Nonlinear Ordinary Differential Equations-Initial-Value Problems

5 Nonlinear Ordinary Differential Equations-Initial-Value Problems 

In this section, we develop numerical solutions for a set of ordinary differential equations in their canonical form  

In order to be able to illustrate these methods graphically, we treaty as a single variable rather than as a vector of variables. The formulas developed for the solution of a single differential equation are readily expandable to those for a set of differential equations, which must be solved simultaneously. This concept is demonstrated in Sec. 5.5.4. 

We begin the development of these methods by first rearranging Eq. (5.27) and integrating 

One method for integrating Eq. (5.55) is to take the left-hand side of this equation and use finite differences for its approximation. This technique works directly with the tangential trajectories of the dependent variable rather than with the areas under the function f x, y). This is the technique applied in Secs. 5.5.1 and 5.5.2. 

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Figure 8 depicts our view of an ideal structure for an applications program. The boxes with the heavy borders represent those functions that are problem specific, while the light-border boxes represent those functions that can be relegated to problem-independent software. This structure is well-suited to problems that are mathematically either systems of nonlinear algebraic equations, ordinary differential equation initial or 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 user must write the subroutine that defines his particular system of equations. However, that subroutine should be able to make calls to problem-independent software to return many of the components that are needed to assemble the governing equations. Specifically, such software could be called to return in-... 

The solution to Eq. 10.79 must be obtained numerically as it is a nonlinear ordinary differential equation. (See Problem 10C.6 for a solution.) The solution is obtained for arbitrary values of L. The thickness profile can be obtained by means of Eq. 10.75 using the initial thickness estimated from die swell data. One must assume that die swell is unaffected by the weight of the hanging parison. The complete solution of the problem can only be obtained using numerical techniques. ... 

This text first presents a fundamental discussion of linear algebra, to provide the necessary foundation to read the applied mathematical literature and progress further on one s own. Next, a broad array of simulation techniques is presented to solve problems involving systems of nonlinear algebraic equations, initial value problems of ordinary differential and differential-algebraic (DAE) systems, optimizations, and boundary value problems of ordinary and partial differential equations. A treatment of matrix eigenvalue analysis is included, as it is fundamental to analyzing these simulation techniques. 

Absorption columns can be modeled in a plate-to-plate fashion (even if it is a packed bed) or as a packed bed. The former model is a set of nonlinear algebraic equations, and the latter model is an ordinary differential equation. Since streams enter at both ends, the differential equation is a two-point boundary value problem, and numerical methods are used (see Numerical Solution of Ordinary Differential Equations as Initial-Value Problems ). 

Ordinary differential Eqs. (13-149) to (13-151) for rates of change of hquid-phase mole fractions are nonlinear because the coefficients of Xij change with time. Therefore, numerical methods of integration with respect to time must be employed. 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 Ordinary Differential Equations, Prentice Hall, Englewood Cliffs, N.J., 1971). The choice of time... 

In ref 146 the authors present a non-standard (nonlinear) two-step explicit P-stable method of fourth algebraic order and 12th phase-lag order for solving second-order linear periodic initial value problems of ordinary differential equations. The proposed method can be extended to be vector-applicable for multi-dimensional problem based on a special vector arithmetic with respect to an analytic function. 

This appendix explains how to use DDAPLUS to solve nonlinear initial-value problems containing ordinary differential equations with or without algebraic equations, or to solve purely algebraic nonlinear equation systems by a damped Newton method. Three detailed examples are given. 

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. 

Steady state mass or heat transfer in solids and current distribution in electrochemical systems involve solving elliptic partial differential equations. The method of lines has not been used for elliptic partial differential equations to our knowledge. Schiesser and Silebi (1997)[1] added a time derivative to the steady state elliptic partial differential equation and applied finite differences in both x and y directions and then arrived at the steady state solution by waiting for the process to reach steady state. [2] When finite differences are applied only in the x direction, we arrive at a system of second order ordinary differential equations in y. Unfortunately, this is a coupled system of boundary value problems in y (boundary conditions defined at y = 0 and y = 1) and, hence, initial value problem solvers cannot be used to solve these boundary value problems directly. In this chapter, we introduce two methods to solve this system of boundary value problems. Both linear and nonlinear elliptic partial differential equations will be discussed in this chapter. We will present semianalytical solutions for linear elliptic partial differential equations and numerical solutions for nonlinear elliptic partial differential equations based on method of lines. 

In [165] a study of a new methodology for development of efficient methods for the numerical solution of second-order periodic initial value problems (IVPs) of ordinary differential equations is presented. The methodology is based on the development of nonlinear numeircal methods. In this paper the authors study the following nonlinear scheme ... 

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... 

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