Nonlinear coupled system

A recent attempt at a direct stochastic theory by Weinberger [94] using the deterministic flow term as an external (precomputed) constraint should be mentioned here. The intractability of a large coupled system of second-order partial differential equations for the generating function is then reduced to a (nonlinearly coupled) system of ordinary differential equations. The price is the loss of proper population regulation and possible extinction. 

A1.6.4.4 NONLINEAR RESPONSE SYSTEMS COUPLED TO AN ENVIRONMENT (A) ECHO SPECTROSCOPY... 

In molecular dynamics applications there is a growing interest in mixed quantum-classical models various kinds of which have been proposed in the current literature. We will concentrate on two of these models the adiabatic or time-dependent Born-Oppenheimer (BO) model, [8, 13], and the so-called QCMD model. Both models describe most atoms of the molecular system by the means of classical mechanics but an important, small portion of the system by the means of a wavefunction. In the BO model this wavefunction is adiabatically coupled to the classical motion while the QCMD model consists of a singularly perturbed Schrddinger equation nonlinearly coupled to classical Newtonian equations, 2.2. 

This modified density Is a more slowly varying function of x than the density. The domain of Interest, 0 < x < h, Is discretized uniformly and the trapezoidal rule Is used to evaluate the Integrals In Equations 8 and 9. This results In a system of nonlinear, coupled, algebraic equations for the nodal values of n and n. Newton s method Is used to solve for n and n simultaneously. The domain Is discretized finely enough so that the solution changes negligibly with further refinement. A mesh size of 0.05a was adopted In our calculations. 

Nonlinear coupling, multidegenerate conditions higher order coupling, complex representations, 243-244 molecular systems, 233-249 adiabatic-to-diabatic transformation, 241— 242... 

Total molecular wave function, permutational symmetry, 661-668, 674-678 Tracing techniques, molecular systems, multidegenerate nonlinear coupling continuous tracing, component phase, 236-241... 

The more interesting problems tend to be neither steady state nor linear, and the reverse Euler method can be applied to coupled systems of ordinary differential equations. As it happens, the application requires solving a system of linear algebraic equations, and so subroutine GAUSS can be put to work at once to solve a linear system that evolves in time. The solution of nonlinear systems will be taken up in the next chapter. 

Let us now consider a system of two nonlinearly coupled Kerr oscillators. Now, we write the Hamiltonian (25) in the form... 

It is readily apparent that the system of equations is a coupled system of nonlinear partial differential equations. The independent variables are time t and the spatial coordinates (e.g., z, r, 0). For the fluid mechanics alone, the dependent variables are mass density, p, pressure p, and V. In addition the energy equation adds either enthalpy h or temperature T. Finally the mass fractions of chemical species are also dependent variables. 

These two ordinary differential equations are a nonlinear, first-order coupled system, with the axial coordinate z as the independent variable. The dependent variables are U and p. 

Numerical Solution Equations 6.40 and 6.41 represent a nonlinear, coupled, boundary-value system. The system is coupled since u and V appear in both equations. The system is nonlinear since there are products of u and V. Numerical solutions can be accomplished with a straightforward finite-difference procedure. Note that Eq. 6.41 is a second-order boundary-value problem with values of V known at each boundary. Equation 6.40 is a first-order initial-value problem, with the initial value u known at z = 0. 

The system of equations in the Von Mises form leads to a coupled system of nonlinear differential-algebraic equations. The transport equations (Eqs. 7.59 and 7.62) have parabolic characteristics, with the axial coordinate z being the timelike direction. The other three equations (Eqs. 7.60, 7.61, and 7.63) are viewed as algebraic constraints—in the sense that they have no timelike derivatives. 

The first term on the right-hand side corresponds to Eq. (2), whereas the second term describes dissipative effects that are induced in the system due to its coupling to the environment. The latter is modeled, as usual [32, 33], as the thermal (temperature T) ensemble of harmonic oscillators, with nonlinear coupling A Qiq) F( thermal bath, expressed in terms of nonlinear molecular and linear environment coupling operators Q(q) and F( qk )- As shown in Ref. 15, it is important to describe the dissipative term in Eq. (10) by making use of the non-Markovian expression... 

Weakly coupled systems of nonlinear elliptic boundary value problems (with K. Zygour-akis). Nonlin. Anal. 6, 555-569 (1982). 

Monotone iteration methods for solving coupled systems of nonlinear boundary value problems (with K. Zygourakis). Comput. Chem. Eng. 7, 183-193 (1983). 

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