the solution of ordinary differential equations for one-dimensional problems) to the conservation equations are of great interest, of course, but they can be obtained only under restricted conditions. When the equations can be rendered linear (e.g., when transport of the conserved quantities of interest is dominated by diffusion rather than convection) analytical solutions are often possible, provided the geometry of the domain and the boundary conditions are not too complicated. When the equations are nonlinear, analytical solutions are sometimes possible, again provided the boundary conditions and geometry are relatively simple. Even when the problem is dominated by diffusive transport and the geometry and boundary conditions are simple, nonlinear constitutive behavior can eliminate the possibility of analytical solution. 

BSE = basis set expansion FDA = finite difference approximation raA = finite element approximation GTF = gauss-ian-type function ODE = ordinary differential equation nD = n-dimensional PDE = partial differential equation PW = partial-wave STF = Slater-type function. 

For certain profiles there are more direct methods for determining ej. In Table 14-2, page 307, the solution for e, is found using separation of variables and the solution of ordinary differential equations. For the clad power-law profiles of Section 14-8, we show that the solutions of the scalar wave equation are expressible as power-series expansions. The cartesian components of e can then be determined as power-series expansions [6]. 

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