Differential equations numerical solution

Batch and bio process modeling often results in a set of ordinary differential equations. Numerical solution techniques for solving these differential equations normally involve approximating the differential equations by difference equations that are solved in a step-by-step marching fashion. 

Many reaction schemes with one or more intermediates have no closed-form solution for concentrations as a function of time. The best approach is to solve these differential equations numerically. The user specifies the reaction scheme, the initial concentrations, and the rate constants. The output consists of concentration-time values. The values calculated for a given model can be compared with the experimental data, and the rate constants or the model revised as needed. Methods to obtain numerical solutions will be given in the last section of this chapter. 

Davidson et al. (13) developed numerical solutions of the differential equation for solute transport for a model that... 

The reason for constructing this rather complex model was that even though the mathematical equations may be easily set up using the dispersion model, the numerical solutions are quite involved and time consuming. Deans and Lapidus were actually concerned with the more complicated case of mass and heat dispersion with chemical reactions. For this case, the dispersion model yields a set of coupled nonlinear partial differential equations whose solution is quite formidable. The finite-stage model yields a set of differential-double-difference equations. These are ordinary differential equations, which are easier to solve than the partial differential equations of the dispersion model. The stirred-tank equations are of an initial-value type rather than the boundary-value type given by the dispersion model, and this fact also simplifies the numerical work. 

In this text all numerical problems involve integration of simultaneous ordinary differential equations or solution of simultaneous algebraic equations. You should have no trouble finding ways to solve algebraic equations with a calculator, a spreadsheet, a personal computer, etc. 

In this section we deal with estimating the parameters p in the dynamical model of the form (5.37). As we noticed, methods of Chapter 3 directly apply to this problem only if the solution of the differential equation is available in analytical form. Otherwise one can follow the same algorithms, but solving differential equations numerically whenever the computed responses are needed. The partial derivations required by the Gauss - Newton type algorithms can be obtained by solving the sensitivity equations. While this indirect method is... 

Mean temperature differences in such flow patterns are obtained by solving the differential equation. Analytical solutions have been found for the simpler cases, and numerical ones for many important complex patterns, whose results sometimes are available in generalized graphical form. 

Three-dimensional models can handle complex cases with fewer empirical features. They require the solution of multidimensional, partial differential equations. These solutions are numerical tools and require the use of a supercomputer, or very long computation times on less sophisticated hardware. 

While the boundary layer equations that were derived in the preceding sections are much simpler than the general equations from which they were derived, they still form a complex set of simultaneous partial differential equations. Analytical solutions to this set of equations have been obtained in a few important cases. For the majority of flows, however, a numerical solution procedure must be adopted. Such solutions are readily obtained today using modest modem computing facilities. This was, however, not always so. For this reason, approximate solutions to the boundary layer equations have in the past been quite widely used. While such methods of solution are less important today, they are still used to some extent. One such approach will, therefore, be considered in the present text. 

This equation defines directly the change in concentration of the spedes AB with given concentrations of the reactants A and B, and the product AB. This is a differential equation whose solution is an expression of the form cAB=f(t, c% eg). The solution involves a process of integration, which is often difficult, and sometimes impossible, at least analytically. In such cases, numerical integration can be used to simulate the time-dependent variation of cAB in an experiment, enabling theoretical data to be obtained even for complex systems. 

In order to demonstrate the physical significance of asymjjtotic nonadiabatic transitions and especially the aiialj-tical theory developed an application is made to the resonant collisional excitation transfer between atoms. This presents a basic physical problem in the optical line broadening [25]. The theoretical considerations were mad( b( for< [25, 27, 28, 29, 25. 30] and their basic id( a has bec n verified experimentally [31]. These theoretical treatments assumed the impact parameter method and dealt with the time-dependent coupled differenticil equations imder the common nuclear trajectory approximation. At that time the authors could not find any analytical solutions and solved the coupled differential equations numerically. The results of calculations for the various cross sections agree well with each other and also with experiments, confirming the physical significance of the asymptotic type of transitions by the dipole-dipole interaction. 

Mathematical models of flow processes are non-linear, coupled partial differential equations. Analytical solutions are possible only for some simple cases. For most flow processes which are of interest to a reactor engineer, the governing equations need to be solved numerically. A brief overview of basic steps involved in the numerical solution of model equations is given in Section 1.2. In this chapter, details of the numerical solution of model equations are discussed. 

The mathematical origin of these concentration discontinuities or weak solutions of Eq. 7.1 has been explained by Courant and Friedrichs [24] and by Lax [25]. The mathematical backgroimd has been reviewed in cormection with the discussion of the numerical solution of the equilibrium-dispersive model given by Rou-chon et al. [26]. In the traditional theory of partial differential equations, a solution should be continuous. Lax [27] generalized the concept of solution to include weak solutions, which are not continuously differentiable. A solution of Eq. 7.1 that includes a continuous part, or diffuse bormdary, and a concentration shock is a weak solution of this equation [1,26-28]. A serious problem then arises, since there is no unique weak solution of Eq. 7.1. It is necessary to define the weak solution that is acceptable for the physical problem in order to achieve the determination of the band profile. This solution must make physical sense and prevent the crossing of the characteristics. Oleinik has suggested a selection rule that can... 

H. A. Deans and L. Lapidus [AJChE J., 6, 656, 663 (I960)] consider the reactor to consist of an assembly of cells, with complete mixing within each cell and with a separate cell associated with each catalyst pellet. The balances then are written as difference equations. Numerical solution of differential equations by machine computation is described in L. Lapidus, Digital Computation for Chemical Engineers, McGraw-Hill Book Company, chap. 4, New York, 1960. 

It has been shown earlier (Chapter 2) that it is possible to use the vaiiation principle to derive a single differential equation whose solutions are the opti-iimm orbitals of a single-determinant wavefunction. We now wish to carry this derivation forward in order to be able to have a practical method of obtaining (approximations to) these optimum orbitals. In particular, since the solution of a highly non-linear differential equation by numerical methods for molecules of arbitrary geometry is out of the question, it is desirable to transform the differential equation into an algebraic equation which will he more amenable to a systematic method of solution independently of the size and shape of the molecule, radical, ion or group of molecules. 

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. 

The three chapters in this volume deal with various aspects of singular perturbations and their numerical solution. The first chapter is concerned with the analysis of some singular perturbation problems that arise in chemical kinetics. In it the matching method is applied to find asymptotic solutions of some dynamical systems of ordinary differential equations whose solutions have multiscale time dependence. The second chapter contains a comprehensive overview of the theory and application of asymptotic approximations for many different kinds of problems in chemical physics, with boundary and interior layers governed by either ordinary or partial differential equations. In the final chapter the numerical difficulties arising in the solution of the problems described in the previous chapters are discussed. In addition, rigorous criteria are proposed for... 

One of the attractive features of Markov models is that numerical results can be obtained without an extensive knowledge of probabihty theory and techniques. The intuitive ideas of probabihty discussed in the previous section are sufficient for the construction of a Markov model. Once the model is constructed, it can be computed by differential equations. The solution does not require any additional knowledge of probabihty. 

In ref. 145 the authors develop two families of explicit and implicit BDF methods (Backward Differentiation Methods), for the accurate integration of differential equations with solutions any linear combinations of exponential of matrices, products of the exponentials by polynomials and products of those matrices by ordinary polynomials. More specifically, the authors study the numerical solution of the problem ... 

We will state in this chapter the mathematical task of parameter identification and discuss the corresponding numerical methods. Techniques from various branches of numerical mathematics are required, e.g. numerical solution of differential equations, numerically solving nonlinear problems especially large-scale constrained nonlinear least squares problem. Thus, some of the methods discussed in the previous chapters will reappear here. We will see how parameter identification problems can be treated efficiently by boundary value problem (BVP) methods and extend the discussion of solution techniques for initial value problems (IVPs) to those for BVPs. 

When integrating a differential equation numerically, one would expect the suggested step size to be relatively small in a region in which the solution curve displays much variation and to be relatively large where the solution curve straightens out to approach a line with a slope of nearly zero. Unfortunately, this is not always the case. The DDEs that make up the mathematical models of most chemical engineering systems usually represent a collection of fast and slow dynamics. For instance, in a typical distillation tower, the liquid mechanics (e.g., flow, hold-up) is considered as fast dynamics (time constant seconds), compared with the tray composition slow dynamics (time constant minutes). Systems with such a collection of fast and slow ODEs are denoted stiff systems. 

Since they can t be solved explicitly for tlie stresses as was the case for the Newtonian fluid, it is not possible to obtain a differential equation such as Eq. 10.70, which can be solved to find R/R directly. To solve these equations one must guess at values of R/R first, solve the set of coupled differential equations numerically, and then determine whether the stress values satisfy Eq. 10.67. One must repeat this process until Eq. 10.67 is satisfied for a given pressure differential. (This approach is used in Problem 10C.3.) Because the solution is obtained numerically, the guess for R/R is made for a small time step over which it is assumed that R/R is constant, and hence R t) = / oCxp(Cr). The thickness distribution is then determined using the calculated value of R(t) and Eqs. 10.58 and 10.54. ... 

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