The Nonlinear Case

Solution of the second-order two-point boundary value problem (5.24) for a second-order reaction 

As explained in Section 1.2.2, the numerical solution of second- or higher-order DEs is generally approached numerically from an equivalent but enlarged first-order system of 

Therefore our first step is to put the second-order differential equation (5.36) into the form of two first-order ones. For this we set x = x and x i = dx/dui. Then 

Let us investigate how MATLAB handles boundary value problems such as (5.24) with the boundary conditions (5.25) and (5.26) for first- and second-order reactions. The MATLAB program linquadbvp.m has been designed for this purpose. 

Output Plots of the solutions (theoretical [if known] and numerical) 

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Crowe, C.M., Reconciliation of Process Flow Rates by Matrix Projection, part 2, The Nonlinear Case," AlChE Journal, 32(4), 1986, 616-62.3. 

Both these equations are of the Mathieu type (6-126) is linear and (6-127) nonlinear on account of the term cx3. It is well known that in the linear case one can eliminate the term bx by the classical transformation of the dependent variable. In the nonlinear case this is impossible and one has to keep the term bx. 

In both the linear and the nonlinear cases the total variation of the residuals is the sum of the random error, plus the departure from linearity. When the data is linear, the variance due to the departure from nonlinearity is effectively zero. For a nonlinear set of data, since the X-difference between adjacent data points is small, the nonlinearity of the function makes minimal contribution to the total difference between adjacent residuals and most of that difference contributing to the successive differences in the numerator of the DW calculation is due to the random noise of the data. The denominator term, on the other hand, is dependent almost entirely on the systematic variation due to the curvature, and for nonlinear data this is much larger than the random noise contribution. Therefore the denominator variance of the residuals is much larger than the numerator variance when nonlinearity is present, and the Durbin-Watson statistic reflects this by assuming a value less than 2. 

Crowe, C. M. (1986). Reconciliation of process flow rates by matrix projection. Part II The nonlinear case. AIChEJ. 32,616-623. 

In this chapter we concentrate on the statement and further solution of the general steady-state data reconciliation problem. Initially, we analyze its resolution for linear plant models, and then the nonlinear case is discussed. 

Notice that in this case, all the terms S have the same known dimension. This feature will not be maintained in the nonlinear case, but it will be treated under a slightly different approach. 

Remark f. Notice that for a linear system the coefficients ao,, ai j,..., in equation (48) represent the coefficients of the characteristic equation of matrix S. For the nonlinear case, these coefficients do not represent a generalization of the Cayley-Hamilton theorem hence the assumption is necessary for the existence of the solution of the NRRP. 

The Fokker-Planck approach does not lead lo a sensible extension for the nonlinear case either. Suppose one writes3... 

It is again impossible to choose ax(q) such that fluctuation spectrum led to physically different results. 4The conclusion is that the clever guess of Langevin does not work in the nonlinear case and that a more fundamental starting point is indispensable. 

Of course, the macroscopic equations cannot actually be derived from the microscopic ones. In practice they are pieced together from general principles and experience. The stochastic mesoscopic description must be obtained in the same way. This semi-phenomenological approach is remarkably successful in the range where the macroscopic equations are linear, see chapter VIII. In the nonlinear case, however, difficulties appear, which can only be resolved by the improved, but still mesoscopic, method of chapter X. 

Exercise. Also find the solutions of (3.4) and (3.5) for the general linear one-step process. Why can they not be solved in the nonlinear case ... 

Students and readers should be very familiar with the nature of the isothermal case before embarking on the nonlinear case and its numerical solution in Section 5.1.3. The... 

Besides, if SHa — oo, then xa w=i.o = xAb This also corresponds to a negligible external mass transfer resistance. In both cases, that of a finite Sherwood number SHa or for SHa — oo, we get a two-point boundary value differential equation. For the nonlinear case this has to be solved numerically. However, as for the axial dispersion model, we will start out with the linear case that can be solved analytically. 

For the nonlinear case, the nonlinear two-point boundary value differential equation(s) for the catalyst pellet can be solved using the same method as used for the axial dispersion model in Section 5.1, i.e., by the orthogonal collocation technique of MATLAB s bvp4c. m boundary value solver. 

In conclusion, the ADM is a more convenient method to obtain approximate solutions for the models with arbitrary reaction orders, n, so that the relationship between effectiveness c and Thiele modulus can be quantitatively understood in more details for the nonlinear cases. 

One may surmise that the low-frequency limit, introduced while discussing the linear relaxation, would also lead to a reliable simplification in the nonlinear case since the process is governed mainly by the relaxation time xio. As we were tempted by this idea, in Ref. 67 we have supposed that the approximate expression... 

For the sake of simplicity, in what follows it will be considered that the double layer potential is sufficiently small to allow the linearization of the Poisson—Boltzmann equation (the Debye—Hiickel approximation). The extension to the nonlinear cases is (relatively) straightforward however, it will turn out that the differences from the DLVO theory are particularly important at high electrolyte concentrations, when the potentials are small. In this approximation, the distribution of charge inside the double layer is given by... 

An important property of the stochastic version of compartmental models with linear rate laws is that the mean of the stochastic version follows the same time course as the solution of the corresponding deterministic model. That is not true for stochastic models with nonlinear rate laws, e.g., when the probability of transfer of a particle depends on the state of the system. However, under fairly general conditions the mean of the stochastic version approaches the solution of the deterministic model as the number of particles increases. It is important to emphasize for the nonlinear case that whereas the deterministic formulation leads to a finite set of nonlinear differential equations, the master equation... 

The material behaviors considered will include linear elasticity plus linear or nonlinear creep behavior. The nonlinear case will be restricted to power-law rheologies. In some cases the elasticity will be idealized as rigid. In ceramics, it is commonly the case that creep occurs by mass transport on the grain boundaries.1 This usually leads to a linear rheology. In the models considered,... 

Carry out the least-squares minimization of the quantity in Eq. (7) according to an appropriate algorithm (presumably normal equations if the observational equations are linear in the parameters to be determined otherwise some other such as Marquardf s ). The linear regression and Solver operations in spreadsheets are especially useful (see Chapter HI). Convergence should not be assumed in the nonlinear case until successive cycles produce no significant change in any of the parameters. 

Going back to the nonlinear case governed by Eq. (145), the distribution in the product stream and the residual are... 

In the nonlinear case the standard Fokker-Planck equation can be regarded as a good approximation only when for a given nonlinearity the heat bath relaxation time is small enough to make the effects of higher-order corrections negligible. 

In the nonlinear case, the function ij t does not vanish (at times intermediate between t = 0 and t = oo). The itinerant oscillator with an effective potential harder than the linear one is shown to result in ri(t) < 0 in accordance with the results of CFP calculations which show its decay after excitation to be faster than the corresponding equilibrium correlation function (see Fig. 11). 

N. Dyn and P.Oswald Univariate subdivision and multiscale transforms The nonlinear case. pp203-247 in Multiscale, Nonlinear and Adaptive Approximation (eds R.DeVore, A.Kunoth), Springer, 2009 M.Sabin Two open questions relating to subdivision. Computing 86 pp213-217, 2009... 

Here Vsoiid is the apparent solid flow rate, HA and ffB describe the slopes of the adsorption isotherm, which are calculated in the nonlinear case by linearization of the adsorption isotherm for the feed concentration Cfeed, . The transformation reflects the fact that, in a counter-current process, it is not the net flow rates that are important but rather their values relative to the apparent solid movement. For this reason, Morbidelli et al. introduced the m factors in their graphical design (known as the triangle theory) (Biressi et al., 2000 and Mazzotti et al., 1997c). 

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