Equations—continued

Thus, from an investigation of the compressibility of a gas we can deduce the values of its critical constants. We observe that, according to van der Waals theory, liquid and gas are really two distant states on the same isotherm, and having therefore the same characteristic equation. Another theory supposes that each state has its own characteristic equation, with definite constants, which however vary with the temperature, so that both equations continuously coalesce at the critical point. The correlation of the liquid and gaseous states effected by van der Waals theory is, however, rightly regarded as one of the greatest achievements of molecular theory. 

This equation continues to conserve mass but is no longer stable. The original upset grows exponentially in magnitude and oscillates in sign. This marching-ahead scheme is clearly unstable in the presence of small blunders or round-off errors. 

Schrodinger equation (Continued) time-dependent equation ... 

In this equation, the superscript ( ) indicates that a term is computed based upon the most recent information, which complies with the ( + l)th time level when all iterative loops have converged. Further, the convective transport and viscous generation of fluctuating kinetic energy have been collected in the explicit term D. The iterative solution procedure for the granular energy equations continues until the convergence criteria... 

A sound decomposition strategy should be applicable to any type of mathematical model of a physical process. Therefore, the set of system equations might include linear or nonlinear equations algebraic, differential, difference, or integral equations continuous or discrete variables with the following restrictions ... 

Substituting this equation continues to simplify the transformation law, which emerges as... 

Understanding the order of the hydrodynamics equations, continuity and momentum, can be somewhat confusing and possibly not the same from problem to problem. The continuity and momentum equations must be viewed as a closely coupled system. Again, it is clear that the momentum equations are second order in velocity and first order in pressure. The continuity equation is first order in density. However, an equation of state requires that density be a function of pressure, and vice versa. Density and pressure must be dependent on each other through an algebraic equation. Therefore a substitution could be done to eliminate either pressure or density. As a result the coupled system is third order, which can present some practical issues for boundary-condition assignment. The first-order behavior must carry information from some portions of the boundary into the domain, but it does not communicate information back. Therefore, over some portions of a problem... 

Overall the system of equations (continuity and momentum) is third order, nonlinear, ordinary-differential equation, boundary-value problem. The boundary conditions require no-slip at the plates and specified wall-injection velocities,... 

If Yi equals 0, the model becomes partially recursive. The first equation becomes a regression which can be estimated by ordinary least squares. However, the second equation continues to fail the order condition. To see the problem, consider that even with the restriction, any linear combination of the two equations has the same variables as the original second eqation. 

The approach through the adjoint equation - Continuous case... 

---