Initial-value problems

Theorem 3.1 guarantees a unique solution to the initial value problem, and the subsequent methods that are discussed can be used to derive general solutions. Given a general solution, the integration constants can be evaluated with the use of the given initial conditions. 

Another procedure, which transforms a differential equation into an algebraic equation, is given below. This is the Laplace transform method. For example, the problem 

To recover the result in terms of the variables with which we started, an inversion of Y s) is needed. 

Therefore, the complete solution to the given initial value problem is 

This method is very efficient when it is applicable, and is one of the many integral transformations that are useful for solving initial value problems. Integral transforms are certain functions that are defined by an integral, such as 

Many chemical/biological engineering problems can be described by differential equations with known initial conditions, i.e., with known or given values of the state variables at the start of the process. 

Different problems are modeled by two-point boundary value differential equations in which the values of the state variables are predetermined at both endpoints of the independent variable. These endpoints may involve a starting and ending time for a time-dependent process or for a space-dependent process, the boundary conditions may apply at the entrance and at the exit of a tubular reactor, or at the beginning and end of a counter-current process, or they may involve parameters of a distributed process with recycle, etc. Boundary value problems (BVPs) are treated in Chapter 5. 

Initial value problems, abbreviated by the acronym IVP, can be solved quite easily, since for these problems all initial conditions are specified at only one interval endpoint for the variable. More precisely, for IVPs the value of the dependent variable(s) are given for one specific value of the independent variable such as the initial condition at one location or at one time. Simple numerical integration techniques generally suffice to solve IVPs. This is so nowadays even for stiff differential equations, since good stiff DE solvers are widely available in software form and in MATLAB. 

Furthermore, IVPs have only one solution for the given initial values if there are sufficiently many initial conditions given, and therefore bifurcation plays no role once the initial conditions are specified. For dynamic systems, different initial conditions may, however, lead to different steady states we refer to the fluidized bed reactor in Section 4.3, for example. 

Armed with techniques for solving linear and nonlinear algebraic systems (Chapters 1 and 2) and the tools of eigenvalue analysis (Chapter 3), we are now ready to treat more complex problems of greater relevance to chemical engineering practice. We begin with the study of initial value problems (IVPs) of ordinary differential equations (ODEs), in which we compute the trajectory in time of a set of N variables Xj(t) governed by the set of first-order ODEs 

We start the simulation, usually at to = 0, at the initial condition, x to) = Such problems arise commonly in the study of chemical kinetics or process dynamics. While we have interpreted above the variable of integration to be time, it might be another variable such as a spatial coordinate. 

Our task will be to develop iterative rules for updating the trajectory by taking small steps forward in time. We would like the numerical trajectory to agree with the exact solution 

Therefore, tiiis problem is closely related to that of numerically computing the values of definite integrals 

we first consider the subject of numerical integration (quadrature). As we can compute /p analytically when /(x) is a polynomial, 

Engineers develop mathematical models to describe processes of interest to them. For example, the process of converting a reactant A to a product B in a batch chemical reactor can be described by a first order, ordinary differential equation with a known initial condition. This type of model is often referred to as an initial value problem (IVP), because the initial conditions of the dependent variables must be known to determine how the dependent variables change with time. In this chapter, we will describe how one can obtain analytical and numerical solutions for linear IVPs and numerical solutions for nonlinear IVPs. 

Many practical problems of wavefield theory require examining propagation of waves generated by a given source in a medium with given material property distributions. This is a typical example of the forward geophysical problem. There are two major types of these problems 1) so-called initial-value problems , and 2) problems without initial values. 

In the framework of the initial-value problems we examine wavefield propagation within a domain of an acoustic or elastic medium starting from a certain moment of time t = to, assuming that we know the initial wavefield distribution at this starting moment. This condition is called a domain initial-value condition. 

Let us consider the case of a homogeneous isotropic elastic medium within a local domain V, bounded by the surface S. Various physical processes may take place on the boundary of the domain V and in the space around it. For example, the oscillation energy can flow freely through the boundary, or energy can stay within the domain, being reflected from the boundary 5, etc. Therefore to make the problem of wavefield propagation of the initial disturbance mathematically specific, we also should determine the boundary conditions of the oscillations of the elastic medium. This condition is called a boundary-value condition. 

In summary, we can characterize the initial-value problems by assigning conditions of two kinds 1) domain initial-value conditions, and 2) boundary-value conditions. Let us consider these conditions in more detail. 

The domain initial-value conditions for an elastic displacement field can be formulated as follows  

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Gear C W 1971 Numerical Initial Value Problems in Ordinary Differential Equations (Englewood Cliffs, NJ Prentice-Hall)... 

The index J can label quantum states of the same or different chemical species. Equation (A3.13.20) corresponds to a generally stiff initial value problem [42, 43]. In matrix notation one may write ... 

This treatment of reaction at the limit of bulk diffusion control is essentially the same as that presented by HugoC 69j. It is attractive computationally, since only a single two-point boundary value problem must be solved, namely that posed by equations (11.15) and conditions (11.16). This must be re-solved each time the size of the pellet is changed, since the pellet radius a appears in the boundary conditions. However, the initial value problem for equations (11.12) need be solved only once as a preliminary to solving (11.15) and (11.16) for any number of different pellet sizes. 

The solution of an Initial value problem for these equations expresses... 

Adding time-dependent terms to the equations the simulation is treated as an initial value problem in which at a given reference time all stresses are zero the steady-state solution can be found iteratively. 

Modelling of steady-state free surface flow corresponds to the solution of a boundary value problem while moving boundary tracking is, in general, viewed as an initial value problem. Therefore, classification of existing methods on the basis of their suitability for boundary value or initial value problems has also been advocated. 

Ritchmyer, R., and K. Morton. Difference Methods for Initial-Value Problems. 2d ed.. Interscience, New York (1967). 

In mathematical language, the propagation problem is known as an initial-value problem (Fig. 3-2). Schematically, the problem is characterized by a differential equation plus an open region in which the equation holds. The solution of the differential equation must satisfy the initial conditions plus any side boundary conditions. 

Equations of the first land are very sensitive to solution errors so that they present severe numerical problems. Volterra equations are similar to initial value problems. 

This integral equation is a Volterra equation of the second land. Thus the initial-value problem is eqmvalent to a Volterra integral equation of the second kind. 

NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS AS INITIAL VALUE PROBLEMS... 

A differential equation for a function that depends on only one variable, often time, is called an ordinary differential equation. The general solution to the differential equation includes many possibilities the boundaiy or initial conditions are needed to specify which of those are desired. If all conditions are at one point, then the problem is an initial valueproblem and can be integrated from that point on. If some of the conditions are available at one point and others at another point, then the ordinaiy differential equations become two-point boundaiy value problems, which are treated in the next section. Initial value problems as ordinary differential equations arise in control of lumped parameter models, transient models of stirred tank reactors, and in all models where there are no spatial gradients in the unknowns. 

Shooting Methods The first method is one that utihzes the techniques for initial value problems but allows for an iterative calculation to satisfy all the boundaiy conditions. Consider the nonlinear boundaiy value problem... 

Parabolic Equations m One Dimension By combining the techniques apphed to initial value problems and boundary value problems it is possible to easily solve parabolic equations in one dimension. The method is often called the method of lines. It is illustrated here using the finite difference method, but the Galerldn finite element method and the orthogonal collocation method can also be combined with initial value methods in similar ways. The analysis is done by example. 

Ordinaiy differential Eqs. (13-149) to (13-151) for rates of change of hquid-phase mole fractious are uouhuear because the coefficients of Xi j change with time. Therefore, numerical methods of integration with respect to time must be enmloyed. Furthermore, the equations may be difficult to integrate rapidly and accurately because they may constitute a so-called stiff system as considered by Gear Numerical Initial Value Problems in Ordinaiy Differential Equations, Prentice Hall, Englewood Cliffs, N.J., 1971). The choice of time... 

C (0). The analytieal solution to Equation 9-34 is rather eomplex for reaetion order n > 1, the (-r ) term is usually non-linear. Using numerieal methods, Equation 9-34 ean be treated as an initial value problem. Choose a value for = C (0) and integrate Equation 9-34. If C (A.) aehieves a steady state value, the eorreet value for C (0) was guessed. Onee Equation 9-34 has been solved subjeet to the appropriate boundary eonditions, the eonversion may be ealeulated from Caouc = Ca(0). 

To calculate the profiles and the differential capacitance of the interface numerically we have to choose a differential equation solver. However, the usual packages require that the problem is posed on a finite interval rather than on a semi-infinite interval as in our problem. In principle, we can transform the semi-infinite interval into a finite one, but the price to pay is a loss of translational invariance of the equations and the point mapped from that at infinity is singular, which may pose a problem on the solver. Most of the solvers are designed for initial-value problems while in our case we deal with a boundary-value problem. To circumvent these inconveniences we follow a procedure strongly influenced by the Lie group description. 

Richtmyer, R. D. and K. W. Morton. 1967. Difference methods for initial value problems. New York Interscience. 

Initial value problems where conditions are specified at some starting value of the independent variable. 

The solution of boundary value problems depends to a great degree on the ability to solve initial value problems.) Any n -order initial value problem can be represented as a system of n coupled first-order ordinary differential equations, each with an initial condition. In general... 

Shooting methods attempt to convert a boundary value problem into an initial value problem. For example, given the preceding example restated as an initial value problem for which... 

U is unknown and must be chosen so that y(L) = 0. The equation may be solved as an initial value problem with predetermined step sizes so that will equal L at the end point. Since y(L) is a function of U, it will be denoted as y,(U) and an appropriate value of U sought so that... 

Given two estimates of the root and Uj, two solutions of the initial value problem are calculated, yL(U,j, ) and yLCU,), a new estimate of U is obtained where... 

Solving (3.14.2.25) for initial value problems and applying pure culture media with a single species (r), gives ... 

Gear, G.W. Numerical Initial Value Problems in ODEs . Prentice-Hall, New Jersey, 1971. 

The extension to multiple reactions is done by writing Equation (3.1) (or the more complicated versions of Equation (3.1) that will soon be developed) for each of the N components. The component reaction rates are found from Equation (2.7) in exactly the same ways as in a batch reactor. The result is an initial value problem consisting of N simultaneous, first-order ODEs that can be solved using your favorite ODE solver. The same kind of problem was solved in Chapter 2, but the independent variable is now z rather than t. 

To find u, it is necessary to use some ancillary equations. As usual in solving initial value problems, we assume that all variables are known at the reactor inlet so that (Ac)i UinPin will be known. Equation (3.2) can be used to calculate m at a downstream location if p is known. An equation of state will give p but requires knowledge of state variables such as composition, pressure, and temperature. To find these, we will need still more equations, but a closed set can eventually be achieved, and the calculations can proceed in a stepwise fashion down the tube. 

They convert the initial value problem into a two-point boundary value problem in the axial direction. Applying the method of lines gives a set of ODEs that can be solved using the reverse shooting method developed in Section 9.5. See also Appendix 8.3. However, axial dispersion is usually negligible compared with radial dispersion in packed-bed reactors. Perhaps more to the point, uncertainties in the value for will usually overwhelm any possible contribution of D. ... 

The forward shooting method seems straightforward but is troublesome to use. What we have done is to convert a two-point boundary value problem into an easier-to-solve initial value problem. Unfortunately, the conversion gives a numerical computation that is ill-conditioned. Extreme precision is needed at the inlet of the tube to get reasonable accuracy at the outlet. The phenomenon is akin to problems that arise in the numerical inversion of matrices and Laplace transforms. 

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