TABLE OF CONTENTS First-order partial differential equations (PDEs) arise in a variety of problems in science and engineering. In this chapter we cover some basic concepts related to the solution of these equations, which are helpful in the analysis of problems of practical interest. These problems are briefly reviewed in the first section of the chapter, where they are cast as special cases of the kinematic wave equation.
We then proceed to discuss the solution of first-order PDEs using the method of characteristics introduced in chapter 5. This leads to solutions in the form of simple waves, including linear, expansion, compression, and centered waves. The occurrence of shocks (i.e., jump discontinuities) in the solution is then examined. This is a peculiar feature of first-order PDEs, and such solutions are sometimes referred to as weak solutions. These various aspects are illustrated in the context of a practical example, involving the dynamic behavior of a countercurrent separation unit.
Finally, in the last section the occurrence of constant-pattern solutions for PDEs is discussed. Although these equations are of order greater than one, their analysis is relevant to this chapter since it provides further support to the existence of weak solutions for first-order PDEs.
In section 5.4 we discussed some examples indicating the importance of first-order PDEs in modeling various physical processes. These equations arise typically when applying a conservation law (e.g., mass) to a flowing system under transient conditions. Let us consider the case of a one-dimensional process which...
