Differential equations, ordinary particular solution

This is not the place for a treatise on the solution of differential equations, ordinary or partial. There are some excellent mathematical texts and some, such as Varma and Morbidelli s Mathematical Methods in Chemical Engineering (Oxford University Press, 1997), are specifically directed at the chemical engineer. What we shall try to do, however, is to explore some of the ad hoc methods that take advantage of peculiar features of particular problems and those that give partial solutions, as well as mentioning a few fall-traps for the unwary. 

The solution of Equations 47, 48, and 49 requires numerical techniques. For such nonlinear equations, it is usually wise to employ a simple numerical integration scheme which is easily understood and pay the price of increased computational time for execution rather than using a complex, efficient, numerical integration scheme where unstable behavior is a distinct possibility. A variety of simple methods are available for integrating a set of ordinary first order differential equations. In particular, the method of Huen, described in Ref. 65, is effective and stable. It is self-starting and consists of a predictor and a corrector step. Let y = f(t,y) be the vector differential equation and let h be the step size. 

Chapters 2-5 are concerned with concrete difference schemes for equations of elliptic, parabolic, and hyperbolic types. Chapter 3 focuses on homogeneous difference schemes for ordinary differential equations, by means of which we try to solve the canonical problem of the theory of difference schemes in which a primary family of difference schemes is specified (in such a case the availability of the family is provided by pattern functionals) and schemes of a desired quality should be selected within the primary family. This problem is solved in Chapter 3 using a particular form of the scheme and its solution leads us to conservative homogeneous schemes. 

In this chapter we concentrate on dynamic, distributed systems described by partial differential equations. Under certain conditions, some of these systems, particularly those described by linear PDEs, have analytical solutions. If such a solution does exist and the unknown parameters appear in the solution expression, the estimation problem can often be reduced to that for systems described by algebraic equations. However, most of the time, an analytical solution cannot be found and the PDEs have to be solved numerically. This case is of interest here. Our general approach is to convert the partial differential equations (PDEs) to a set of ordinary differential equations (ODEs) and then employ the techniques presented in Chapter 6 taking into consideration the high dimensionality of the problem. 

We look for its particular solution of the form 0o =/(x0 — x3). On direct check, we become convinced of the fact that this function satisfies the first two equations of system (43) and that, the third one reduces to the ordinary differential equation... 

Inserting (52) into (46) yields a system of ordinary differential equations for the functions ( ). If we succeed in constructing its general or particular solution, then substituting it into (52) gives an exact solution of the Yang-Mills equations (46). However, the so-constructed solution will have an unpleasant feature of being asymmetric in the variables , while Eqs. (46) are symmetric in these. 

Clearly, efficiency of the symmetry reduction procedure is subject to our ability to integrate the reduced systems of ordinary differential equations. Since the reduced equations are nonlinear, it is not at all clear that it will be possible to construct their particular or general solutions. That it why we devote the first part of this subsection to describing our technique for integrating the reduced systems of nonlinear ordinary differential equations (further details can be found in Ref. 33). 

Systems (87) and (91) contain 12 nonlinear second-order ordinary differential equations with variable coefficients. That is why there is little hope for constructing their general solutions. Nevertheless, it is possible to obtain particular solutions of system (87), whose coefficients are given by formulas 2-4 from (91). 

Figure E.l represents a highly simplified view of an ideal structure for an application program. The boxes with the rounded borders represent those functions that are problem specific, while the square-comer boxes represent those functions that can be relegated to problem-independent software. This structure is well-suited to problems that are mathematically systems of nonlinear algebraic equations, ordinary differential equation initiator boundary-value problems, or parabolic partial differential equations. In these cases the problem-independent mathematical software is usually written in the form of a subroutine that in turn calls a user-supplied subroutine to define the system of equations. Of course, the analyst must write the subroutine that describes the particular system of equations. Moreover, for most numerical-solution algorithms, the system of equations must be written in a discrete form (e.g., a finite-volume representation). However, the equation-defining sub-...

Remark. The following difference with ordinary, non-stochastic differential equations needs to be emphasized. All solutions of a non-stochastic equation are obtained by imposing at an arbitrary t0 an initial condition u t0) = a, and then considering all possible values of a. For a stochastic differential equation, however, one gets in this way only a subclass of solutions, namely those that happen to have no dispersion at this particular t0. 

For a qualitative description of kinetics, of great interest are both asymptotical (t —> oo) solutions independent on initial conditions and how particular solutions approach them. Complicated systems can reveal several such asymptotic solutions. Initial conditions define a choice of one of several possible asymptotic solutions. A general scheme for investigating a set of ordinary differential equations was very well described in a number of monographs [4, 7, 14-16] it includes ... 

The above reasoning allows a conclusion that once a researcher has decided upon the particular ideal kinetic model of polycondensation, he or she will be able to readily calculate any statistical characteristics of its products. The only thing he or she is supposed to do is to find the solution of a set of several ordinary differential equations for the concentrations of functional groups, using then the expressions known from literature. 

The Method of Lines or MOL is not so much a particular method as a way of approaching numerical solutions of pdes. It is described well by Hartree [295] as the replacement of the second-order (space) derivative by a finite difference that is, leaving the first (time) derivative as it is, thus forming from, say, the diffusion equation a set of ordinary differential equations, to be solved in an unspecified manner. Thus, a system such as (9.17) on page 151, can be written in the general vector-matrix form... 

Finite element methods — The finite element method is a powerful and flexible numerical technique for the approximate solution of (both ordinary and partial) differential equations involving replacing the continuous problem with unknown solution by a system of algebraic equations. The method was first introduced by Richard Courant in 1943 [i], and over the next three decades, and particularly in the 1960s, a comprehensive mathematical framework was developed to underpin the method. 

Still be very sensitive to a particular variable. On the other hand, an unstable condition is such that the least perturbation will lead to a finite change and such a condition may be regarded as infinitely sensitive to any operating variable. Sensitivity can be fully explored in terms of steady state solutions. A complete discussion of stability really requires the study of the transient equations. For the stirred tank this was possible since we had only to deal with ordinary differential equations for the tubular reactor the full treatment of the partial differential equations is beyond our scope here. Nevertheless, just as much could be learned about the stability of a stirred tank from the heat generation and removal diagram, so here something may be learned about stability from features of the steady state solution. 

The basis for any numerical reactor performance simulation is a mathematical model, consisting of a set of ordinary differential-, partial differential-or integro-differential equations and initial and boundary conditions. An optimized solution method must then be designed for this particular set of equations. 

A first step to understand the nature of these fronts is to consider (4.16) on the infinite line and particularize it to the class of solutions (4.18). One obtains a second order ordinary differential equation for... 

Ordinary differential equations have two classes of solutions —the complete integral and the singular solution. Particular solutions are only varieties of the complete integral. Three... 

Certain types of differential equations occur sufficiently often to justify the use of formulas for the corresponding particular solutions. The following set of Tables I to XIV covers all first, second, and th order ordinary linear differential equations with constant coefficients for which the right members are of the form F[x)e sinsx or P x)e cossx, where r and s are constants and P(x) is a polynomial of degree n. 

These boundary conditions are particularly convenient to evaluate the integration constants, as illustrated below. The mass transfer equation corresponds to a second-order linear ordinary differential equation with constant coefficients. The analytical solution for I a is... 

The term microkinetic analysis has been applied " to attempts to synthesise information from a variety of sources into a coherent reaction model for the hydrogenation of ethene. The input includes steady-state kinetics (most importantly the temperature-dependence of reaction orders ), isotopic tracing, vibrational spectroscopy and TPD it uses deterministic methods, i.e. the solution of ordinary differential equations, for estimating kinetic parameters. It selects a somewhat eclectic set of elementary reactions, and in particular the model... 

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