
6
Analytic solutions of variational problems
This chapter presents a handful of analytic methods for solving variational
problems. They include the methods of Laplace transformation, separation
of variables, complete integrals, and Poisson’s integral formula. The method
of gradients, with high relevance to engineering optimization, concludes the
chapter.
6.1 Laplace transform solution
The first method we discuss in this chapter transforms the original variational
problem by applying the Laplace transform and producing an auxiliary dif-
ferential equation.
Let us consider the variational problem of
I(t, x)=
f(t, x)dt = extremum,
and apply the Laplace tra