ABSTRACT
Exact solutions for problems in the calculus of variations are available only for relatively simple problems. Fortunately, accurate approximate solutions often suffice for many applications. One approach to approximate solutions is to approximately solve the boundary-value problem (BVP) which determines the admissible extremals; the minimum value of J, if it exists, must be attained at such an extremal. Some basic numerical methods for BVPs of ODEs will be discussed in the Appendix of this book. Perturbation and asymptotic methods for these problems will be illustrated in Chapter 14 [see also Kevorkian and Cole (1981) and references therein]. We can also obtain an approximate solution by solving a discrete analogue of the basic problem similar to that formulated at the end of Section 3, Chapter 1. For a fine subdivision of the interval [a,b], the result is expected to provide a good approximation for the exact solution of the variational problem.
