Integration by numerical methods

In order to find the evolution of species concentration or temperature with time, the above equations must be integrated. For complex reaction mechanisms this usually means integration by numerical methods. There are a large number of schemes for the numerical integration of coupled sets of differential equations, but not all will be suitable for the types of mechanisms we are discussing. Chemical systems form a difficult problem because of the differences in reaction time-scale between each of the... 

This equation can only be integrated by numerical methods. The expression (6.32) gives an approximate analytical solution, valid for B A02 L [23] ... 

For the study of time-resolved processes such as discussed in Chapter 6, but with many strongly coupled states in the manifold ipi, a closed-form solution cannot be carried through analytically and approximate treatments are necessary. This is not the case when solving Equation 6.58 for many states, which can always be integrated by numerical methods. All require modern, high-speed computers for their execution and the development of numerical recipes to handle large determinants. Furthermore, numerical solutions may often be obtained much more easily than closed-form solutions and may be sufficiently accurate, as the physical situation warrants. On the other hand, the closed-form solution given by Equation 6.58 serves as a convenient introduction to pursue much more difficult problems when possible. 

Simultaneous integration of these equations by numerical methods can provide the concentrations of A,B1,B2 and B3 as a function of time and the concentrations of o-methylol, p-methylol and methylene ether (note earlier definition) groups can be calculated as follows, where CONC is the initial monomer concentration. 

For kinetic expressions which are just a little more complicated than first oder kinetics, the rate cannot be integrated by hand. Very complex kinetic expressions are easy to treat by numerical methods, but analytical treatment is only possible for extremely simple kinetic expressions. 

One important use of the stream function is for the visualization of flow fields that have been determined from the solution of Navier-Stokes equations, usually by numerical methods. Plotting stream function contours (i.e., streamlines) provides an easily interpreted visual picture of the flow field. Once the velocity and density fields are known, the stream function field can be determined by solving a stream-function-vorticity equation, which is an elliptic partial differential equation. The formulation of this equation is discussed subsequently in Section 3.13.1. Solution of this equation requires boundary values for l around the entire domain. These can be evaluated by integration of the stream-function definitions, Eqs. 3.14, around the boundaries using known velocities on the boundaries. For example, for a boundary of constant z with a specified inlet velocity u(r),... 

Slater-type orbitals were introduced in Section 5.2 (Eq. (5.2)) as the basis functions used in extended Hiickel theory. As noted in that discussion, STOs have a number of attractive features primarily associated with the degree to which they closely resemble hydrogenic atomic orbitals. In ab initio HF theory, however, they suffer from a fairly significant limitation. There is no analytical solution available for the general four-index integral (Eq. (4.56)) when the basis functions are STOs. The requirement that such integrals be solved by numerical methods severely limits their utility in molecular systems of any significant size. 

Product Distribution at the Riser Exit With X or calculated, YQ, YE, Y and Y at the riser exit must be now be computed. This can be done by integrating the kinetic equations by numerical methods with the following boundary conditions YA = YAo and YQ = Yg = YQ = Y = 0. 

Since equation (5.27) can hardly be integrated in quadratures, calculations are carried out with the use of different approximate methods. For example, the piecemeal-analytical method may successfully be employed.332 It is based on dividing the examined time range into a finite number of sufficiently short intervals and the subsequent application of equation (5.29) to each of them. Equation (5.27) can be transformed into a transcendental equation239 which is then solved by numerical methods. 

Thus, if a2 is known as a function of x2, the definite integral may be evaluated by numerical methods, yielding aj,. 

Melln, J.W., Numerical Integration by Beta Method, ASCE Conference on Electronic Computation, Kansas City, Mo., I968. (N)... 

It was possible to solve this polymerization problem in closed form because we were dealing with algebraic equations, that is, with an ideal stirred-tank reactor. The same problem in a tubular-flow reactor would require the solution of a series of integral equations, which is possible only by numerical methods. 

Although analytical solutions of Eqs. (12.31) and (12.32) exist,the integration is usually carried out by numerical methods. 

The integral in the integral equation (9.14) can be computed only by numerical methods, but analytical solutions are possible in three Umiting cases for the case that A kfiT,... 

This is the scheme of a polymerization reaction by addition of radicals. Although this system is complex and usually solved by numerical methods, the general solution using the integral method will be shown here. This is the easiest way to identify the kinetic parameters involved and indicate a general solving method for complex reactions of this type, although the numerical solution is more appropriate. We should start from a batch system (constant volume), whose equations for the rates of reactants and products are described as follows ... 

The integration of the system of differential Equations (4.9-4.12) for concurrent flow can be obtained by numerical methods. When a stripping solution is used as the stripping agent, an instantaneous reaction is assumed to occur on the outside of the fiber, leading to Cj = 0 and C/= 0. In this case, the solution to Equations (4.9-4.12)... 

The problem in interpretation of EPR spectra of textured paramagnetic complexes in liquid crystals, or radicals in prolated polymeric compounds was already discussed in literature. We can find some experimental and theoretical papers on this problem [1-3]. From the point of view of an experimental technique the EPR spectra acquisition and mathematical simulation algorithm for paramagnetic centers in a fiat surface film do not differ from centres in oriented liquid crystal solutions or inserted in prolated polymeric compounds. Solving the problem in a general form requires triple integration and is possible only by numerical methods. Attempts to solve this problem analytically were successful only in very particular cases. The techniques proposed in these papers based on theoretical tables and plots is not obvious enough. 

The integral equation for the elastic boundary tractions and displacements is solved by numerical methods. The boundary is divided Into N finite length elements. In this paper the surface tractions and displacements are assumed to change linearly over each of the boundary elements. Figure 2 shows a typical boundary the surface tractions are prescribed on part of the boundary and the displacements are prescribed on the remaining part of the boundary. At each node point on the boundary there are two components of traction and two components of displacement. Thus, for N elements and N nodes there are 2N unknowns in the discretized system. The boundary Integral equation for the elasticity problem Is rewritten as below ... 

The value of the constant incorporated into (8.120) is dictated by the condition of periodicity. The resulting ordinary differential equation is readily solved by numerical methods, with the constants of integration being determined by the constraint of periodicity and conservation of mass. 

The present study shows that It is possible to evaluate the variability of statically determinate and statically indeterminate structures due to spatial variation of elastic properties without resort to finite element analysis. If a Green s function formulation is used, the mean square statistics of the indeterminate forces are obtained in a simple Integral form which is evaluated by numerical methods in negligible computer time. It was shown that the response variability problem becomes a problem Involving only few random variables, even if the material property is considered to constitute stochastic fields. The response variability was estimated using two methods, the First-Order Second Moment method, and the Monte Carlo simulation technique. 

The most common algorithms in use today are discretized PI and PID algorithms in which the continuous functions of integration and differentiation are approximated by numerical methods. A PID controller that is approximated by simple rectangular integration is ... 

The left-hand side of Eq. (20) is reducible to an exponential integral function of the pth order, so the exact solution of the integral transcendental equation (23) for Ox can be obtained only by numerical methods and only if the function /f Ox) is known. Nevertheless, it follows from the general form of Eq. (23) that is independent of v and cfe and is determined only by the values ofp and aox- The latter is involved in Eq. (23) through ... 

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