Nonclassical Hamilton’s actions and the numerical performance of variational methods for some dissipative problems

Abstract

The use of variational methods for the construction of sufficiently accurate approximate solutions of a given system requires the existence of the corresponding variational principle - a solution of the inverse problems of the calculus of variations. In the frame of the Euler’s functionals there may not exist variational principles. But if we extend the class of functionals then it could allow to get the variational formulations of the given problems. There naturally arises the problem of the constructive determination of the corresponding functionals - nonclassical Hamilton’s actions - and their application for the search of approximate solutions of the given boundary value problems. The main goal of the paper is to present a scheme for the construction of indirect variational formulations for given evolutionary problems and to demonstrate the effective use of the nonclassical Hamilton’s action for the construction of approximate solutions with the high accuracy for the given dissipative problem.