ABSTRACT
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 first method the authors discuss in this chapter transforms the original variational problem by applying the Laplace transform and producing an auxiliary differential equation. The method of gradients, with high relevance to engineering optimization, concludes the chapter. This is the analytic solution to the problem of the compression of a unit length beam along its longitudinal axis. The method discussed here has a resemblance to the Laplace transform solution since it also uses a transformation of the solution. For certain types of problems a complete integral solution is available. The complete integral form presents a parametric family of general solutions. The chapter also considers the Laplace's equation in two dimensions that plays a fundamental role in mathematical physics.
