Overview

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 x _j( t) governed by the set of first-order ODEs

We start the simulation, usually at t ₀ = 0, at the initial condition, x( t ₀) = x ^[0]. 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, this problem is closely related to that of numerically computing the values of definite integrals

Thus, we first consider the subject of numerical integration (quadrature). As we can compute I _F analytically when f( x) is a polynomial,

our first topic will be polynomial interpolation, the representation of an arbitrary function f( x) by an approximating polynomial.

Following a discussion of polynomial interpolation and numerical integration, a survey is...