Nonlinear numerical problems

The partial differential equations describing the catalyst particle are discretized with central finite difference formulae with respect to the spatial coordinate [50]. Typically, around 10-20 discretization points are enough for the particle. The ordinary differential equations (ODEs) created are solved with respect to time together with the ODEs of the bulk phase. Since the system is stiff, the computer code of Hindmarsh [51] is used as the ODE solver. In general, the simulations progressed without numerical problems. The final values of the rate constants, along with their temperature dependencies, can be obtained with nonlinear regression analysis. The differential equations were solved in situ with the backward... 

Keller, H. B. Numerical Solution of Bifurcation and Nonlinear Eigenvalue Problems, In Applications of Bifurcation Theory, Rablnowltz,P., Ed. Academic Press New York, 1977. 

Consequently, modeling of a two-phase flow system is subject to both the constraints of the hydrodynamic equations and the constraint of minimizing N. Such modeling is a nonlinear optimization problem. Numerical solution on a computer of this mathematical system yields the eight parameters ... 

The partial differential equations representing material and energy balances of a reaction in a packed bed are rarely solvable by analytical means, except perhaps when the reaction is of zero or first order. Two examples of derivation of the equations and their analytical solutions are P8.0.1.01 and P8.01.02. In more complex cases analytical, approximations can be made (by "Collocation" or "Perturbation", for instance), but these usually are quite sophisticated to apply. Numerical solutions, on the other hand, are simple in concept and are readily implemented on a computer. Two such methods that are suited to nonlinear kinetics problems will be described. 

If/(x) has a simple closed-form expression, analytical methods yield an exact solution, a closed form expression for the optimal x, x. Iff(x) is more complex, for example, if it requires several steps to compute, then a numerical approach must be used. Software for nonlinear optimization is now so widely available that the numerical approach is almost always used. For example, the Solver in the Microsoft Excel spreadsheet solves linear and nonlinear optimization problems, and many FORTRAN and C optimizers are available as well. General optimization software is discussed in Section 8.9. 

Although in this chapter we have chosen to linearize the mathematical system after reduction to a system of ordinary differential equations, the linearization can be performed prior to or after the reduction of the partial differential equations to ordinary differential equations. The numerical problem is identical in either case. For example, linearization of the nonlinear partial differential equations to linear partial differential equations followed by application of orthogonal collocation results in the same linear ordinary differential equation system as application of orthogonal collocation to the nonlinear partial differential equations followed by linearization of the resulting nonlinear ordinary differential equations. The two processes are shown ... 

In a strict sense parameter estimation is the procedure of computing the estimates by localizing the extremum point of an objective function. A further advantage of the least squares method is that this step is well supported by efficient numerical techniques. Its use is particularly simple if the response function (3.1) is linear in the parameters, since then the estimates are found by linear regression without the inherent iteration in nonlinear optimization problems. 

Nonlinear Systems and Optimization for the Chemical Engineer Solving Numerical Problems... 

Glowinski, R. (1983). Numerical Methods for Nonlinear Variational Problems (Springer-Verlag, Berlin). 

H. Keller, 1977, Numerical solution of bifurcation and nonlinear eigenvalue problems. In P. Rabinowitz (Ed.), Application of Bifurcation Theory,... 

S. G. Nash, in Numerical Optimization 1984, P. T. Boggs, R. H. Byrd, and R. B. Schnabel, Eds., pp. 119-136, SIAM, Philadelphia, 1985. Solving Nonlinear Programming Problems Using Truncated-Newton Techniques. 

The principal numerical problem associated with the solution of (7) is that lengthy calculations are required to integrate several coupled nonlinear equations in three dimensions. However, models based on a fixed coordinate approach may be used to predict pollutant concentrations at all points of interest in the airshed at any time. This is in contrast to moving cell methods, wherein predictions are confined to the paths along which concentration histories are computed. 

Difficulties arise even in forward modeling because of the huge size of the numerical problem to be solved for adequate representation of the complex 3-D distribution of EM parameters in the media. As a re.sult, computer simulation time and memory requirements could be excessive even for practically realistic models. Additional difficulties are related to EM imaging which is based on EM inverse problems. These problems are nonlinear and ill-posed, because, in general cases, the solutions can be unstable and/or nonunique. In order to overcome these difficulties one should... 

When more than one reaction is considered, which is the usual situation faced in applications, we require numerical methods to find the equilibrium composition. Two approaches to this problem were presented. We either solve a set of nonlinear algebraic equations or solve a nonlinear optimization problem subject to constraints. If optimization software is available, the optirnization approach is more powerful and provides more insight. 

Some of the numerical problems in nonlinear regression seem to be able to be partially avoidable by utilizing special features of a given transfer function, as proposed by Mixon, Whitaker, and Orcutt [ 104] and extended by 0ster-gaard and Michelsen [ 105]. This is best illustrated by considering the transfer function of the axial dispersion model with semiinfinite boundary conditions ... 

Vol. Ill Buzzi-Ferraris and Manenti (2014) Nonlinear Systems and Optimization for the Chemical Engineer. Solving Numerical Problems, Wiley-VCH Verlag GmbH, Weinheim, Germany. 

While 0, some numerical problems may arise if the matrix A is singular (differential-algebraic systems). For this reason, in the general case, we cannot force convergence by simply decreasing the integration step. Instead, it is necessary to use more sophisticated nonlinear system solution programs. 

Although DAE problems of any index should be solved theoretically, serious numerical problems may arise in a general program. As seen in Chapter 2, in the case of ODEs too, the explicit form has special features that can be exploited to solve, in a stable manner, the corresponding nonlinear system ... 

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