TABLE OF CONTENTS Mathematical descriptions of systems often are based on prescribed values of dependent variables on the boundaries of the system. The vibrations of a clamped string is an example of such a system. For this case, no string motions occur at the clamped or endpoints. A second example is that of an object placed in a uniform fluid flow; we expect the flow velocity to be constant at large distances from the object. The purpose of this chapter is to consider some of the mathematical properties of systems modeled by second-order differential equations for which the solutions take particular values at their boundaries. A large class of systems are characterized mathematically as being Sturm-Liouville problems. We introduce the various issues related to this situation by first examining in detail the vibrating string. Next, we state, without proof, several important theorems which form the background knowledge needed to understand Sturm-Liouville problems. We show that the differential equations and associated boundary conditions for the special (polynomial) functions are Sturm-Liouville problems and indicate how Green's functions can be calculated. The chapter ends with a discussion of how the asymptotic behavior of certain types of second-order, linear differential equations can be determined. The method is illustrated by application to Bessel functions.
As stated above, we do not provide proofs of any of the stated theorems and related results. However, in many instances such proofs are straightforward to obtain. The books listed in the bibliography to this chapter do contain these proofs and...
