Ricatti equations

These equations reduce to a 3 x 3 matrix Ricatti equation in this case. In the appendix of [20], the efficient iterative solution of this nonlinear system is considered, as is the specialization of the method for linear and planar molecules. In the special case of linear molecules, the SHAKE-based method reduces to a scheme previously suggested by Fincham[14]. 

LHSFs are determined at the center p of each shell. These LHSFs are then used to obtain the coupling matrix V i /nr(p p) given in Eq. (102). The coupled hyperradial equations in Eq. (101) are transformed into the coupled first-order nonlinear Bessel-Ricatti logarithmic matrix differential equation... 

Another approach is known as the local optimization method. Here local means that maximization of the objective function J is carried out at each time, i.e., locally in time between 0 and tf. There are several methods for deriving an expression for the optimal laser pulse by local optimization. One is to use the Ricatti expression for a linear time-invariant system in which a differential equation of a function connecting [r(t) and (f) is solved, instead of directly solving for these two functions. Another method... 

Ricatti equations can be solved if F, Q, and R are constant. The solution is then... 

The numerical procedure used in association with the p, ax coordinates is as follows (12). The internal configuration space is divided into a number of spherical hyperradial sectors. The two-dimensional LHSFs are then determined at the center p of each sector and used to obtain the coupling matrix V7111 (p p) over the entire sector. Equation (178) is transformed into the firstorder nonlinear Bessel-Ricatti matrix differential equation... 

Let us now introduce p via /3(/3 + 1) = a. Weakly attractive potentials correspond to -1/2 < /I < 0 and repulsive potentials to > 0. In both cases, the solution of the Schrodinger equation at large r can be written as a linear combination of the Ricatti-Bessel functions /(kr) and h/s(kr), and the order of the corresponding cylinder functions v = p + 1/2 remains positive. Using analytic properties of the Bessel functions as kr 0, to lowest order in k,... 

This is exactly like the special form of Ricatti s equation with Q = 0,7 = x, so let... 

We applied the Ricatti transformation to a nonlinear equation in Example 2.10 and arrived at the linear Airy equation... 

This is achieved by the solution of two algebraic Ricatti equations, while in classical structural control one such equation arises. 

Thus we have obtained a set of coupled second-order differential equations in the diffraction channels. The equations (5.5) may be solved efficiently by a number of methods, such as the log-derivative method, in which 4>c = (d In c/dz) = (d G/dz) G is propagated instead of g [126]. Thus substitution transforms the second-order differential equation to a first-order nonlinear Ricatti equation. 

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