Applied Calculus of Variations for Engineers, Third edition, 3rd Edition

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Analytic solutions

This chapter presents a handful of analytic methods for solving variational problems. They include the methods of Laplace transformation, d’Alembert’s separation of variables techniques, the complete integrals and Poisson’s integral formula. The method of gradients, with an illustrative example, concludes the chapter.

 

6.1Laplace transform solution

The first method we discuss in this chapter transforms the original variational problem by applying Laplace transformation and producing an auxiliary differential equation [12].

Let us consider the variational problem of

I(t,x)=∫f(t,x)dt=extremum,

and apply the Laplace transform to the function as

∫0∞e−stf(t,x)dt.

During this transform we regard t as the independent variable ...