Boundary value problem, conditionally stable

The difference boundary-value problem associated with the difference equation (7) of second order can be solved by the standard elimination method, whose computational algorithm is stable, since the conditions Ai 0, Ci > Ai -f Tj+i are certainly true for cr > 0. 

The system (2.30), and a more complicated one, can be found in [5]. HI. BOUNDARY VALUE PROBLEM A. Conditionally Stable Case... 

Indeed, for a certain range of operating conditions, three steady-state profiles are possible with the same feed conditions, as is shown in Fig. 11.6-2. The outer two of these steady state profiles are stable, at least to small perturbations, whereas the middle one is unstable. Which steady-state profile will be predicted by steady-state computations depends on the initial guesses of Ca and T involved in the integration of this two-point boundary value problem. Physically, this means that the steady state actually experienced depends on the initial profile in the reactor. For all situations where the initial values are different from the feed conditions, transient equations have to be considered to make sure that the correct steady-state profile is predicted. To avoid those transient computations when they are unnecessary, it is useful to know a priori if more than one steady-state profile is possible. From Fig. 11.6-2 it is seen that a necessary and sufficient condition for uniqueness of the steady-state profile in an adiabatic reactor is that the curve Tq = f[1 I)] has no hump. 

The following (initial and) boundary conditions allow the solution of the botmdary value problem for a reversible charge transfer involving stable soluble species (the so-called uncomplicated charge transfer process ) ... 

Thus, for two free surfaces, the eigenvalue problem for a is reduced to finding a nontrival solution of Eq. (12-199), subject to the six boundary conditions, (12-200), (12-203), and (12-204). In particular, let us suppose that we specify Pr, Gr, and a2 (the wave number of the normal model of perturbation). There is then a single eigenvalue for a such that / / 0. If Real(er) < 0 for all a, the system is stable to infinitesimal disturbances. On the other hand, if Real(er) > 0 for any a, it is unconditionally unstable. Stated in another way, the preceding statements imply that for any Pr there will be a certain value of Gr such that all disturbances of any a decay. The largest such value of Gr is called the critical value for linear stability. 

The particular boundary conditions to be satisfied at the top and bottom surfaces of the fluid layer determine the form of the z dependence of the solution, i.e., the functions T z), n (z), etc. These are then used to locate the marginal state and thereby the critical point of the system, by recalling that the parametric space (defined by the parameters of the differential equations) contains both stable and imstable regions and hence a boundary between these two domains which is termed the marginal state or the state of neutral stability. Furthermore, in most stability problems, all the parameters of the system are fixed a priori except for one which is allowed to vary continuously over some range this one parameter then possesses some critical value that separates the stable from the unstable portions of the range. 

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