Wavelets in Optimisation and Approximations

We give the explicit time description of four the following problems: dynamics and optimal dynamics for some important electromechanical system, Galerkin approximation for beam equation, computations of Melnikov function for perturbed Hamiltonian systems. All these problems are reduced to the problem of the solving of the systems of diierential equations with polynomial nonlinearities and with or without some constraints. The rst main part of our construction is some variational approach to this problem, which reduces initial problem to the problem of the solution of functional equations at the rst stage and some algebraical problems at the second stage. We consider also two private cases of our general construction. In the rst case (particular) we have the solution as a series on shifted Legendre polynomials, which is parameterized by the solution of reduced algebraical system of equations. In the second case (general) we have the solution in a compactly supported wavelet basis. Multiresolution expansion is the second main part of our construction. The solution is parameterized by solutions of two reduced algebraical problems, the rst one is the same as in the rst case and the second one is some linear problem, which is obtained from anyone of the next wavelet construction: Fast Wavelet Transform, Stationary Subdivision Schemes, the method of Connection Coeecients. We give the explicit time description of the following problems: dynamics and optimal dynamics for nonlinear dynamical systems and Galerkin approximation for some class of partial differential equations, computations of Melnikov function for perturbed Hamiltonian systems. All these problems are reduced to the problem of the solving of the systems of diierential equations with polynomial nonlinearities with or without some constraints. The rst main part of our construction is some variational approach to this problem, which reduces initial problem to the problem of the solution of functional equations at the rst stage and some algebraical problems at the second stage. We consider also two private cases of our general construction. In the rst case (particular) we have for Riccati type equations the solution as a series on shifted Legendre polynomials, which is parametrized by the solution of reduced algebraical (also Riccati) system of equations 1]{{5]. In the second case (general polynomial systems) we have the solution in a compactly supported wavelet basis 6]{{8]. Multiresolution expansion is the second main part of our construction. In this case the solution is parametrized by solutions of two reduced algebraic problems, one …