Nonlinear problems, intractability

In the previous three chapters, we described various analytical techniques to produce practical solutions for linear partial differential equations. Analytical solutions are most attractive because they show explicit parameter dependences. In design and simulation, the system behavior as parameters change is quite critical. When the partial differential equations become nonlinear, numerical solution is the necessary last resort. Approximate methods are often applied, even when an analytical solution is at hand, owing to the complexity of the exact solution. For example, when an eigenvalue expression requires trial-error solutions in terms of a parameter (which also may vary), then the numerical work required to successfully use the analytical solution may become more intractable than a full numerical solution would have been. If this is the case, solving the problem directly by numerical techniques is attractive since it may be less prone to human error than the analytical counterpart. 

In cases of different dispersion coefficients (laminar flow), the analytical problem is close to intractable. For nonlinear reactions the axial dispersion model leads to a set of two-point boundary value problems which must be solved by an appropriate iterative numerical scheme. This is a great disadvantage of the model. We conclude that the axial dispersion model is cumbersome in reactor type calculations and should be abandoned. The reasons for this can be stated as follows. 

If the profile n(r. A) does not satisfy the linear-dispersion condition of Eq. (3-13), then we have nonlinear dispersion. The separable term 2A(A)/(r) in Eq. (3-12) is replaced by F (A, r), which is not expressible in separable form, e.g. the clad power-law profiles of Eq. (3-15) when the exponent is wavelength dependent. The determination of the optimal profile is then a virtually intractable problem by analytical methods. However, if we make certain simplifying assumptions about the form of the transit time, we can pose the problem in a different and more tractable way. 

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