Bi-variational Evolutionary Systems and Approximate Solutions

The use of variational methods for the construction of sufficiently accurate approximate solutions of a given system requires the existence of the correspondent variational principle - a solution of the inverse problem 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 - in general nonclassical Hamiltonian actions - and their applications for the search of approximate solutions of the given boundary value problems. Its solution may not be unique. In particular, there can exists a bi-variational system, i.e. generated by two different Hamiltonian actions. The main aim 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 non-classical Hamiltonian actions for the construction of approximate solutions with the high accuracy for the given dissipative problem. In the paper there are used notions and methods of nonlinear functional analysis and of modern calculus of variations.